FACTORIZABLE SHEAVES AND QUANTUM GROUPS

ROMAN BEZRUKAVNIKOV, MICHAEL FINKELBERG, AND VADIM SCHECHTMAN

Table of contents

Acknowledgement 7

Part 0. OVERVIEW

1. Introduction 8

Chapter 1. Local

2. The category C 13

3. Braiding local systems 17

4. Factorizable sheaves 22

5. Tensor product 24

6. Vanishing cycles 27

Chapter 2. Global (genus 0)

7. Cohesive local systems 31

8. Gluing 34

9. Semiinfinite cohomology 36

10. Conformal blocks (genus 0) 37

11. Integration 40

12. Regular representation 41

13. Regular sheaf 43

Chapter 3. Modular

14. Heisenberg local system 45

15. Fusion structures on FS 52

16. Conformal blocks (higher genus) 54 Part I. INTERSECTION COHOMOLOGY OF REAL ARRANGEMENTS

1. Introduction 56

2. Topological preliminaries 57

3. Vanishing cycles functors 61

4. Computations for standard sheaves 71

Part II. CONFIGURATION SPACES AND QUANTUM GROUPS

1. Introduction 80

Chapter 1. Algebraic discussion

2. Free algebras and bilinear forms 82

3. Hochschild complexes 92

4. Symmetrization 95

5. Quotient algebras 98

Chapter 2. Geometric discussion

6. Diagonal stratification and related algebras 100

7. Principal stratification 106

8. Standard sheaves 113

Chapter 3. Fusion

9. Additivity theorem 124

10. Fusion and tensor products 127

Chapter 4. Category C

11. Simply laced case 131

12. Non-simply laced case 134 Part III. TENSOR CATEGORIES ARISING FROM CONFIGURATION

SPACES

1. Introduction 138

Chapter 1. Category FS

2. Space A 140

3. Braiding local system I 142

4. Factorizable sheaves 144

5. Finite sheaves 147

6. Standard sheaves 149

Chapter 2. Tensor structure

7. Marked disk operad 151

8. Cohesive local systems ^{K}I 155

9. Factorizable sheaves over ^{K}A 157

10. Gluing 160

11. Fusion 163

Chapter 3. Functor Φ

12. Functor Φ 169

13. Main properties of Φ 179

Chapter 4. Equivalence

14. Truncation functors 183

15. Rigidity 187

16. Steinberg sheaf 189

17. Equivalence 191

18. The case of generic ζ 192 Part IV. LOCALIZATION OVER ℙ^{1}

1. Introduction 195

Chapter 1. Gluing over ℙ^{1}

2. Cohesive local system 196

3. Gluing 199

Chapter 2. Semiinifinite cohomology

4. Semiinfinite functors Ext and T or in C 203

5. Some calculations 207

Chapter 3. Global sections

6. Braiding and balance in C and FS 212

7. Global sections over A(K) 213

8. Global sections over P 214

9. Application to conformal blocks 217

Part V. MODULAR STRUCTURE ON THE CATEGORY FS

1. Introduction 222

Chapter 1. Heisenberg local system

2. Notations and statement of the main result 224

3. The scheme of construction 227

4. The universal line bundle 229

5. The universal local system 233

6. Factorization isomorphisms 243 Chapter 2. The modular property of the Heisenberg system

7. Degeneration of curves: recollections and notations 247

8. Proof of Theorem 7.6(a) 249

9. Proof of Theorem 7.6(b) 253 Chapter 3. Regular representation

10. A characterization of the regular bimodule 261

11. The adjoint representation 264 Chapter 4. Quadratic degeneration in genus zero

12. I-sheaves 266

13. Degenerations of quadrics 267

14. The I-sheaf R 268

15. Convolution 271 Chapter 5. Modular functor

16. Gluing over C 274

17. Degeneration of factorizable sheaves 279

18. Global sections over C 280 Chapter 6. Integral representations of conformal blocks

19. Conformal blocks in arbitrary genus 282

References 285

Acknowledgement

During the work on this project, the first author visited Independent Moscow University, the second author visited Harvard University, Stony Brook University and Tel-Aviv University, and the third author visited Harvard University and Max-Planck-Institut für Mathematik. We thank these institutions for the hospitality.

The second author was partially supported by the grants from AMS, INTAS94-4720, CRDF RM1-265. The third author was partially supported by NSF.

Part 0. OVERVIEW

1. Introduction

1.1. Let (I, ·) be a Cartan datum of finite type and (Y, X, …) the simply connected root datum of type (I, ·), cf. [L1].

Let l > 1 be an integer. Set ℓ = l/(l, 2); to simplify the exposition, we will suppose in this Introduction that d_{i} := i · i/2 divides ℓ for all i ∈ I; we set ℓ_{i} := ℓ/d_{i}. We suppose that all ℓ_{i} > 3. Let ρ ∈ X be the half-sum of positive roots; let ρ_{ℓ} ∈ X be defined by ⟨i, ρ_{ℓ}⟩ = ℓ_{i} − 1 (i ∈ I). We define a lattice Y_{ℓ} ⊂ Y by Y_{ℓ} = {μ ∈ X| for all μ′ ∈ X, μ · μ′ ∈ ℓℤ}, and set d_{ℓ} = card(X/Y_{ℓ}).

We fix a base field k of characteristic not dividing l. We suppose that k contains a primitive l-th root of unity ζ, and fix it. Starting from these data, one defines certain category C. Its objects are finite dimensional X-graded k-vector spaces equipped with an action of Lusztig’s ”small” quantum group 𝔲 (cf. [L2]) such that the action of its Cartan subalgebra is compatible with the X-grading. Variant: one defines certain algebra 𝔲^{•} which is an ”X-graded” version of 𝔲 (see 12.2), and an object of C is a finite dimensional 𝔲^{•}-module. For the precise definition of C, see 2.11, 2.13. For l prime and chark = 0, the category C was studied in [AJS], for chark > 0 and arbitrary l, C was studied in [AW]. The category C admits a remarkable structure of a ribbon^{1} category (Lusztig).

1.2. The main aim of this work is to introduce certain tensor category FS of geometric origin, which is equivalent to C. Objects of FS are called (finite) factorizable sheaves.

A notion of a factorizable sheaf is the main new concept of this paper. Let us give an informal idea, what kind of an object is it. Let D be the unit disk in the complex plane^{2}. Let D denote the space of positive Y -valued divisors on D. Its points are formal linear combinations ∑ ν · x (ν ∈ Y^{+} := ℕ[I], x ∈ D). This space is a disjoint union

D = ∐_{ν∈Y^{+}} D^{ν}

where D^{ν} is the subspace of divisors of degree ν. Variant: D^{ν} is the configuration space of ν points running on D; its points are (isomorphism clases of) pairs of maps

(J ⟶ D, j ↦ x_{j}; π : J ⟶ I, j ↦ i_{j}),

J being a finite set. We say that we have a finite set {x_{j}} of (possibly equal) points of D, a point x_{j} being ”coloured” by the colour i_{j}. The sum (in Y) of colours of all points should be equal to ν. The space D^{ν} is a smooth affine analytic variety; it carries a canonical stratification defined by various intersections of hypersurfaces {x_{j} = 0} and {x_{j′} = x_{j′′}}. The open stratum (the complement of all the hypersurfaces above) is denoted by A^{ν∘}.

1in other terminology, braided balanced rigid tensor

2One could also take a complex affine line or a formal disk; after a suitable modification of the definitions, the resulting categories of factorizable sheaves are canonically equivalent.

One can imagine D as a Y^{+}-graded smooth stratified variety. Let A = ∐ A^{ν} ⊂ D be the open Y^{+}-graded subvariety of positive Y -valued divisors on D − {0}. We have an open Y^{+}-graded subvariety A^{∘} = ∐ A^{ν∘} ⊂ A.

Let us consider the (Y^{+})^{2}-graded variety

A × A = ∐ A^{ν_{1}} × A^{ν_{2}};

we define another (Y^{+})^{2}-graded variety { \atop A × A}, together with two maps

(a) A × A ⟵^{p} { \atop A × A} ⟶^{m} A

respecting the Y^{+}-gradings^{3}; the map p is a homotopy equivalence. One can imagine the diagram (a) above as a ”homotopy multiplication”

m_{A} : A × A ⟶ A;

this ”homotopy map” is ”homotopy associative”; the meaning of this is explained in the main text. We say that A is a (Y^{+}-graded) ”homotopy monoid”; A^{∘} ⊂ A is a ”homotopy submonoid”.

The space D is a ”homotopy A-space”: there is a ”homotopy map”

m_{D} : D × A ⟶ D

which is, as above, a diagram of usual maps between Y^{+}-graded varieties

(b) D × A ⟵^{p} { \atop D × A} ⟶^{m} D,

p being a homotopy equivalence.

For each μ ∈ X, ν ∈ Y^{+}, one defines a one-dimensional k-local system I_{μ}^{ν} over A^{ν∘}. Its monodromies are defined by variuos scalar products of ”colours” of running points and the colour μ of the origin 0 ∈ D. These local systems have the following compatibility. For each μ ∈ X, ν_{1}, ν_{2} ∈ Y^{+}, a factorization isomorphism

ϕ_{μ}(ν_{1}, ν_{2}) : m^{∗}I_{μ}^{ν_{1}+ν_{2}} ⟶^{∼} p^{∗}(I_{μ}^{ν_{1}} ⊠ I_{μ−ν_{1}}^{ν_{2}})

is given (where m, p are the maps in the diagram (a) above), these isomorphisms satisfying certain (co)associativity property. The collection of local systems {I_{μ}^{ν}} is an X-graded local system I = ⊕ I_{μ} over A^{∘}. One could imagine the collection of factorization isomorphisms {ϕ_{μ}(ν_{1}, ν_{2})} as an isomorphism

ϕ : m_{A^{∘}}^{∗}I ⟶^{∼} I ⊠ I

We call I the braiding local system.

3a (Y^{+})^{2}-graded space is considered as a Y^{+}-graded by means of the addition (Y^{+})^{2} ⟶ Y^{+}.

Let I^{•} be the perverse sheaf on A which is the Goresky-MacPherson extension of the local system I. The isomorphism ϕ above induces a similar isomorphism

ϕ^{•} : m_{A}^{∗}I^{•} ⟶^{∼} I^{•} ⊠ I^{•}

This sheaf, together with ϕ^{•}, looks like a ”coalgebra”; it is an incarnation of the quantum

group 𝔲^{•}.

A factorizable sheaf is a couple

(μ ∈ X (”the highest weight”); a perverse sheaf X over D, smooth along the canonical stratification).

Thus, X is a collection of sheaves X^{ν} over D^{ν}. These sheaves should be connected by factorization isomorphisms

ψ(ν_{1}, ν_{2}) : m^{∗}X^{ν_{1}+ν_{2}} ⟶^{∼} p^{∗}(X^{ν_{1}} ⊠ I_{μ−ν_{1}}^{ν_{2}•})

satisfying an associativity property. Here m, p are as in the diagram (b) above. One could imagine the whole collection {ψ(ν_{1}, ν_{2})} as an isomorphism

ψ : m_{D}^{∗}X ⟶^{∼} I^{•} ⊠ X

satisfying a (co)associativity property. We impose also certain finiteness (of singularities) condition on X. So, this object looks like a ”comodule” over I^{•}. It is an incarnation of an 𝔲^{•}-module.

We should mention one more important part of the structure of the space D. It comes from natural closed embeddings ι_{ν} : D ↪ D[ν] (were [ν] denotes the shift of the grading); these mappings define certain inductive system, and a factorizable sheaf is a sheaf on its inductive limit.

The latter inductive limit is an example of a ”semiinfinite space”.

For the precise definitions, see Sections 3, 4 below.

The category FS has a structure of a braided balanced tensor category coming from geometry. The tensor structure on FS is defined using the functors of nearby cycles. The tensor equivalence

Φ : FS ⟶^{∼} C

is defined using vanishing cycles functors. It respects the braidings and balances. This is the contents of Parts I–III.

1.3. Factorizable sheaves are local objects. It turns out that one can ”glue” them along complex curves. More precisely, given a finite family {X_{a}}_{a∈A} of factorizable sheaves and a smooth proper curve C together with a family {x_{a}, τ_{a}}_{A} of distinct points x_{a} ∈ C with non-zero tangent vectors τ_{a} at them (or, more generally, a family of such objects over a smooth base S), one can define a perverse sheaf, denoted by

× _{A}^{(C)} X_{a}

on the (relative) configuration space C^{ν}. Here ν ∈ Y^{+} is defined by

(a) ν = ∑_{a∈A} μ_{a} + (2g − 2)ρ_{ℓ}

where g is the genus of C, μ_{a} is the highest weight of X_{a}. We assume that the right hand side of the equality belongs to Y^{+}.

One can imagine this sheaf as an ”exterior tensor product” of the family {X_{a}} along C. It is obtained by ”planting” the sheaves X_{a} into the points x_{a}. To glue them together, one needs a ”glue”. This glue is called the Heisenberg local system H; it is a sister of the braiding local system I. Let us describe what it is.

For a finite set A, let M_{A} denote the moduli stack of punctured curves (C, {x_{a}, τ_{a}}_{A}) as above; let η : C_{A} ⟶ M_{A} be the universal curve. Let λ_{A} = det(Rη_{∗}O_{C_{A}}) be the determinant line bundle on M_{A}, and M_{A;λ} ⟶ M_{A} be the total space of λ_{A} with the zero section removed. For ν ∈ Y^{+}, let η^{ν} : C_{A}^{ν} ⟶ M_{A} be the relative configuration space of ν points running on C_{A}; let C_{A}^{ν∘} ⊂ C_{A}^{ν} be the open stratum (where the running points are distinct and distinct from the punctures x_{a}). The complementary subscript ()_{λ} will denote the base change under M_{A;λ} ⟶ M_{A}. The complementary subscript ()_{g} will denote the base change under M_{A,g} ↪ M_{A}, M_{A,g} being the substack of curves of genus g.

The Heisenberg local system is a collection of local systems H_{μ⃗;A,g} over the stacks C_{A,g;λ}^{ν∘}. Here μ⃗ is an A-tuple {μ_{a}} ∈ X^{a}; ν = ν(μ⃗; g) is defined by the equality (a) above. We assume that μ⃗ is such that the right hand side of this equality really belongs to Y^{+}. The dimension of H_{μ⃗;A,g} is equal to d_{ℓ}^{g}; the monodromy around the zero section of the determinant line bundle is equal to

(b) c = (−1)^{card(I)}ζ^{−12ρ·ρ}.

These local systems have a remarkable compatibility (”fusion”) property which we do not specify here, see Section 15.

1.4. In Part IV we study the sheaves × _{A}^{(C)} X_{a} for C = ℙ^{1}. Their cohomology, when expressed algebraically, turn out to coincide with certain ”semiinfinite” Tor spaces in the category C introduced by S.M.Arkhipov. Due to the results of Arkhipov, this enables one to prove that ”spaces of conformal blocks” in WZW models are the natural subquotients of such cohomology spaces.

1.5. In Part V we study the sheaves × _{A}^{(C)} X_{a} for arbitrary smooth families of punctured curves. Let × _{A,g} X_{a} denotes the ”universal exterior product” living on C_{A,g;λ}^{ν} where ν = ν(μ⃗; g) is as in 1.3 (a) above, μ⃗ = {μ_{a}}, μ_{a} being the highest weight of X_{a}. Let us integrate it: consider

∫_{C^{ν}} × _{A,g} X_{a} := Rη_{∗}^{ν}( × _{A,g} X_{a});

it is a complex of sheaves over M_{A,g;λ} with smooth cohomology; let us denote it ⟨⊗_{A} X_{a}⟩_{g}. The cohomology sheaves of such complexes define a fusion structure^{4} on FS (and hence on C).

The classical WZW model fusion category is, in a certain sense, a subquotient of one of them. The number c, 1.3 (b), coincides with the ”multiplicative central charge” of the model (in the sense of [BFM]).

As a consequence of this geometric description and of the Purity Theorem, [BBD], the local systems of conformal blocks (in arbitrary genus) are semisimple. The Verdier duality induces a canonical non-degenerate Hermitian form on them (if k = ℂ).

1.6. This work in a sense completes the program outlined in [S] (no fusion structure in higher genus was mentioned there). We should mention a very interesting related work of G.Felder and collaborators. The idea of realizing quantum groups’ modules in the cohomology of configuration spaces appeared independently in [FW]. The Part V of the present work arose from our attempts to understand [CFW].

We are grateful to D.Kazhdan for the encouragement. The second author was strongly influenced by the ideas and suggestions of A.Beilinson, P.Deligne, B.Feigin, G.Felder and D.Kazhdan; he feels greatly indebted to all of them. Also, he is happy to thank L.Positselski whose friendly help came at the crucial moment. We are very obliged to G.Lusztig for the permission to use his unpublished results.

1.7. This work consists of 6 parts. Part 0 is an overview of the Parts I–V. More precisely, Chapter 0.1 surveys the Parts I–III; Chapter 0.2 is an exposition of Part IV, and Chapter 0.3 is an exposition of Part V. The proofs are given in the Parts I–V. Part 0 contains no proofs, but we hope that it clarifies the general picture to some extent. Each part starts with a separate Introduction. The references within one Part look like 1.1.1, and the references to other parts look like I.1.1.1.

4actually a family of such structures (depending on the degree of cohomology)

Chapter 1. Local.

2. The category C

2.1. We will follow Lusztig’s terminology and notations concerning root systems, cf. [L1].

We fix an irreducible Cartan datum (I, ·) of finite type. Thus, I is a finite set together with a symmetric ℤ-valued bilinear form ν_{1}, ν_{2} ↦ ν_{1} · ν_{2} on the free abelian group ℤ[I], such that for any i ∈ I, i · i is even and positive, for any i ≠ j in I, 2i · j/i · i ≤ 0 and the I × I-matrix (i · j) is positive definite. We set d_{i} = i · i/2.

Let d = max_{i∈I} d_{i}. This number is equal to the least common multiple of the numbers d_{i}, and belongs to the set {1, 2, 3}. For a simply laced root datum d = 1.

We set Y = ℤ[I], X = Hom(Y, ℤ); ⟨, ⟩ : Y × X ⟶ ℤ will denote the obvious pairing. The obvious embedding I ↪ Y will be denoted by i ↦ i. We will denote by i ↦ i′ the embedding I ↪ X given by ⟨i, j′⟩ = 2i · j/i · i. (Thus, (Y, X, …) is the simply connected root datum of type (I, ·), in the terminology of Lusztig.)

The above embedding I ⊂ X extends by additivity to the embedding Y ⊂ X. We will regard Y as the sublattice of X by means of this embedding. For ν ∈ Y, we will denote by the same letter ν its image in X. We set Y^{+} = ℕ[I] ⊂ Y.

We will use the following partial order on X. For μ_{1}, μ_{2} ∈ X, we write μ_{1} ≤ μ_{2} if μ_{2} − μ_{1} belongs to Y^{+}.

2.2. We fix a base field k, an integer l > 1 and a primitive root of unity ζ ∈ k as in the Introduction.

We set ℓ = l if l is odd and ℓ = l/2 if l is even. For i ∈ I, we set ℓ_{i} = ℓ/(ℓ, d_{i}). Here (a, b) stands for the greatest common divisor of a, b. We set ζ_{i} = ζ^{d_{i}}. We will assume that ℓ_{i} > 1 for any i ∈ I and ℓ_{i} > −⟨i, j′⟩ + 1 for any i ≠ j in I.

We denote by ρ (resp. ρ_{ℓ}) the element of X such that ⟨i, ρ⟩ = 1 (resp. ⟨i, ρ_{ℓ}⟩ = ℓ_{i} − 1) for all i ∈ I.

For a coroot β ∈ Y, there exists an element w of the Weyl group W of our Cartan datum and i ∈ I such that w(i) = β. We set ℓ_{β} := \frac{ℓ}{(ℓ,d_{i})}; this number does not depend on the choice of w and i.

We have ρ = \frac{1}{2} ∑ α; ρ_{ℓ} = \frac{1}{2} ∑ (ℓ_{α} − 1)α, the sums over all positive roots α ∈ X.

For a ∈ ℤ, i ∈ I, we set [a]_{i} = 1 − ζ_{i}^{−2a}.

2.3. We use the same notation μ_{1}, μ_{2} ↦ μ_{1} · μ_{2} for the unique extension of the bilinear form on Y to a ℚ-valued bilinear form on Y ⊗_{ℤ} ℚ = 𝕏 ⊗_{ℤ} ℚ.

We define a lattice Y_{ℓ} = {λ ∈ X| for all μ ∈ X, λ · μ ∈ ℓℤ}.

2.4. Unless specified otherwise, a ”vector space” will mean a vector space over k; ⊗ will denote the tensor product over k. A ”sheaf” (or a ”local system”) will mean a sheaf (resp. local system) of k-vector spaces.

If (T, S) is an open subspace of the space of complex points of a separate scheme of finite type over ℂ, with the usual topology, together with an algebraic stratification S satisfying the properties [BBD] 2.1.13 b), c), we will denote by M(T; S) the category of perverse sheaves over T lisse along S, with respect to the middle perversity, cf. [BBD] 2.1.13, 2.1.16.

2.5. Let ′𝔣 be the free associative k-algebra with 1 with generators θ_{i} (i ∈ I). For ν = ∑ ν_{i}i ∈ ℕ[I], let ′𝔣_{ν} be the subspace of ′𝔣 spanned by the monomials θ_{i_{1}} · . . . · θ_{i_{a}} such that ∑_{j} i_{j} = ν in ℕ[I].

Let us regard′𝔣⊗′𝔣 as a k-algebra with the product (x_{1}⊗x_{2})(y_{1}⊗y_{2}) = ζ^{ν·μ}x_{1}y_{1}⊗x_{2}y_{2} (x_{2} ∈ ′𝔣_{ν}, y_{1} ∈ ′𝔣_{μ}). Let r denote a unique homomorphism of k-algebras ′𝔣 ⟶ ′𝔣 ⊗ ′𝔣 carrying θ_{i} to 1 ⊗ θ_{i} + θ_{i} ⊗ 1 (i ∈ I).

2.6. Lemma-definition. There exists a unique k-valued bilinear form (·, ·) on ′𝔣 such that

(i) (1, 1) = 1; (θ_{i}, θ_{j}) = δ_{ij} (i, j ∈ I); (ii) (x, yy′) = (r(x), y ⊗ y′) for all x, y, y′ ∈ ′𝔣.

This bilinear form is symmetric. □

In the right hand side of the equality (ii) we use the same notation (·, ·) for the bilinear form on ′𝔣 ⊗ ′𝔣 defined by (x_{1} ⊗ x_{2}, y_{1} ⊗ y_{2}) = (x_{1}, y_{1})(x_{2}, y_{2}).

The radical of the form ′𝔣 is a two-sided ideal of ′𝔣.

2.7. Let us consider the associative k-algebra 𝔲 (with 1) defined by the generators ϵ_{i}, θ_{i} (i ∈ I), K_{ν} (ν ∈ Y) and the relations (a) — (e) below.

(a) K_{0} = 1, K_{ν} · K_{μ} = K_{ν+μ} (ν, μ ∈ Y);

(b) K_{ν}ϵ_{i} = ζ^{⟨ν,i′⟩}ϵ_{i}K_{ν} (i ∈ I, ν ∈ Y);

(c) K_{ν}θ_{i} = ζ^{−⟨ν,i′⟩}θ_{i}K_{ν} (i ∈ I, ν ∈ Y);

(d) ϵ_{i}θ_{j} − ζ^{i·j}θ_{j}ϵ_{i} = δ_{ij}(1 − K̃_{i}^{−2}) (i, j ∈ I).

Here we use the notation K̃_{ν} = ∏_{i} K_{d_{i}ν_{i}i} (ν = ∑ ν_{i}i).

(e) If f(θ_{i}) ∈ ′𝔣 belongs to the radical of the form (·, ·) then f(θ_{i}) = f(ϵ_{i}) = 0 in 𝔲.

2.8. There is a unique k-algebra homomorphism Δ : 𝔲 ⟶ 𝔲 ⊗ 𝔲 such that

Δ(ϵ_{i}) = ϵ_{i} ⊗ 1 + K̃_{i}^{−1} ⊗ ϵ_{i}; Δ(θ_{i}) = θ_{i} ⊗ 1 + K̃_{i}^{−1} ⊗ θ_{i}; Δ(K_{ν}) = K_{ν} ⊗ K_{ν}

for any i ∈ I, ν ∈ Y. Here 𝔲 ⊗ 𝔲 is regarded as an algebra in the standard way.

There is a unique k-algebra homomorphism e : 𝔲 ⟶ k such that e(ϵ_{i}) = e(θ_{i}) = 0, e(K_{ν}) = 1 (i ∈ I, ν ∈ Y).

2.9. There is a unique k-algebra homomorphism A : 𝔲 ⟶ 𝔲^{opp} such that

A(ϵ_{i}) = −ϵ_{i}K̃_{i}; A(θ_{i}) = −θ_{i}K̃_{i}; A(K_{ν}) = K_{−ν} (i ∈ I, ν ∈ Y).

There is a unique k-algebra homomorphism A′ : 𝔲 ⟶ 𝔲^{opp} such that

A′(ϵ_{i}) = −K̃_{i}ϵ_{i}; A′(θ_{i}) = −K̃_{i}θ_{i}; A′(K_{ν}) = K_{−ν} (i ∈ I, ν ∈ Y).

2.10. The algebra 𝔲 together with the additional structure given by the comultiplication Δ, the counit e, the antipode A and the skew-antipode A′, is a Hopf algebra.

2.11. Let us define a category C as follows. An object of C is a 𝔲-module M which is finite dimensional over k, with a given direct sum decomposition M = ⊕_{λ∈X} M_{λ} (as a vector space) such that K_{ν}x = ζ^{⟨ν,λ⟩}x for any ν ∈ Y, λ ∈ X, x ∈ M_{λ}. A morphism in C is a 𝔲-linear map respecting the X-gradings.

Alternatively, an object of C may be defined as an X-graded finite dimensional vector space M = ⊕ M_{λ} equipped with linear operators

θ_{i} : M_{λ} ⟶ M_{λ−i′}, ϵ_{i} : M_{λ} ⟶ M_{λ+i′} (i ∈ I, λ ∈ X)

such that

(a) for any i, j ∈ I, λ ∈ X, the operator ϵ_{i}θ_{j} − ζ^{i·j}θ_{j}ϵ_{i} acts as the multiplication by δ_{ij}[⟨i, λ⟩]_{i} on M_{λ}.

Note that [⟨i, λ⟩]_{i} = [⟨d_{i}i, λ⟩] = [i′ · λ].

(b) If f(θ_{i}) ∈ ′𝔣 belongs to the radical of the form (·, ·) then the operators f(θ_{i}) and f(ϵ_{i}) act as zero on M.

2.12. In [L2] Lusztig defines an algebra u_{B} over the ring B which is a quotient of ℤ[v, v^{−1}] (v being an indeterminate) by the l-th cyclotomic polynomial. Let us consider the kalgebra u_{k} obtained from u_{B} by the base change B ⟶ k sending v to ζ. The algebra u_{k} is generated by certain elements E_{i}, F_{i}, K_{i} (i ∈ I). Here E_{i} = E_{α_{i}}^{(1)}, F_{i} = F_{α_{i}}^{(1)} in the notations of loc.cit.

Given an object M ∈ C, let us introduce the operators E_{i}, F_{i}, K_{i} on it by

E_{i} = \frac{ζ_{i}}{1 − ζ_{i}^{−2}} ϵ_{i}K̃_{i}, F_{i} = θ_{i}, K_{i} = K_{i}.

2.13. Theorem. The above formulas define the action of the Lusztig’s algebra u_{k} on an object M.

This rule defines an equivalence of C with the category whose objects are X-graded finite dimensional u_{k}-modules M = ⊕M_{λ} such that K_{i}x = ζ^{⟨i,λ⟩}x for any i ∈ I, λ ∈ X, x ∈ M_{λ}. □

2.14. The structure of a Hopf algebra on 𝔲 defines canonically a rigid tensor structure on C (cf. [KL]IV, Appendix).

The Lusztig’s algebra u_{k} also has an additional structure of a Hopf algebra. It induces the same rigid tensor structure on C.

We will denote the duality in C by M ↦ M^{∗}. The unit object will be denoted by 1.

2.15. Let 𝔲^{−} (resp. 𝔲^{+}, 𝔲^{0}) denote the k-subalgebra generated by the elements θ_{i} (i ∈ I) (resp. ϵ_{i} (i ∈ I), K_{ν} (ν ∈ Y)). We have the triangular decomposition 𝔲 = 𝔲^{−}𝔲^{0}𝔲^{+} = 𝔲^{+}𝔲^{0}𝔲^{−}.

We define the ”Borel” subalgebras 𝔲^{≤0} = 𝔲^{−}𝔲^{0}, 𝔲^{≥0} = 𝔲^{+}𝔲^{0}; they are the Hopf subalgebras of 𝔲.

Let us introduce the X-grading 𝔲 = ⊕𝔲_{λ} as a unique grading compatible with the structure of an algebra such that θ_{i} ∈ 𝔲_{−i′}, ϵ_{i} ∈ 𝔲_{i′}, K_{ν} ∈ 𝔲_{0}. We will use the induced gradings on the subalgebras of 𝔲.

2.16. Let C^{≤0} (resp. C^{≥0}) be the category whose objects are X-graded finite dimensional 𝔲^{≤0}- (resp. 𝔲^{≥0}-) modules M = ⊕M_{λ} such that K_{ν}x = ζ^{⟨ν,λ⟩}x for any ν ∈ Y, λ ∈ X, x ∈ M_{λ}. Morphisms are 𝔲^{≤0}- (resp. 𝔲^{≥0}-) linear maps compatible with the X-gradings.

We have the obvious functors C ⟶ C^{≤0} (resp. C ⟶ C^{≥0}). These functors admit the exact left adjoints ind_{𝔲^{≤0}}^{𝔲} : C^{≤0} ⟶ C (resp. ind_{𝔲^{≥0}}^{𝔲} : C^{≥0} ⟶ C).

For example, ind_{𝔲^{≥0}}^{𝔲}(M) = 𝔲 ⊗_{𝔲^{≥0}} M. The triangular decomposition induces an isomorphism of graded vector spaces ind_{𝔲^{≥0}}^{𝔲}(M) ≅ 𝔲^{−} ⊗ M.

2.17. For λ ∈ X, let us consider on object k^{λ} ∈ C^{≥0} defined as follows. As a graded vector space, k^{λ} = k_{λ}^{λ} = k. The algebra 𝔲^{≥0} acts on k^{λ} as follows: ϵ_{i}x = 0, K_{ν}x = ζ^{⟨ν,λ⟩}x (i ∈ I, ν ∈ Y, x ∈ k^{λ}).

The object M(λ) = ind_{𝔲^{≥0}}^{𝔲}(k^{λ}) is called a (baby) Verma module. Each M(λ) has a unique irreducible quotient object, to be denoted by L(λ). The objects L(λ) (λ ∈ X) are mutually non-isomorphic and every irreducible object in C is isomorphic to one of them. Note that the category C is artinian, i.e. each object of C has a finite filtration with irreducible quotients.

For example, L(0) = 1.

2.18. Recall that a braiding on C is a collection of isomorphisms

R_{M,M′} : M ⊗ M′ ⟶^{∼} M′ ⊗ M (M, M′ ∈ C)

satisfying certain compatibility with the tensor structure (see [KL]IV, A.11).

A balance on C is an automorphism of the identity functor b = {b_{M} : M ⟶^{∼} M (M ∈ C)} such that for every M, N ∈ C, b_{M⊗N} ∘ (b_{M} ⊗ b_{N})^{−1} = R_{N,M} ∘ R_{M,N} (see loc. cit.).

2.19. Let δ denote the determinant of the I ×I-matrix (⟨i, j′⟩). From now on we assume that char(k) does not divide 2δ, and k contains an element ζ′ such that (ζ′)^{2δ} = ζ; we fix such an element ζ′. For a number q ∈ \frac{1}{2δ}ℤ, ζ^{q} will denote (ζ′)^{2δq}.

2.20. Theorem. (G. Lusztig) There exists a unique braided structure {R_{M,N}} on the tensor category C such that for any λ ∈ X and M ∈ C, if μ ∈ X is such that M_{μ′} ≠ 0 implies μ′ ≤ μ, then

R_{L(λ),M}(x ⊗ y) = ζ^{λ·μ}y ⊗ x

for any x ∈ L(λ), y ∈ M_{μ}. □

2.21. Let n : X ⟶ \frac{1}{2δ}ℤ be the function defined by

n(λ) = \frac{1}{2}λ · λ − λ · ρ_{ℓ}.

2.22. Theorem. There exists a unique balance b on C such that for any λ ∈ X, b_{L(λ)} = ζ^{n(λ)}. □

2.23. The rigid tensor category C, together with the additional structure given by the above braiding and balance, is a ribbon category in the sense of Turaev, cf. [K] and references therein.

3. Braiding local systems

3.1. For a topological space T and a finite set J, T^{J} will denote the space of all maps J ⟶ T (with the topology of the cartesian product). Its points are J-tuples (t_{j}) of points of T. We denote by T^{Jo} the subspace consisting of all (t_{j}) such that for any j′ ≠ j′′ in J, t_{j′} ≠ t_{j′′}.

Let ν = ∑ ν_{i}i ∈ Y^{+}. Let us call an unfolding of ν a map of finite sets π : J ⟶ I such that card(π^{−1}(i)) = ν_{i} for all i. Let Σ_{π} denote the group of all automorphisms σ : J ⟶^{∼} J such that π ∘ σ = π.

The group Σ_{π} acts on the space T^{J} in the obvious way. We denote by T^{ν} the quotient space T^{J}/Σ_{ν}. This space does not depend, up to a unique isomorphism, on the choice of an unfolding π. The points of T^{ν} are collections (t_{j}) of I-colored points of T, such that for any I, there are ν_{i} points of color i. We have the canonical projection T^{J} ⟶ T^{ν}, also to be denoted by π. We set T^{νo} = π(T^{Jo}). The map π restricted to T^{Jo} is an unramified Galois covering with the Galois group Σ_{π}.

3.2. For a real r > 0, let D(r) denote the open disk on the complex plane {t ∈ ℂ| |t| < r} and D̄(r) its closure. For r_{1} < r_{2}, denote by A(r_{1}, r_{2}) the open annulus D(r_{2}) − D̄(r_{1}).

Set D = D(1). Let A denote the punctured disk D − {0}.

3.3. For an integer n ≥ 1, consider the space

E_{n} = {(r_{0}, …, r_{n}) ∈ ℝ^{n+1}| 0 = r_{0} < r_{1} < … < r_{n} = 1}.

Obviously, the space E_{n} is contractible.

Let J_{1}, …, J_{n} be finite sets. Set J = ∐_{a=1}^{n} J_{a}. Note that D^{J} = D^{J_{1}} × … × D^{J_{n}}. Let us untroduce the subspace D^{J_{1},... ,J_{n}} ⊂ E_{n} × D^{J}. By definition it consists of points ((r_{a}); (t_{j}^{a}) ∈ D^{J_{a}}, a = 1, …, n) such that t_{j}^{a} ∈ A(r_{a−1}, r_{a}) for a = 1, …, n. The canonical projection E_{n} × D^{J} ⟶ D^{J} induces the map

m(J_{1}, …, J_{n}) : D^{J_{1},... ,J_{n}} ⟶ D^{J}.

The image of the above projection lands in the subspace D^{J_{1}} × A^{J_{2}} × … × A^{J_{n}} ⊂ D^{J_{1}} × … × D^{J_{n}} = D^{J}. The induced map

p(J_{1}, …, J_{n}) : D^{J_{1},... ,J_{n}} ⟶ D^{J_{1}} × A^{J_{2}} × … × A^{J_{n}}

is homotopy equivalence.

Now assume that we have maps π_{a} : J_{a} ⟶ I which are unfoldings of the elements ν_{a}. Then their sum π : J ⟶ I is an unfolding of ν = ν_{1} + … + ν_{n}. We define the space D^{ν_{1},... ,ν_{n}} ⊂ E_{n} × D^{ν} as the image of D^{J_{1},... ,J_{n}} under the projection Id × π : E_{n} × D^{J} ⟶ E_{n} × D^{ν}. The maps m(J_{1}, …, J_{n}) and p(J_{1}, …, J_{n}) induce the maps

m(ν_{1}, …, ν_{n}) : D^{ν_{1},... ,ν_{n}} ⟶ D^{ν_{1}+...+ν_{n}}

and

p(ν_{1}, …, ν_{n}) : D^{ν_{1},... ,ν_{n}} ⟶ D^{ν_{1}} × A^{ν_{2}} × … × A^{ν_{n}}

respectively, the last map being homotopy equivalence.

3.4. We define the open subspaces

A^{ν_{1},... ,ν_{n}} = D^{ν_{1},... ,ν_{n}} ∩ (E_{n} × A^{ν})

and

A^{ν_{1},... ,ν_{n}o} = D^{ν_{1},... ,ν_{n}} ∩ (E_{n} × A^{νo}).

We have the maps

m_{a}(ν_{1}, …, ν_{n}) : D^{ν_{1},... ,ν_{n}} ⟶ D^{ν_{1},... ,ν_{a−1},ν_{a}+ν_{a+1},ν_{a+2},... ,ν_{n}}

and

p_{a}(ν_{1}, …, ν_{n}) : D^{ν_{1},... ,ν_{n}} ⟶ D^{ν_{1},... ,ν_{a}} × A^{ν_{a+1},... ,ν_{n}}

(a = 1, …, n − 1) defined in an obvious manner. They induce the maps

m_{a}(ν_{1}, …, ν_{n}) : A^{ν_{1},... ,ν_{n}} ⟶ A^{ν_{1},... ,ν_{a−1},ν_{a}+ν_{a+1},ν_{a+2},... ,ν_{n}}

and

p_{a}(ν_{1}, …, ν_{n}) : A^{ν_{1},... ,ν_{n}} ⟶ A^{ν_{1},... ,ν_{a}} × A^{ν_{a+1},... ,ν_{n}}

and similar maps between ”o”-ed spaces. All the maps p are homotopy equivalences.

All these maps satisfy some obvious compatibilities. We will need the following particular case.

3.5. The rhomb diagram below commutes.

[figure: D^{ν_{1}+ν_{2}+ν_{3}}; D^{ν_{1},ν_{2}+ν_{3}}; D^{ν_{1}+ν_{2},ν_{3}}; D^{ν_{1}} × A^{ν_{1}+ν_{2}}; D^{ν_{1},ν_{2},ν_{3}}; D^{ν_{1}+ν_{2}} × A^{ν_{3}}; D^{ν_{1}} × A^{ν_{1},ν_{2}}; D^{ν_{1},ν_{2}} × A^{ν_{3}}; D^{ν_{1}} × A^{ν_{2}} × A^{ν_{3}}; m; m; p; m; m; p; m; p; p; m; p; p]

3.6. We will denote by A^{o} and call an open I-coloured configuration space the collection of all spaces {A^{ν_{1},... ,ν_{n}o}} together with the maps {m_{a}(ν_{1}, …, ν_{n}), p_{a}(ν_{1}, …, ν_{n})} between their various products.

We will call a local system over A^{o}, or a braiding local system a collection of data (a), (b) below satisfying the property (c) below.

(a) A local system I_{μ}^{ν} over A^{νo} given for any ν ∈ Y^{+}, μ ∈ X.

(b) An isomorphism ϕ_{μ}(ν_{1}, ν_{2}) : m^{∗}I_{μ}^{ν_{1}+ν_{2}} ⟶^{∼} p^{∗}(I_{μ}^{ν_{1}} ⊠ I_{μ−ν_{1}}^{ν_{2}}) given for any ν_{1}, ν_{2} ∈

Y^{+}, μ ∈ X.

Here p = p(ν_{1}, ν_{2}) and m = m(ν_{1}, ν_{2}) are the arrows in the diagram A^{ν_{1}o} × A^{ν_{2}o} ⟵^{p}

A^{ν_{1},ν_{2}o} ⟶^{m} A^{ν_{1}+ν_{2}o}.

The isomorphisms ϕ_{μ}(ν_{1}, ν_{2}) are called the factorization isomorphisms.

(c) (The associativity of factorization isomorphisms.) For any ν_{1}, ν_{2}, ν_{3} ∈ Y^{+}, μ ∈ X, the octagon below commutes. Here the maps m, p are the maps in the rhombic diagram above, with D, A replaced by A^{o}.

[figure: m^{∗}m_{1}^{∗}I_{μ}^{ν_{1}+ν_{2}+ν_{3}}; m^{∗}m_{2}^{∗}I_{μ}^{ν_{1}+ν_{2}+ν_{3}}; m^{∗}p^{∗}(I_{μ}^{ν_{1}} ⊠ I_{μ−ν_{1}}^{ν_{2}+ν_{3}}); m^{∗}p^{∗}(I_{μ}^{ν_{1}+ν_{2}} ⊠ I_{μ−ν_{1}−ν_{2}}^{ν_{3}}); p^{∗}m^{∗}(I_{μ}^{ν_{1}} ⊠ I_{μ−ν_{1}}^{ν_{2}+ν_{3}}); p^{∗}m^{∗}(I_{μ}^{ν_{1}+ν_{2}} ⊠ I_{μ−ν_{1}−ν_{2}}^{ν_{3}}); p_{1}^{∗}p^{∗}(I_{μ}^{ν_{1}} ⊠ I_{μ−ν_{1}}^{ν_{2}} ⊠ I_{μ−ν_{1}−ν_{2}}^{ν_{3}}); p_{2}^{∗}p^{∗}(I_{μ}^{ν_{1}} ⊠ I_{μ−ν_{1}}^{ν_{2}} ⊠ I_{μ−ν_{1}−ν_{2}}^{ν_{3}}); ϕ_{μ}(ν_{1}, ν_{2} + ν_{3}); ϕ_{μ}(ν_{1} + ν_{2}, ν_{3}); ϕ_{μ−ν_{1}}(ν_{2}, ν_{3}); ϕ_{μ}(ν_{1}, ν_{2})]

Written more concisely, the axiom (c) reads as a ”cocycle” condition

(c)′ ϕ_{μ}(ν_{1}, ν_{2}) ∘ ϕ_{μ}(ν_{1} + ν_{2}, ν_{3}) = ϕ_{μ−ν_{1}}(ν_{2}, ν_{3}) ∘ ϕ_{μ}(ν_{1}, ν_{2} + ν_{3}).

3.7. Correctional lemma. Assume we are given the data (a), (b) as above, with one-dimensional local systems I_{μ}^{ν}. Then there exist a collection of constants c_{μ}(ν_{1}, ν_{2}) ∈ k^{∗} (μ ∈ X, ν_{1}, ν_{2} ∈ Y^{+}) such that the corrected isomorphisms ϕ′_{μ}(ν_{1}, ν_{2}) = c_{μ}(ν_{1}, ν_{2})ϕ_{μ}(ν_{1}, ν_{2}) satisfy the associativity axiom (c). □

3.8. The notion of a morphism between two braiding local systems is defined in an obvious way. This defines the category of braiding local systems, Bls.

Suppose that I = {I_{μ}^{ν}; ϕ_{μ}(ν_{1}, ν_{2})} and J = {J_{μ}^{ν}; ψ_{μ}(ν_{1}, ν_{2})} are two braiding local systems. Their tensor product I ⊗ J is defined by (I ⊗ J)_{μ}^{ν} = I_{μ}^{ν} ⊗ J_{μ}^{ν}, the factorization isomorphisms being ϕ_{μ}(ν_{1}, ν_{2}) ⊗ ψ_{μ}(ν_{1}, ν_{2}). This makes Bls a tensor category. The subcategory of one-dimensional braiding local systems is a Picard category.

3.9. Example. Sign local system. Let L be a one-dimensional vector space. Let ν ∈ Y^{+}. Let π : J ⟶ I be an unfolding of ν, whence the canonical projection π : A^{J∘} ⟶ A^{ν∘}. Pick a point x̃ = (x_{j}) ∈ A^{J∘} with all x_{j} being real; let x = π(x̃). The choice of base points defines the homomorphism π_{1}(A^{ν∘}; x) ⟶ Σ_{π}. Consider its composition with the sign map Σ_{π} ⟶ μ_{2} ↪ k^{∗}. Here μ_{2} = {±1} is the group of square roots of 1 in k^{∗}. We get a map s : π_{1}(A^{ν∘}; x) ⟶ k^{∗}. It defines a local system Sign^{ν}(L) over A^{ν∘} whose stalk at x is equal to L. This local system does not depend (up to the unique isomorphism) on the choices made.

Given a diagram

A^{ν_{1}∘} × A^{ν_{2}∘} ⟵^{p} A^{ν_{1},ν_{2}∘} ⟶^{m} A^{ν_{2}+ν_{2}∘}

we can choose base point x_{i} ∈ A^{ν_{i}} and x ∈ A^{ν_{1}+ν_{2}∘} such that there exists y ∈ A^{ν_{1},ν_{2}} with p(y) = (x_{1}, x_{2}) and m(y) = x. Let

ϕ^{L_{1},L_{2}}(ν_{1}, ν_{2}) : m^{∗}Sign^{ν_{1}+ν_{2}}(L_{1} ⊗ L_{2}) ⟶^{∼} p^{∗}(Sign^{ν_{1}}(L_{1}) ⊠ Sign^{ν_{2}}(L_{2}))

be the unique isomorphism equal to Id_{L_{1}⊗L_{2}} over y. These isomorphisms satisfy the associativity condition (cf. (c)′ above).

Let us define the brading local system Sign by Sign_{μ}^{ν} = Sign^{ν}(1), ϕ_{μ}(ν_{1}, ν_{2}) = ϕ^{1,1}(ν_{1}, ν_{2}). Here 1 denotes the standard vector space k.

3.10. Example. Standard local systems J, I. Let ν ∈ Y^{+}, let π : J ⟶ I be an unfolding of ν, n = card(J), π : A^{J∘} ⟶ A^{ν∘} the corresponding projection. Let L be a one-dimensional vector space. For each isomorphism σ : J ⟶^{∼} {1, …, n}, choose a point x(σ) = (x_{j}) ∈ A^{J∘} such that all x_{j} are real and positive and for each j′, j′′ such that σ(j′) < σ(j′′), we have x_{j′} < x_{j′′}.

Let us define a local system J_{μ}^{π}(L) over A^{J∘} as follows. We set J_{μ}^{π}(L)_{x(σ)} = L for all σ. We define the monodromies

T_{γ} : J_{μ}^{π}(L)_{x(σ)} ⟶ J_{μ}^{π}(L)_{x(σ′)}

along the homotopy classes of paths γ : x_{σ} ⟶ x_{σ′} which generate the fundamental groupoid. Namely, let x_{j′} and x_{j′′} be some neighbour points in x(σ), with x_{j′} < x_{j′′}. Let γ(j′, j′′)^{+} (resp. γ(j′, j′′)^{−}) be the paths corresponding to the movement of x_{j′′} in the upper (resp. lower) hyperplane to the left from x_{j′} position. We set

T_{γ(j′,j′′)^{±}} = ζ^{∓π(j′)·π(j′′)}.

Let x_{j} be a point in x(σ) closest to 0. Let γ(j) be the path corresponding to the counterclockwise travel of x_{j} around 0. We set

T_{γ(j)} = ζ^{2μ·π(j)}.

The point is that the above fromulas give a well defined morphism from the fundamental groupoid to the groupoid of one-dimensional vector spaces.

The local system J_{μ}^{π}(L) admits an obvious Σ_{π}-equivariant structure. This defines the local system J_{μ}^{ν}(L) over A^{ν∘}. Given a diagram

A^{ν_{1}∘} × A^{ν_{2}∘} ⟵^{p} A^{ν_{1}+ν_{2}∘} ⟶^{m} A^{ν_{1}+ν_{2}∘},

let

ϕ_{μ}^{L_{1},L_{2}}(ν_{1}, ν_{2}) : m^{∗}J_{μ}^{ν_{1}+ν_{2}}(L_{1} ⊗ L_{2}) ⟶^{∼} p^{∗}(J_{μ}^{ν_{1}}(L_{1}) ⊠ J_{μ−ν_{1}}^{ν_{2}}(L_{2}))

be the unique isomorphism equal to Id_{L_{1}⊗L_{2}} over compatible base points. Here the compatibility is understood in the same sense as in the previous subsection. These isomorphisms satisfy the associativity condition.

We define the braiding local system J by J_{μ}^{ν} = J_{μ}^{ν}(1), ϕ_{μ}(ν_{1}, ν_{2}) = ϕ_{μ}^{1,1}(ν_{1}, ν_{2}).

We define the braiding local system I by I = J ⊗ Sign. The local systems J, I are called the standard local systems. In the sequel we will mostly need the local system I.

4. Factorizable sheaves

4.1. In the sequel for each ν ∈ Y^{+}, we will denote by S the stratification on the space D^{ν} the closures of whose strata are various intersections of hypersurfaces given by the equations t_{j} = 0, t_{j′} = t_{j′′}. The same letter will denote the induced stratifications on its subspaces.

4.2. Set I_{μ}^{ν•} = j_{!∗}I_{μ}^{ν}[dim A^{ν∘}] ∈ M(A^{ν}; S) where j : A^{ν∘} ↪ A^{ν} is the embedding.

Given a diagram

A^{ν_{1}} × A^{ν_{2}} ⟵^{p} A^{ν_{1},ν_{2}} ⟶^{m} A^{ν_{1}+ν_{2}},

the structure isomorphisms ϕ_{μ}(ν_{1}, ν_{2}) of the local system I induce the isomorphisms, to be denoted by the same letters,

ϕ_{μ}(ν_{1}, ν_{2}) : m^{∗}I^{ν_{1}+ν_{2}•} ⟶^{∼} p^{∗}(I^{ν_{1}•} ⊠ I^{ν_{2}•}).

Obviously, these isomorphisms satisfy the associativity axiom.

4.3. Let us fix a coset c ∈ X/Y. We will regard c as a subset of X. We will call a factorizable sheaf M supported at c a collection of data (w), (a), (b) below satisfying the axiom (c) below.

(w) An element λ = λ(M) ∈ c.

(a) A perverse sheaf M^{ν} ∈ M(D^{ν}; S) given for each ν ∈ Y^{+}.

(b) An isomorphism ψ(ν_{1}, ν_{2}) : m^{∗}M^{ν_{1}+ν_{2}} ⟶^{∼} p^{∗}(M^{ν_{1}} ⊠ I_{λ−ν_{1}}^{ν_{2}•}) given for any ν_{1}, ν_{2} ∈ Y^{+}.

Here p, m denote the arrows in the diagram D^{ν_{1}} × A^{ν_{2}} ⟵^{p} D^{ν_{1},ν_{2}} ⟶^{m} D^{ν_{1}+ν_{2}}.

The isomorphisms ψ(ν_{1}, ν_{2}) are called the factorization isomorphisms.

(c) For any ν_{1}, ν_{2}, ν_{3} ∈ Y^{+}, the following associativity condition is fulfilled:

ψ(ν_{1}, ν_{2}) ∘ ψ(ν_{1} + ν_{2}, ν_{3}) = ϕ_{λ−ν_{1}}(ν_{2}, ν_{3}) ∘ ψ(ν_{1}, ν_{2} + ν_{3}).

We leave to the reader to draw the whole octagon expressing this axiom.

4.4. Let M = {M^{ν}; ψ(ν_{1}, ν_{2})} be a factorizable sheaf supported at a coset c ∈ X/Y, λ = λ(M). For each λ′ ≥ λ, ν ∈ Y^{+}, define a sheaf M_{λ′}^{ν} ∈ M(D^{ν}; S) by

M_{λ′}^{ν} = \{ \begin{matrix}ι(λ′ − λ)_{∗}M^{ν−λ′+λ} & if ν − λ′ + λ ∈ Y^{+} \\ 0 & otherwise. \end{matrix}

Here

ι(ν′) : D^{ν} ⟶ D^{ν+ν′}

denotes the closed embedding adding ν′ points sitting at the origin. The factorization isomorphisms ψ(ν_{1}, ν_{2}) induce similar isomorphisms

ψ_{λ′}(ν_{1}, ν_{2}) : m^{∗}M_{λ′}(ν_{1} + ν_{2}) ⟶^{∼} p^{∗}(M_{λ′}^{ν_{1}} ⊠ I_{λ′−ν_{1}}^{ν_{2}}) (λ′ ≥ λ)

4.5. Let M, N be two factorizable sheaves supported at c. Let λ ∈ X be such that λ ≥ λ(M) and λ ≥ λ(N). For ν ≥ ν′ in Y^{+}, consider the following composition

τ_{λ}(ν, ν′) : Hom(M_{λ}^{ν}, N_{λ}^{ν}) ⟶^{m_{∗}} Hom(m^{∗}M_{λ}^{ν}, m^{∗}N_{λ}^{ν}) ⟶^{∼}^{ψ(ν′,ν−ν′)_{∗}}

⟶^{∼} Hom(p^{∗}(M_{λ}^{ν′} ⊠ I_{λ−ν′}^{ν−ν′•}), p^{∗}(N_{λ}^{ν′} ⊠ I_{λ−ν′}^{ν−ν′•})) = Hom(M_{λ}^{ν′}, N_{λ}^{ν′}).

Let us define the space of homomorphisms Hom(M, N) by

Hom(M, N) = lim_{→}_{λ}lim_{←}_{ν}Hom(M_{λ}^{ν}, N_{λ}^{ν})

Here the inverse limit is taken over ν ∈ Y^{+}, the transition maps being τ_{λ}(ν, ν′) and the direct limit is taken over λ ∈ X such that λ ≥ λ(M), λ ≥ λ(N), the transition maps being induced by the obvious isomorphisms

Hom(M_{λ}^{ν}, N_{λ}^{ν}) = Hom(M_{λ+ν′}^{ν+ν′}, N_{λ+ν′}^{ν+ν′}) (ν′ ∈ Y^{+}).

With these spaces of homomorphisms and the obvious compositions, the factorizable sheaves supported at c form the category, to be denoted by F̃S_{c}. By definition, the category F̃S is the direct product ∏_{c∈X/Y} F̃S_{c}.

4.6. Finite sheaves. Let us call a factorizable sheaf M = {M^{ν}} ∈ F̃S_{c} finite if there exists only a finite number of ν ∈ Y^{+} such that the conormal bundle of the origin O ∈ A^{ν} is contained in the singular support of M^{ν}. Let FS_{c} ⊂ F̃S_{c} be the full subcategory of finite factorizable sheaves. We define the category FS by FS = ∏_{c∈X/Y} FS_{c}. One proves (using the lemma below) that FS is an abelian category.

4.7. Stabilization Lemma. Let M, N ∈ FS_{c}, μ ∈ X_{c}, μ ≥ λ(M), μ ≥ λ(N). There exists ν_{0} ∈ Y^{+} such that for all ν ≥ ν_{0} the canonical maps

Hom(M, N) ⟶ Hom(M_{μ}^{ν}, N_{μ}^{ν})

are isomorphisms. □

4.8. Standard sheaves. Given μ ∈ X, let us define the ”standard sheaves” M(μ), DM(μ) and L(μ) supported at the coset μ + Y, by λ(M(μ)) = λ(DM(μ)) = λ(L(μ)) = μ;

M(μ)^{ν} = j_{!}I_{μ}^{ν•}; DM(μ)^{ν} = j_{∗}I_{μ}^{ν•}; L(μ)^{ν} = j_{!∗}I_{μ}^{ν•},

j being the embedding A^{ν} ↪ D^{ν}. The factorization maps are defined by functoriality from the similar maps for I^{•}.

One proves that all these sheaves are finite.

5. Tensor product

In this section we will give (a sketch of) the construction of the tensor structure on the category F̃S. We will make the assumption of 2.19^{5}.

5.1. For z ∈ ℂ and a real positive r, let D(z; r) denote the open disk {t ∈ ℂ| |t − z| < r} and D̄(z; r) its closure.

5.2. For ν ∈ Y^{+}, let us define the space D^{ν}(2) as the product A × D^{ν}. Its points will be denoted (z; (t_{j})) where z ∈ A, (t_{j}) ∈ D^{ν}. Let us define the open subspaces

A^{ν}(2) = {(z; (t_{j})) ∈ D^{ν}(2)| t_{j} ≠ 0, z for all j}; A^{ν}(2)^{∘} = A^{ν}(2) ∩ (A × A^{ν∘}).

For ν, ν′ ∈ Y^{+}, let us define the space D^{ν,ν′}(2) as the subspace of ℝ_{>0}×D^{ν+ν′}(2) consisting of all elements (r; z; (t_{j})) such that |z| < r < 1 and ν of the points t_{j} live inside the disk D(r) and ν′ of them inside the annulus A(r, 1).

We have a diagram

(a) D^{ν}(2) × A^{ν′} ⟵^{p} D^{ν,ν′}(2) ⟶^{m} D^{ν+ν′}(2).

Here p((r; z; (t_{j}))) = ((z; (t_{j′})), (t_{j′′})) where t_{j′} (resp. t_{j′′}) being the points from the collection (t_{j}) lying in D(r) (resp. in A(r, 1)); m((r; z; (t_{j}))) = (z; (t_{j})). The map p is a homotopy equivalence.

For ν_{1}, ν_{2}, ν ∈ Y^{+}, let D^{ν_{1};ν_{2};ν}(2) be the subspace of ℝ_{>0} × ℝ_{>0} × D^{ν_{1}+ν_{2}+ν}(2) consisting of all elements (r_{1}; r_{2}; z; (t_{j})) such that D̄(r_{1}) ∪ D̄(z; r_{2}) ⊂ D; D̄(r_{1}) ∩ D̄(z; r_{2}) = ∅; ν_{1} of the points (t_{j}) lie inside D(r_{1}), ν_{2} of them lie inside D(z; r_{2}) and ν of them lie inside D − (D̄(r_{1}) ∪ D̄(z; r_{2})).

We have a diagram

(b) D^{ν_{1}} × D^{ν_{2}} × A^{ν}(2) ⟵^{p} D^{ν_{1};ν_{2};ν}(2) ⟶^{m} D^{ν_{1}+ν_{2}+ν}(2).

Here p((r_{1}; r_{2}; z; (t_{j}))) = ((t_{j′}); (t_{j′′} − z); (t_{j′′′})) where t_{j′} (resp. t_{j′′}, t_{j′′′}) are the points lying inside D(r_{1}) (resp. D(z; r_{2}), D − (D̄(r_{1}) ∪ D̄(z; r_{2}))); m((r_{1}; r_{2}; z; (t_{j}))) = (z; (t_{j})). The map p is a homotopy equivalence.

5Note that these assumptions are not necessary for the construction of the tensor structure. They are essential, however, for the construction of braiding.

5.3. We set A^{ν,ν′}(2) = D^{ν,ν′}(2)∩(ℝ_{>0}×A^{ν+ν′}(2)); A^{ν,ν′}(2)^{∘} = D^{ν,ν′}(2)∩(ℝ_{>0}×A^{ν+ν′}(2)^{∘}); A^{ν_{1};ν_{2};ν}(2) = D^{ν_{1};ν_{2};ν}(2) ∩ (ℝ_{>0} × ℝ_{>0} × A^{μ_{1}+ν_{2}+ν}(2)); A^{ν_{1};ν_{2};ν}(2)^{∘} = D^{ν_{1};ν_{2};ν}(2) ∩ (ℝ_{>0} × ℝ_{>0} × A^{ν_{1}+ν_{2}+ν}(2)^{∘}).

5.4. Given μ_{1}, μ_{2} ∈ X, ν ∈ Y^{+}, choose an unfolding π : J ⟶ I of the element ν. In the same manner as in 3.10, we define the one-dimensional local system J_{μ_{1},μ_{2}}^{ν} over A^{ν}(2)^{∘} with the following monodromies: the monodromy around a loop corresponding to the counterclockwise travel of the point z around 0 (resp. t_{j} around 0, t_{j} around z, t_{j′} around t_{j′′}) is equal to the multiplication by ζ^{−2μ_{1}·μ_{2}} (resp. ζ^{2μ_{1}·π(j)}, ζ^{2μ_{2}·π(j)}, ζ^{−2π(j′)·π(j′′)}).

As in loc. cit., one defines isomorphisms

(a) ϕ_{μ_{1},μ_{2}}(ν, ν′) : m^{∗}J_{μ_{1},μ_{2}}^{ν+ν′} ⟶^{∼} p^{∗}(J_{μ_{1},μ_{2}}^{ν} ⊠ J_{μ_{1}+μ_{2}−ν}^{ν′})

where p, m are the morphisms in the diagram 5.2 (a) (restricted to the A^{∘}-spaces) and

(b) ϕ_{μ_{1},μ_{2}}(ν_{1}; ν_{2}; ν) : m^{∗}J_{μ_{1},μ_{2}}^{ν_{1}+ν_{2}+ν} ⟶^{∼} p^{∗}(J_{μ_{1}}^{ν_{1}} ⊠ J_{μ_{2}}^{ν_{2}} ⊠ J_{μ_{1}−ν_{1},μ_{2}−ν_{2}}^{ν})

where p, m are the morphisms in the diagram 5.2 (b) (restricted to the A^{∘}-spaces), which satisfy the cocycle conditions

(c) ϕ_{μ_{1},μ_{2}}(ν, ν′) ∘ ϕ_{μ_{1},μ_{2}}(ν + ν′, ν′′) = ϕ_{μ_{1}+μ_{2}−ν}(ν′, ν′′) ∘ ϕ_{μ_{1},μ_{2}}(ν, ν′ + ν′′)

(d) (ϕ_{μ_{1}}(ν_{1}, ν′_{1}) ⊠ ϕ_{μ_{2}}(ν_{2}, ν′_{2})) ∘ ϕ_{μ_{1},μ_{2}}(ν_{1} + ν′_{1}; ν_{2} + ν′_{2}; ν) =

= ϕ_{μ_{1}−ν_{1},μ_{2}−ν_{2}}(ν′_{1}; ν′_{2}; ν) ∘ ϕ_{μ_{1},μ_{2}}(ν_{1}; ν_{2}; ν + ν′_{1} + ν′_{2})

(we leave to the reader the definition of the corresponding spaces).

5.5. Let us consider the sign local systems introduced in 3.9. We will keep the same notation Sign^{ν} for the inverse image of the local system Sign^{ν} under the forgetting of z map A^{ν}(2)^{∘} ⟶ A^{ν∘}. We have the factorization isomorphisms

(a) ϕ^{Sign}(ν, ν′) : m^{∗}Sign^{ν+ν′} ⟶^{∼} p^{∗}(Sign^{ν} ⊠ Sign^{ν′});

(b) ϕ^{Sign}(ν_{1}; ν_{2}; ν) : m^{∗}Sign^{ν_{1}+ν_{2}+ν} ⟶^{∼} p^{∗}(Sign^{ν_{1}} ⊠ Sign^{ν_{2}} ⊠ Sign^{ν})

which satisfy the cocycle conditions similar to (c), (d) above.

5.6. We define the local systems I_{μ_{1},μ_{2}}^{ν} over the spaces A^{ν}(2)^{∘} by

I_{μ_{1},μ_{2}}^{ν} = J_{μ_{1},μ_{2}}^{ν} ⊗ Sign^{ν}.

The collection of local systems {I_{μ_{1},μ_{2}}^{ν}} together with the maps ϕ_{μ_{1},μ_{2}}^{I}(ν, ν′) = ϕ_{μ_{1},μ_{2}}^{J}(ν, ν′) ⊗ ϕ^{Sign}(ν, ν′) and ϕ_{μ_{1},μ_{2}}^{I}(ν_{1}; ν_{2}; ν) = ϕ_{μ_{1},μ_{2}}^{J}(ν_{1}; ν_{2}; ν) ⊗ ϕ^{Sign}(ν_{1}; ν_{2}; ν), forms an object I(2) which we call a standard braiding local system over the configuration space A(2)^{∘} = {A^{ν}(2)^{∘}}. It is unique up to a (non unique) isomorphism. We fix such a local system.

5.7. We set I_{μ_{1},μ_{2}}^{ν•} = j_{!∗}I_{μ_{1},μ_{2}}^{ν}[dim A^{ν}(2)^{∘}] where j : A^{ν}(2)^{∘} ↪ A^{ν}(2) is the open embedding. It is an object of the category M(A^{ν}(2); S) where S is the evident stratification. The factorization isomorphisms for the local system I induce the analogous isomorphisms between these sheaves, to be denoted by the same letter. The collection of these sheaves and factorization isomorphisms will be denoted I(2)^{•}.

5.8. Suppose we are given two factorizable sheaves M, N. Let us call their gluing, and denote by M ⊠ N, the collection of perverse sheaves (M ⊠ N)^{ν} over the spaces D^{ν}(2) (ν ∈ Y^{+}) together with isomorphisms

ψ(ν_{1}; ν_{2}; ν) : m^{∗}(M ⊠ N)^{ν} ⟶^{∼} p^{∗}(M^{ν_{1}} ⊠ N^{ν_{2}} ⊠ I_{λ(M)−ν_{1},λ(N)−ν_{2}}^{ν•}),

p, m being the maps in the diagram 5.2 (b), which satisfy the cocycle condition

(ψ^{M}(ν_{1}, ν′_{1}) ⊠ ψ^{N} (ν_{2}, ν′_{2})) ∘ ψ(ν_{1} + ν′_{1}; ν_{2} + ν′_{2}; ν) =

= ϕ_{λ(M)−ν_{1},λ(N)−ν_{2}}(ν′_{1}; ν′_{2}; ν) ∘ ψ(ν_{1}; ν_{2}; ν + ν′_{1} + ν′_{2})

for all ν_{1}, ν′_{1}, ν_{2}, ν′_{2}, ν ∈ Y^{+}.

Such a gluing exists and is unique, up to a unique isomorphism. The factorization isomorphisms ϕ_{μ_{1},μ_{2}}(ν_{1}; ν_{2}; ν) for I(2)^{•} and the ones for M, N, induce the isomorphisms

ψ^{M⊠N}(ν, ν′) : m^{∗}(M ⊠ N)^{ν+ν′} ⟶^{∼} p^{∗}((M ⊠ N)^{ν} ⊠ I_{λ(M)+λ(N)−ν}^{ν′•}).

satisfying the obvious cocycle condition.

5.9. Now we can define the tensor product M ⊗ N ∈ F̃S. Namely, set λ(M ⊗ N) = λ(M) + λ(N). For each ν ∈ Y^{+}, set

(M ⊗ N)^{ν} = Ψ_{z→0}((M ⊠ N)^{ν}).

Here Ψ_{z→0} : M(D^{ν}(2)) ⟶ M(D^{ν}) denotes the functor of nearby cycles for the function D^{ν}(2) ⟶ D sending (z; (t_{j})) to z. Note that

Ψ_{z→0}(I_{μ_{1},μ_{2}}^{ν}) = I_{μ_{1}+μ_{2}}^{ν}.

The factorization isomorphisms ψ^{M⊠N} induce the factorization isomorphisms between the sheaves (M ⊗ N)^{ν}. This defines a factorizable sheaf M ⊗ N.

One sees at once that this construction is functorial; thus it defines a functor of tensor product ⊗ : F̃S × F̃S ⟶ F̃S.

The subcategory FS ⊂ F̃S is stable under the tensor product. The functor ⊗ : FS × FS ⟶ FS extends uniquely to a functor ⊗ : FS ⊗ FS ⟶ FS (for the discussion of the tensor product of abelian categories, see [D2] 5).

5.10. The half-circle travel of the point z around 0 from 1 to −1 in the upper halfplane defines the braiding isomorphisms

R_{M,N} : M ⊗ N ⟶^{∼} N ⊗ M.

We will not describe here the precise definition of the associativity isomorphisms for the tensor product ⊗. We just mention that to define them one should introduce into the game certain configuration spaces D^{ν}(3) whose (more or less obvious) definition we leave to the reader.

The unit of this tensor structure is the sheaf 1 = L(0) (cf. 4.8).

Equipped with these complementary structures, the category F̃S becomes a braided tensor category.

6. Vanishing cycles

GENERAL GEOMETRY

6.1. Let us fix a finite set J, and consider the space D^{J}. Inside this space, let us consider the subspaces D_{ℝ}^{J} = D^{J} ∩ ℝ^{J} and D^{J+} = D^{J} ∩ ℝ_{≥0}^{J}.

Let H be the set (arrangement) of all real hyperplanes in D_{ℝ}^{J} of the form H_{j} : t_{j} = 0 or H_{j′j′′} : t_{j′} = t_{j′′}. An edge L of the arrangement H is a subspace of D_{ℝ}^{J} which is a nonempty intersection ⋂ H of some hyperplanes from H. We denote by L^{∘} the complement L−⋃ L′, the union over all edges L′ ⊂ L of smaller dimension. A facet of H is a connected component F of some L^{∘}. We call a facet positive if it lies entirely inside D^{J+}.

For example, we have a unique smallest facet O — the origin. For each j ∈ J, we have a positive one-dimensional facet F_{j} given by the equations t_{j′} = 0 (j′ ≠ j); t_{j} ≥ 0.

Let us choose a point w_{F} on each positive facet F. We call a flag a sequence of embedded positive facets F : F_{0} ⊂ … F_{p}; we say that F starts from F_{0}. To such a flag we assign the simplex ΔF — the convex hull of the points w_{F_{0}}, …, w_{F_{p}}.

To each positive facet F we assign the following two spaces: D_{F} = ⋃ ΔF, the union over all flags F starting from F, and S_{F} = ⋃ ΔF′, the union over all flags F′ starting from a facet which properly contains F. Obviously, S_{F} ⊂ D_{F}.

6.2. Given a complex K ∈ D(D^{J}; S) and a positive facet F, we introduce a complex of vector spaces Φ_{F} (K) by

Φ_{F} (K) = RΓ(D_{F} , S_{F}; K)[− dim F].

This is a well defined object of the bounded derived category D(∗) of finite dimensional vector spaces, not depending on the choice of points w_{F}. It is called the complex of vanishing cycles of K across F.

6.3. Theorem. We have canonically Φ_{F} (DK) = DΦ_{F} (K) where D denotes the Verdier duality in the corresponding derived categories. □

6.4. Theorem. If M ∈ M(D^{J}; S) then H^{i}(Φ_{F} (M)) = 0 for i ≠ 0. Thus, Φ_{F} induces an exact functor

Φ_{F} : M(D^{J}; S) ⟶ Vect

to the category Vect of finite dimensional vector spaces. □

6.5. Given a positive facet E and K ∈ D(D^{J}, S), we have S_{E} = ⋃_{F ∈F^{1}(E)} D_{F}, the union over the set F^{1}(E) of all positive facets F ⊃ E with dim F = dim F + 1, and

RΓ(S_{E}, ⋃_{F ∈F^{1}(E)} S_{F}; K) = ⊕_{F ∈F^{1}(E)} RΓ(D_{F} , S_{F}; K).

6.6. For two positive facets E and F ∈ F^{1}(E), and K(D^{J}; S), let us define the natural map

u = u_{E}^{F}(K) : Φ_{F} (K) ⟶ Φ_{E}(K)

called canonical, as the composition

RΓ(D_{F} , S_{F}; K)[−p] ⟶ RΓ(S_{E}, ⋃_{F′∈F^{1}(E)} S_{F′}; K)[−p] ⟶ RΓ(S_{E}; K)[−p] ⟶

⟶ RΓ(D_{E}, S_{E})[−p + 1]

where p = dim F, the first arrow being induced by the equality in 6.5, the last one being the coboundary map.

Define the natural map

v = v_{F}^{E}(K) : Φ_{E}(K) ⟶ Φ_{F} (K)

called variation, as the map dual to the composition

DΦ_{F} (K) = Φ_{F} (DK) ⟶^{u(DK)} Φ_{E}(DK) = DΦ_{E}(K).

BACK TO FACTORIZABLE SHEAVES

6.7. Let ν ∈ Y. We are going to give two equivalent definitions of an exact functor, called vanishing cycles at the origin

Φ : M(D^{ν}; S) ⟶ Vect.

First definition. Let f : D^{ν} ⟶ D be the function f((t_{j})) = ∑ t_{j}. For an object K ∈ D(D^{ν}; S), the Deligne’s complex of vanishing cycles Φ_{f}(K) (cf. [D3]) is concentrated at the origin of the hypersurface f^{−1}(0). It is t-exact with respect to the middle t-structure. We set by definition, Φ(M) = H^{0}(Φ_{f}(M)) (M ∈ M(D^{ν}; S)).

Second definition. Choose an unfolding of ν, π : J ⟶ I. Let us consider the canonical projection π : D^{J} ⟶ D^{ν}. For K ∈ D(D^{ν}; S), the complex π^{∗}K is well defined as an element of the Σ_{π}-equivariant derived category, hence Φ_{O}(π^{∗}(K)) is a well defined object of the Σ_{π}-equivariant derived category of vector spaces (O being the origin facet in D^{J}). Therefore, the complex of Σ_{π}-invariants Φ_{O}(π^{∗}K)^{Σ_{π}} is a well defined object of D(∗). If K ∈ M(D^{ν}; S) then all the cohomology of Φ_{O}(π^{∗}K)^{Σ_{π}} in non-zero degree vanishes. We set

Φ(M) = H^{0}(Φ_{O}(π^{∗}M)^{Σ_{π}})

(M ∈ M(D^{ν}; S)). The equivalence of the two definitions follows without difficulty from the proper base change theorem. In computations the second definition is used.^{6}

6.8. Let M be a factorizable sheaf supported at c ∈ X/Y, λ = λ(M). For ν ∈ Y^{+}, define a vector space Φ(M)_{λ−ν} by

Φ(M)_{λ−ν} = Φ(M^{ν}).

If μ ∈ X, μ ≰ λ, set Φ(M)_{μ} = 0. One sees easily that this way we get an exact functor Φ from F̃S_{c} to the category of X-graded vector spaces with finite dimensional components. We extend it to the whole category F̃S by additivity.

6.9. Let M ∈ F̃S_{c}, λ = λ(M), ν = ∑ ν_{i}i ∈ Y^{+}. Let i ∈ I be such that ν_{i} > 0. Pick an unfolding of ν, π : J ⟶ I. For each j ∈ π^{−1}(i), the restriction of π, π_{j} : J − {j} ⟶ I, is an unfolding of ν − i.

For each j ∈ π^{−1}(i), we have canonical and variation morphisms

u_{j} : Φ_{F_{j}}(π^{∗}M^{ν}) ⇌ Φ_{O}(π^{∗}M^{ν}) : v_{j}

(the facet F_{j} has been defined in 6.1). Taking their sum over π^{−1}(i), we get the maps

∑ u_{j} : ⊕_{j∈π^{−1}(i)}Φ_{F_{j}}(π^{∗}M^{ν}) ⇌ Φ_{O}(π^{∗}M) : ∑ v_{j}

Note that the group Σ_{π} acts on both sides and the maps respect this action. After passing to Σ_{π}-invariants, we get the maps

(a) Φ_{F_{j}}(π^{∗}M^{ν})^{Σ_{π_{j}}} = (⊕_{j′∈π^{−1}(i)}Φ_{F_{j′}}(π^{∗}M^{ν}))^{Σ_{π}} ⇌ Φ_{O}(π^{∗}M^{ν})^{Σ_{π}} = Φ(M)_{ν}.

Here j ∈ π^{−1}(i) is an arbitrary element. Let us consider the space

F_{j}^{⊥} = {(t_{j′}) ∈ D^{J}|t_{j} = r, t_{j′} ∈ D(r′) for all j′ ≠ j}

where r, r′ are some fixed real numbers such that 0 < r′ < r < 1. The space F_{j}^{⊥} is transversal to F_{j} and may be identified with D^{J−{j}}. The factorization isomorphism induces the isomorphism

π^{∗}M^{ν}|_{F_{j}^{⊥}} ≅ π_{j}^{∗}M^{ν−i} ⊗ (I_{λ−ν+i}^{i})_{{r}} = π^{∗}M^{ν−i}

which in turn induces the isomorphism

Φ_{F_{j}}(π^{∗}M^{ν}) ≅ Φ_{O}(π_{j}^{∗}M^{ν−i}).

which is Σ_{π_{j}}-equivariant. Taking Σ_{π_{j}}-invariants and composing with the maps (a), we get the maps (b) ϵ_{i} : Φ(M)_{ν−i} = Φ_{O}(π_{j}^{∗}M^{ν−i})^{Σ_{j}} ⇌ Φ(M)_{ν} : θ_{i}

6its independence of the choice of an unfolding follows from its equivalence to the first definition.

which do not depend on the choice of j ∈ π^{−1}(i).

6.10. For an arbitrary M ∈ F̃S, let us define the X-graded vector space Φ(M) as Φ(M) = ⊕_{λ∈X}Φ(M)_{λ}.

6.11. Theorem. The operators ϵ_{i}, θ_{i} (i ∈ I) acting on the X-graded vector space Φ(M) satisfy the relations 2.11 (a), (b). □

6.12. A factorizable sheaf M is finite iff the space Φ(M) is finite dimensional. The previous theorem says that Φ defines an exact functor

Φ : FS ⟶ C.

One proves that Φ is a tensor functor.

6.13. Example. For every λ ∈ X, the factorizable sheaf L(λ) (cf. 4.8) is finite. It is an irreducible object of FS, and every irreducible object in FS is isomorphic to some L(λ). We have Φ(L(λ)) = L(λ).

The next theorem is the main result of the present work.

6.14. Theorem. The functor Φ is an equivalence of braided tensor categories. □

6.15. Remark. As a consequence, the category FS is rigid. We do not know a geometric construction of the rigidity; it would be very interesting to find one.

Chapter 2. Global (genus 0)

7. Cohesive local systems

7.1. From now on until the end of the paper we make the assumptions of 2.19. In the operadic notations below we partially follow [BD].

7.2. Operad D. For a nonempty finite set J, let D(J) denote the space whose points are J-tuples {x_{j}, τ_{j}} (j ∈ J) where x_{j} ∈ D and τ_{j} is a non-zero tangent vector at x_{j}, such that all points x_{j} are distinct.

Let D̃(J) be the space whose points are J-tuples {ϕ_{j}} of holomorphic maps ϕ_{j} : D ⟶ D (j ∈ J), each ϕ_{j} having the form ϕ_{j}(z) = x_{j} + τ_{j}z (x_{j} ∈ D, τ_{j} ∈ ℂ^{∗}), such that ϕ_{j}(D)∩ϕ_{j′}(D) = ∅ for j ≠ j′. We shall identify the j-tuple {ϕ_{j}} with the J-tuple {x_{j}, τ_{j}}, and consider τ_{j} as a non-zero tangent vector from T_{x_{j}}D, thus identifying T_{x_{j}}D with ℂ using the local coordinate z − x_{j}. So, τ_{j} is the image under ϕ_{j} of the unit tangent vector at 0. We have an obvious map p(J) : D̃(J) ⟶ D(J) which is a homotopy equivalence.

If ρ : K ⟶ J is an epimorphic map of finite sets, the composition defines a holomorphic map

m(ρ) : ∏_{J} D̃(K_{j}) × D̃(J) ⟶ D̃(K)

where K_{j} := ρ^{−1}(j). If L ⟶^{σ} K ⟶^{ρ} J are two epimorphisms of finite sets, the square

∏_{K} D̃(L_{k}) × ∏_{J} D̃(K_{j}) × D̃(J) ⟶^{m(σ)} ∏_{K} D̃(L_{k}) × D̃(K)
∏ m(σ_{j}) ↓  ↓ m(σ)
∏_{J} D̃(L_{j}) × D̃(J)  ⟶^{m(ρσ)}  D̃(L)

commutes. Here σ_{j} : L_{j} ⟶ K_{j} are induced by σ.

Let ∗ denote the one element set. The space D̃(∗) has a marked point, also to be denoted by ∗, corresponding to the identity map ϕ : D ⟶ D.

If ρ : J′ ⟶^{∼} J is an isomorphism, it induces in the obvious way an isomorphism ρ^{∗} : D̃(J) ⟶^{∼} D̃(J′) (resp. D(J) ⟶^{∼} D(J′)). The first map coincides with m(ρ) restricted to (∏_{J} ∗) × D̃(J). In particular, for each J, the group Σ_{J} of automorphisms of the set J, acts on the spaces D̃(J), D(J).

The map m(J ⟶ ∗) restricted to ∗ × D̃(J) is the identity of D̃(J).

We will denote the collection of the spaces and maps {D̃(J), m(ρ)} by D, and call it the operad of disks with tangent vectors.

7.3. Coloured local systems over D. If ρ : K ⟶ J is an epimorphism of finite sets and π : K ⟶ X is a map of sets, we define the map ρ_{∗}π : J ⟶ X by ρ_{∗}π(j) = ∑_{K_{j}} π(k). For j ∈ J, we denote K_{j} := ρ^{−1}(j) as above, and π_{j} : K_{j} ⟶ X will denote the restriction of π.

Let us call an X-coloured local system J over D a collection of local systems J (π) over the spaces D̃(J) given for every map π : J ⟶ X, J being a non-empty finite set, together with factorization isomorphisms

ϕ(ρ) : m(ρ)^{∗}J (π) ⟶^{∼} × _{J}J (π_{j}) ⊠ J (ρ_{∗}π)

given for every epimorphism ρ : K ⟶ J and π : K ⟶ X, which satisfy the properties (a), (b) below.

(a) Associativity. Given a map π : L ⟶ X and a pair of epimorphisms L ⟶^{σ} K ⟶^{ρ} J, the square below commutes.

m(ρ)^{∗}m(σ)^{∗}J (π)  ⟶^{ϕ(σ)}  × _{K}J (π_{k}) ⊠ m(ρ)^{∗}J (σ_{∗}π)
ϕ(ρσ) ↓  ↓ ϕ(ρ)
× _{J}m(σ_{j})^{∗}J (π_{j}) ⊠ J (ρ_{∗}σ_{∗}π) ⟶^{⊠ϕ(σ_{j})} × _{K}J (π_{k}) ⊠ × _{J}J ((σ_{∗}π)_{j}) ⊠ J (ρ_{∗}σ_{∗}π)

Note that π_{j∗}σ_{j} = (σ_{∗}π)_{j}.

For μ ∈ X, let π_{μ} : ∗ ⟶ X be defined by π_{μ}(∗) = μ. The isomorphisms ϕ(id_{∗}) restricted to the marked points in D(∗), give the isomorphisms J (π_{μ})_{∗} ⟶^{∼} k (and imply that the local systems J (π_{μ}) are one-dimensional).

(b) For any π : J ⟶ X, the map ϕ(id_{J}) restricted to (∏_{J} ∗) × D̃(J), equals id_{J (π)}.

The map ϕ(J ⟶ ∗) restricted to ∗ × D̃(J), equals id_{J (π)}.

7.4. The definition above implies that the local systems J (π) are functorial with respect to isomorphisms. In particular, the action of the group Σ_{π} on D̃(J) lifts to J (π).

7.5. Standard local system over D. Let us define the ”standard” local systems J (π) by a version of the construction 3.10.

We will use the notations of loc. cit. for the marked points and paths. We can identify D(J) = D^{J∘} × (ℂ^{∗})^{J} where we have identified all tangent spaces T_{x}D (x ∈ D) with ℂ using the local coordinate z − x.

We set J (π)_{x(σ)} = k, the monodromies being T_{γ(j′,j′′)^{±}} = ζ^{∓π(j′)·π(j′′)}, and the monodromy T_{j} corresponding to the counterclockwise circle of a tangent vector τ_{j} is ζ^{−2nπ(j)}. The factorization isomorphisms are defined by the same condition as in loc. cit.

This defines the standard local system J over D. Below, the notation J will be reserved for this local system.

7.6. Set P = ℙ^{1}(ℂ). We pick a point ∞ ∈ P and choose a global coordinate z : 𝔸^{1}(ℂ) = P − {∞} ⟶^{∼} ℂ. This gives local coordinates: z − x at x ∈ 𝔸^{1}(ℂ) and 1/z at ∞.

For a non-empty finite set J, let P (J) denote the space of J-tuples {x_{j}, τ_{j}} where x_{j} are distinct points on P, and τ_{j} is a non-zero tangent vector at x_{j}. Let P̃(J) denote the space whose points are J-tuples of holomorphic embeddings ϕ_{j} : D ⟶ P with non-intersecting images such that each ϕ_{j} is a restriction of an algebraic morphism P ⟶ P. We have the 1-jet projections P̃(J) ⟶ P (J). We will use the notation P (n) for P ({1, …, n}), etc.

An epimorphism ρ : K ⟶ J induces the maps

m_{P} (ρ) : ∏_{J} D̃(K_{j}) × P̃(J) ⟶ P̃(K)

and

m̄_{P} (ρ) : ∏_{J} D(K_{j}) × P̃(J) ⟶ P (K).

For a pair of epimorphisms L ⟶^{σ} K ⟶^{ρ} J, the square

∏_{K} D̃(L_{k}) × ∏_{J} D̃(K_{j}) × P̃(J) ⟶^{m_{P} (σ)} ∏_{K} D̃(L_{k}) × P̃(K)
∏ m(σ_{j}) ↓  ↓ m_{P} (σ)
∏_{J} D̃(L_{j}) × P̃(J)  ⟶^{m_{P} (ρσ)}  P̃(L)

commutes.

If ρ : J′ ⟶^{∼} J is an isomorphism, it induces in the obvious way an isomorphism ρ^{∗} : P̃(J) ⟶^{∼} P̃(J′) (resp. P (J) ⟶^{∼} P (J′)). This last map coincides with m_{P} (ρ) (resp. m̄_{P} (ρ)) restricted to (∏_{J} ∗) × P̃(J) (resp. (∏_{J} ∗) × P (J)).

The collection of the spaces and maps {P̃(J), m_{P} (ρ)} form an object P̃ called a right module over the operad D.

7.7. Fix an element μ ∈ X. Let us say that a map π : J ⟶ X has level μ if ∑_{J} π(j) = μ.

A cohesive local system H of level μ over P is a collection of local systems H(π) over P̃(J) given for every π : J ⟶ X of level μ, together with factorization isomorphisms

ϕ_{P} (ρ) : m_{P} (ρ)^{∗}H(π) ⟶^{∼} × _{J}J (π_{j}) ⊠ H(ρ_{∗}π)

given for every epimorphism ρ : K ⟶ J and π : K ⟶ X of level μ, which satisfy the properties (a), (b) below.

(a) Associativity. Given a map π : L ⟶ X and a pair of epimorphisms L ⟶^{σ} K ⟶^{ρ} J, the square below commutes.

m_{P} (ρ)^{∗}m_{P} (σ)^{∗}H(π)  ⟶^{ϕ_{P} (σ)}  × _{K}J (π_{k}) ⊠ m_{P} (ρ)^{∗}H(σ_{∗}π)
ϕ_{P} (ρσ) ↓  ↓ ϕ_{P} (ρ)
× _{J}m(σ_{j})^{∗}J (π_{j}) ⊠ H(ρ_{∗}σ_{∗}π) ⟶^{⊠ϕ(σ_{j})} × _{K}J (π_{k}) ⊠ × _{J}J ((σ_{∗}π)_{j}) ⊠ H(ρ_{∗}σ_{∗}π)

(b) For any π : J ⟶ X, the map ϕ_{P} (id_{J}) restricted to (∏_{J} ∗) × P̃(J), equals id_{H(π)}.

7.8. The definition above implies that the local systems H(π) are functorial with respect to isomorphisms. In particular, the action of the group Σ_{π} on P̃(J), lifts to H(π).

7.9. Theorem. For each μ ∈ X such that μ ≡ 2ρ_{ℓ} modY_{ℓ}, there exists a unique up to an isomorphism one-dimensional cohesive local system H = H^{(μ)} of level μ over P. □

The element ρ_{ℓ} ∈ X is defined in 2.2 and the lattice Y_{ℓ} in 2.3.

¿From now on, let us fix such a local system H^{(μ)} for each μ as in the theorem.

7.10. Note that the obvious maps p(J) : P̃(J) ⟶ P (J) are homotopy equivalences. Therefore the local systems H(π) (π : J ⟶ X) descend to the unique local systems over P (J), to be denoted by the same letter.

8. Gluing

8.1. Let us fix a finite set K. For ν ∈ Y^{+}, pick an unfolding of ν, π : J ⟶ I. Consider the space P^{ν}; its points are formal linear combinations ∑_{J} π(j)x_{j} (x_{j} ∈ P). We define the space P^{ν}(K) = P (K) × P^{ν}; its points are tuples {y_{k}, τ_{k}, ∑ π(j)x_{j}} (k ∈ K, y_{k} ∈ P, τ_{k} ≠ 0 in T_{y_{k}}P), all y_{k} being distinct. Let P^{ν}(K)^{∘} (resp. P^{ν}(K)^{•}) be the subspace whose points are tuples as above with all x_{j} distinct from y_{k} and pairwise distinct (resp. all x_{j} distinct from y_{k}). We will use the notation P^{ν}(n) for P^{ν}({1, …, n}), etc.

Let ν′ ∈ Y^{+} and ν⃗ = {ν_{k}} ∈ (Y^{+})^{K} be such that ∑_{K} ν_{k} + ν′ = ν. Define the space P^{ν⃗;ν′} ⊂ P̃(K) × P^{ν} consisting of tuples {ϕ_{k}, ∑ π(j)x_{j}} such that for each k, ν_{k} of the points x_{j} lie inside ϕ_{k}(D) and ν′ of them lie outside all closures of these disks.

Let 0 ∈ (Y^{+})^{K} be the zero K-tuple. Define the space P̃^{ν}(K) := P^{0;ν}. We have obvious maps

(a) ∏_{K} D^{ν_{k}} × P̃^{ν′}(K) ⟵^{p(ν⃗;ν′)} P^{ν⃗;ν′} ⟶^{m(ν⃗;ν′)} P^{ν}(K).

Let ν^{1}, ν^{2} ∈ (Y^{+})^{K} be such that ν^{1} + ν^{2} = ν⃗. Define the space

P^{ν^{1};ν^{2};ν} ⊂ P̃(K) × ℝ_{>0}^{K} × P̃(K) × P^{ν} consisting of all tuples {ϕ_{k}; r_{k}; ϕ′_{k}; ∑ π(j)x_{j}} such that r_{k} < 1; ϕ_{k}(z) = ϕ′_{k}(r_{k}z), and ν_{k}^{1} (resp. ν_{k}^{2}, ν′) from the points x_{j} lie inside ϕ_{k}(D) (resp. inside the annulus ϕ′_{k}(D) − \overline{ϕ_{k}(D)}, inside P − ⋃ \overline{ϕ′_{k}(D)}). We set P̃^{ν⃗;ν′} := P^{0;ν⃗;ν′}, cf. 5.2.

We have a commutative romb (cf. 3.5):

(b)

[figure: P^{ν}(K); P^{ν^{1};ν^{2}+ν′}; P^{ν^{1}+ν^{2};ν′}; ∏ D^{ν_{k}^{1}} × P̃^{ν^{2}+ν′}(K); P^{ν^{1};ν^{2};ν′}; ∏ D^{ν_{k}^{1}+ν_{k}^{2}} × P̃^{ν′}(K); ∏ D^{ν_{k}^{1}} × P̃^{ν^{2};ν′}; ∏ D^{ν_{k}^{1},ν_{k}^{2}} × P̃^{ν′}(K); ∏ D^{ν_{k}^{1}} × ∏ A^{ν_{k}^{2}} × P̃^{ν′}(K); m; m; p; m; m; p; m; p; p; m; p; p]

Here ν^{2} := ∑ ν_{k}^{2}.

8.2. Let P (K; J) be the space consisting of tuples {y_{k}; τ_{k}; x_{j}} (k ∈ K, j ∈ J, y_{k}, x_{j} ∈ P, τ_{k} ≠ 0 in T_{y_{k}}P), where all points x_{k}, y_{j} are distinct. We have P^{ν}(K)^{∘} = P (K; J)/Σ_{π}. We have an obvious projection P (K ∐ J) ⟶ P (K; J).

8.3. Let {M_{k}} be a K-tuple of factorizable sheaves supported at some cosets in X/Y; let μ_{k} = λ(M_{k}).

Let π̃ : K ∐ J ⟶ X be a map defined by π̃(k) = μ_{k}, π̃(j) = −π(j) ∈ I ↪ X. The local system H(π̃) over P (K ∐ J) descends to P (K; J) since ζ^{2n(−i)} = 1 for all i ∈ I; this one in turn descends to the unique local system H̃ν ; μ⃗ over P^{ν}(K)^{∘}, due to Σ_{π}-equivariance. Let us define the local system Hν ; μ⃗ := H̃ν ; μ⃗ ⊗ Sign^{ν}. Here Sign^{ν} denotes the inverse image of the sign local system on P^{ν∘} (defined in the same manner as for the disk, cf. 3.9) under the forgetful map P^{ν}(K)^{∘} ⟶ P^{ν∘}.

Let Hν• ; μ⃗ be the perverse sheaf over P^{ν}(K)^{•} which is the middle extension of Hν ; μ⃗[dim P^{ν}(K)]. Let us denote by the same letter the inverse image of this perverse sheaf on the space P̃^{ν}(K) with respect to the evident projection P̃^{ν}(K) ⟶ P^{ν}(K)^{•}.

8.4. Let us call an element ν ∈ Y^{+} admissible (for a K-tuple {μ_{k}}) if ∑ μ_{k} − ν ≡ 2ρ_{ℓ} modY_{ℓ}, see 2.3.

8.5. Theorem - definition. For each admissible ν, there exists a unique, up to a unique isomorphism, perverse sheaf, denoted by × _{K}^{(ν)} M_{k}, over P^{ν}(K), equipped with isomorphisms

ψ(ν⃗; ν′) : m^{∗} × _{K}^{(ν)} M_{k} ⟶^{∼} p^{∗}( × _{K} M_{k}^{ν_{k}} ⊠ H_{μ⃗−ν⃗}^{ν′•})

given for every diagram 8.1 (a) such that for each rhomb 8.1 (b) the cocycle condition

ϕ(ν^{2}; ν′) ∘ ψ(ν^{1}; ν^{2} + ν′) = ( × _{K} ψ^{M_{k}}(ν_{k}^{1}, ν_{k}^{2})) ∘ ψ(ν^{1} + ν^{2}; ν′)

holds. □

8.6. The sheaf × _{K}^{(ν)} M_{k} defines for each K-tuple of y⃗ = {y_{k}, τ_{k}} of points of P with

non-zero tangent vectors, the sheaf × (ν) ; y⃗ M_{k} over P^{ν}, to be called the gluing of the factorizable sheaves M_{k} into the points (y_{k}, τ_{k}).

8.7. Example. The sheaf × _{K}^{(ν)} L(μ_{k}) is equal to the middle extension of the sheaf Hν• ; μ⃗.

9. Semiinfinite cohomology

In this section we review the theory of semiinfinite cohomology in the category C, due to S. M. Arkhipov, cf. [Ark].

9.1. Let C_{r} be a category whose objects are right 𝔲-modules N, finite dimensional over k, with a given X-grading N = ⊕_{λ∈X} N_{λ} such that xK_{ν} = ζ^{−⟨ν,λ⟩}x for any ν ∈ Y, λ ∈ X, x ∈ N_{λ}. All definitions and results concerning the category C given above and below, have the obvious versions for the category C_{r}.

For M ∈ C, define M^{∨} ∈ C_{r} as follows: (M^{∨})_{λ} = (M_{−λ})^{∗} (the dual vector space); the action of the operators θ_{i}, ϵ_{i} being the transpose of their action on M. This way we get an equivalence ^{∨} : C^{opp} ⟶^{∼} C_{r}.

9.2. Let us call an object M ∈ C 𝔲^{−}- (resp. 𝔲^{+}-) good if it admits a filtration whose successive quotients have the form ind_{𝔲^{≥0}}^{𝔲}(M′) (resp. ind_{𝔲^{≤0}}^{𝔲}(M′′)) for some M′ ∈ C^{≥0} (resp. M′′ ∈ C^{≤0}) (cf. 2.16). These classes of objects are stable with respect to the tensor multiplication by an arbitrary object of C.

If M is 𝔲^{−}- (resp. 𝔲^{+}-) good then M^{∗} is 𝔲^{−}- (resp. 𝔲^{+}-) good.

If M is 𝔲^{−}-good and M′ is 𝔲^{+}-good then M ⊗ M′ is a projective object in C.

9.3. Let us say that a complex M^{•} in C is concave (resp. convex) if

(a) there exists μ ∈ X such that all nonzero components M_{λ}^{•} have the weight λ ≥ μ (resp. λ ≤ μ);

(b) for any λ ∈ X, the complex M_{λ}^{•} is finite.

9.4. For an object M ∈ C, we will call a left (resp. right) 𝔲^{±}-good resolution of M an exact complex

… ⟶ P^{−1} ⟶ P^{0} ⟶ M ⟶ 0

(resp.

0 ⟶ M ⟶ R^{0} ⟶ R^{1} ⟶ …)

such that all P^{i} (resp. R^{i}) are 𝔲^{±}-good.

9.5. Lemma. Each object M ∈ C admits a convex 𝔲^{−}-good left resolution, a concave 𝔲^{+}-good left resolution, a concave 𝔲^{−}-good right resolution and a convex 𝔲^{+}-good right resolution. □

9.6. For N ∈ C_{r}, M ∈ C, define a vector space N ⊗_{C} M as the zero weight component of the tensor product N ⊗_{𝔲} M (which has an obvious X-grading).

For M, M′ ∈ C, we have an obvious perfect pairing

(a) Hom_{C}(M, M′) ⊗ (M^{′∨} ⊗_{C} M) ⟶ k.

9.7. M, M′ ∈ C, N ∈ C_{r} and n ∈ ℤ, define the semiinifinite Ext and Tor spaces

Ext_{C}^{\frac{∞}{2} +n}(M, M′) = H^{n}(Hom_{C}(R_{↘}^{•}(M), R_{↗}^{•}(M′)))

where R_{↘}^{•}(M) (resp. R_{↗}^{•}(M′)) is an arbitrary 𝔲^{+}-good convex right resolution of M (resp. 𝔲^{−}-good concave right resolution of M′),

Tor_{\frac{∞}{2} +n}^{C}(N, M) = H^{−n}(P_{↙}^{•}(N) ⊗_{C} R_{↘}^{•}(M))

where P_{↙}^{•}(N) is an arbitrary 𝔲^{−}-good convex left resolution of N.

This definition does not depend, up to a unique isomorphism, upon the choice of resolutions, and is functorial.

These spaces are finite dimensional and are non-zero only for finite number of degrees n.

The pairing 9.6 (a) induces perfect pairings

Ext_{C}^{\frac{∞}{2} +n}(M, M′) ⊗ Tor_{\frac{∞}{2} +n}^{C}(M^{′∨}, M) ⟶ k (n ∈ ℤ).

10. Conformal blocks (genus 0)

In this section we suppose that k = ℂ.

10.1. Let M ∈ C. We have a canonical embedding of vector spaces

Hom_{C}(1, M) ↪ M

which identifies Hom_{C}(1, M) with the maximal trivial subobject of M. Here ”trivial” means ”isomorphic to a sum of a few copies of the object 1”. Dually, we have a canonical epimorhism

M ⟶ Hom_{C}(M, 1)^{∗}

which identifies Hom_{C}(M, 1)^{∗} with the maximal trivial quotient of M. Let us denote by ⟨M⟩ the image of the composition

Hom_{C}(1, M) ⟶ M ⟶ Hom_{C}(M, 1)^{∗}

Thus, ⟨M⟩ is canonically a subquotient of M.

One sees easily that if N ⊂ M is a trivial direct summand of M which is maximal, i.e. not contained in greater direct summand, then we have a canonical isomorphism ⟨M⟩ ⟶^{∼} N. For this reason, we will call ⟨M⟩ the maximal trivial direct summand of M.

10.2. Let γ_{0} ∈ Y denote the highest coroot and β_{0} ∈ Y denote the coroot dual to the highest root (γ_{0} = β_{0} for a simply laced root datum).

Let us define the first alcove Δ_{ℓ} ⊂ X by

Δ_{ℓ} = {λ ∈ X| ⟨i, λ + ρ⟩ > 0 for all i ∈ I; ⟨γ_{0}, λ + ρ⟩ < ℓ}

if d does not divide ℓ, i.e. if ℓ_{i} = ℓ for all i ∈ I, and by

Δ_{ℓ} = {λ ∈ X| ⟨i, λ + ρ⟩ > 0 for all i ∈ I; ⟨β_{0}, λ + ρ⟩ < ℓ_{β_{0}}}

otherwise, cf. [AP] 3.19 (d is defined in 2.1, and ℓ_{β_{0}} in 2.2). Note that ℓ_{β_{0}} = ℓ/d.

10.3. For λ_{1}, …, λ_{n} ∈ Δ_{ℓ}, define the space of conformal blocks by

⟨L(λ_{1}), …, L(λ_{n})⟩ := ⟨L(λ_{1}) ⊗ … ⊗ L(λ_{n})⟩.

In fact, due to the ribbon structure on C, the right hand side is a local system over the space P (n) := P ({1, …, n}) (cf. [D1]). It is more appropriate to consider the previous equality as the definition of the local system of conformal blocks over P (n).

10.4. Theorem. (Arkhipov) For each λ_{1}, …, λ_{n} ∈ Δ_{ℓ}, the space of conformal blocks ⟨L(λ_{1}), …, L(λ_{n})⟩ is naturally a subquotient of the space

Tor_{\frac{∞}{2} +0}^{C}(1_{r}, L(λ_{1}) ⊗ … ⊗ L(λ_{n}) ⊗ L(2ρ_{ℓ})).

More precisely, due to the ribbon structure on C, the latter space is a stalk of a local system over P (n + 1), and inverse image of the local system ⟨L(λ_{1}), …, L(λ_{n})⟩ under the projection onto the first coordinates P (n + 1) ⟶ P (n), is a natural subquotient of this local system.

Here 1_{r} is the unit object in C_{r}.

Examples, also due to Arkhipov, show that the local systems of conformal blocks are in general proper subquotients of the corresponding Tor local systems.

This theorem is an immediate consequence of the next lemma, which in turn follows from the geometric theorem 11.2 below, cf. 11.3.

10.5. Lemma. We have Tor_{\frac{∞}{2} +n}^{C}(1_{r}, L(2ρ_{ℓ})) = k if n = 0, and 0 otherwise. □

10.6. Let 𝔤 be the simple Lie algebra (over k) associated with our Cartan datum; let 𝔤̂ be the corresponding affine Lie algebra.

Let MS denote the category of integrable 𝔤̂-modules of central charge κ − h. Here κ is a fixed positive integer, h is the dual Coxeter number of our Cartan datum. MS is a semisimple abelian category whose irreducible objects are 𝔏(λ), λ ∈ Δ_{ℓ} where l = 2dκ, i.e. ℓ = dκ (we are grateful to Shurik Kirillov who pointed out the necessity of the factor d here) and 𝔏(λ) is the highest weight module with a highest vector v whose ”top” part 𝔤 · v is the irreducible 𝔤-module of the highest weight λ.

According to Conformal field theory, MS has a natural structure of a ribbon category, cf. [MS], [K].

The usual local systems of conformal blocks in the WZW model may be defined as

⟨𝔏(λ_{1}), …, 𝔏(λ_{𝔫})⟩ = Hom_{MS}(1, 𝔏(λ_{1}) ⊗ … ⊗ 𝔏(λ_{𝔫}))

the structure of a local system on the right hand side is due to the ribbon structure on MS.

10.7. Let ζ = exp(π\sqrt{−1}/dκ). We have an exact functor

ϕ : MS ⟶ C

sending 𝔏(λ) to L(λ). This functor identifies MS with a full subcategory of C.

The functor ϕ does not respect the tensor structures. It admits the left and right adjoints, ϕ^{♭}, ϕ^{♯}. For M ∈ C, let ⟨M⟩_{MS} denotes the image of the composition

ϕ ∘ ϕ^{♭}(M) ⟶ M ⟶ ϕ ∘ ϕ^{♯}(M).

We have the following comparison theorem.

10.8. Theorem. We have naturally

ϕ(𝔐 ⊗ 𝔐′) = ⟨ϕ(𝔐) ⊗ ϕ(𝔐′)⟩_{MS}.

This follows from the combination of the results of [AP], [KL], [L3] and [F2].

10.9. Corollary. For any λ_{1}, …, λ_{n} ∈ Δ_{ℓ}, the functor ϕ induces an isomorphism of local systems

⟨𝔏(λ_{1}), …, 𝔏(λ_{𝔫})⟩ = ⟨𝔏(λ_{1}), …, 𝔏(λ_{𝔫})⟩. □

11. Integration

We keep the notations of the previous section.

11.1. Let K be a finite set, m = card(K), {M_{k}} a K-tuple of finite factorizable sheaves, M_{k} ∈ FS_{c_{k}}, μ_{k} := λ(M_{k}). Assume that ν := ∑_{K} μ_{k} − 2ρ_{ℓ} belongs to Y^{+}.

Let η : P^{ν}(K) ⟶ P (K) be the projection.

11.2. Theorem. We have canonical isomorphisms of local systems over P (K)

R^{a−2m}η_{∗}( × _{K}^{(ν)} M_{k}) = Tor_{\frac{∞}{2} −a}^{C}(1_{r}, ⊗_{K} Φ(M_{k})) (a ∈ ℤ),

the structure of a local system on the right hand side being induced by the ribbon structure on C. □

11.3. Proof of Lemma 10.5. Apply the previous theorem to the case when the K-tuple consists of one sheaf L(2ρ_{ℓ}) and ν = 0. □

11.4. From now until the end of the section, k = ℂ and ζ = exp(π\sqrt{−1}/dκ). Let λ_{1}, …, λ_{n} ∈ Δ_{ℓ}. Let ν = ∑_{m=1}^{n} λ_{m}; assume that ν ∈ Y^{+}. Set μ⃗ = {λ_{1}, …, λ_{n}, 2ρ_{ℓ}}.

Let η be the projection P^{ν}(n+1) := P^{ν}({1, …, n+1}) ⟶ P (n+1) and p : P (n+1) ⟶ P (n) be the projection on the first coordinates.

Let Hν♯ ; μ⃗ denote the middle extension of the sheaf Hν• ; μ⃗. By the Example 8.7,

Hν♯ ; μ⃗ = × _{1≤a≤n}^{(ν)}L(λ_{a}) ⊠ L(2ρ_{ℓ}).

11.5. Theorem. The local system p^{∗}⟨L(λ_{1}), …, L(λ_{n})⟩ is canonically a subquotient of the local system

R^{−2n−2}η_{∗}Hν♯ ; μ⃗.

This theorem is an immediate corollary of the previous one and of the Theorem 10.4.

11.6. Corollary. In the notations of the previous theorem, the local system ⟨L(λ_{1}), …, L(λ_{n})⟩ is semisimple.

Proof. The local system p^{∗}⟨L(λ_{1}), …, L(λ_{n})⟩ is a subquotient of the geometric local system R^{−2n−2}η_{∗}Hν♯ ; μ⃗, and hence is semisimple by the Beilinson-Bernstein-Deligne-Gabber Decomposition theorem, [BBD], Théorème 6.2.5. Therefore, the local system ⟨L(λ_{1}), …, L(λ_{n})⟩ is also semisimple, since the map p induces the surjection on the fundamental groups. □

11.7. For a sheaf F, let F̄ denote the sheaf obtained from F by the complex conjugation on the coefficients.

If a perverse sheaf F on P^{ν} is obtained by gluing some irreducible factorizable sheaves into some points of P then its Verdier dual DF is canonically isomorphic to F̄. Therefore, the Poincaré-Verdier duality induces a perfect hermitian pairing on RΓ(P^{ν}; F).

Therefore, in notations of theorem 11.5, The Poincaré-Verdier duality induces a nondegenerate hermitian form on the local system R^{−2n−2}η_{∗}Hν♯ ; μ⃗.

By a little more elaborated argument using fusion, one can introduce a canonical hermitian form on the systems of conformal blocks.

Compare [K], where a certain hermitian form on the spaces of conformal blocks (defined up to a positive constant) has been introduced.

11.8. By the similar reasons, the Verdier duality defines a hermitian form on all irreducible objects of C (since the Verdier duality commutes with Φ, cf. Theorem 6.3).

12. Regular representation

12.1. From now on we are going to modify slightly the definition of the categories C and FS. Let X_{ℓ} be the lattice

X_{ℓ} = {μ ∈ X ⊗ ℚ| μ · 𝕐_{ℓ} ∈ ℓℤ}

We have obviously X ⊂ X_{ℓ}, and X = X_{ℓ} if d|ℓ.

In this Chapter we will denote by C a category of X_{ℓ}-graded (instead of X-graded) finite dimensional vector spaces M = ⊕_{λ∈X_{ℓ}}M_{λ} equipped with linear operators θ_{i} : M_{λ} ⟶ M_{λ−i′}, ϵ_{i} : M_{λ} ⟶ M_{λ+i′} which satisfy the relations 2.11 (a), (b). This makes sense since ⟨d_{i}i, λ⟩ = i′ · λ ∈ ℤ for each i ∈ I, λ ∈ X_{ℓ}.

Also, in the definition of FS we replace X by X_{ℓ}. All the results of the previous Chapters hold true verbatim with this modification.

We set d_{ℓ} = card(X_{ℓ}/Y_{ℓ}); this number is equal to the determinant of the form μ_{1}, μ_{2} ↦ \frac{1}{ℓ}μ_{1} · μ_{2} on Y_{ℓ}.

12.2. Let 𝔲̃ ⊂ 𝔲 be the k-subalgebra generated by K̃_{i}, ϵ_{i}, θ_{i} (i ∈ I). Following the method of [L1] 23.1, define a new algebra 𝔲^{•} (without unit) as follows. If μ′, μ′′ ∈ X_{ℓ}, we set

μ′ 𝔲̃_{μ′′} = 𝔲̃/(∑_{i∈I}(K̃_{i} − ζ^{i·μ′})𝔲̃ + ∑_{i∈I} 𝔲̃(K̃_{i} − ζ^{i·μ′′})); 𝔲^{•} = ⊕_{μ′,μ′′∈X_{ℓ}}(_{μ′} 𝔲̃_{μ′′}).

Let π_{μ′,μ′′} : 𝔲̃ ⟶ _{μ′} 𝔲̃_{μ′′} be the canonical projection. We set 1_{μ} = π_{μ,μ}(1) ∈ 𝔲^{•}. The structure of an algebra on 𝔲^{•} is defined as in loc. cit. As in loc. cit., the category C may be identified with the category of finite dimensional (over k) (left) 𝔲^{•}-modules M which are unital, i.e.

(a) for every x ∈ M, ∑_{μ∈X_{ℓ}} 1_{μ}x = x. If M is such a module, the X_{ℓ}-grading on M is defined by M_{μ} = 1_{μ}M. Let 𝔲′ denote the quotient algebra of the algebra 𝔲̃ by the relations K̃_{i}^{l_{i}} = 1 (i ∈ I). Here l_{i} := \frac{l}{(l,d_{i})}. We have an isomorphism of vector spaces 𝔲^{•} = 𝔲′ ⊗ k[Y_{ℓ}], cf. 12.5 below.

12.3. Let a : C ⟶^{∼} C_{r} be an equivalence defined by aM = M (M ∈ C) as an X_{ℓ}-vector space, mx = A(x)m (x ∈ 𝔲, m ∈ M). Here A : 𝔲 ⟶ 𝔲^{opp} is the antipode. We will use the same notation a for a similar equivalence C_{r} ⟶^{∼} C. Let us consider the category C ⊗C (resp. C ⊗C_{r}) which may be identified with the category of finite dimensional 𝔲^{•} ⊗ 𝔲^{•}- (resp. 𝔲^{•} ⊗ (𝔲^{•})^{opp}-) modules satisfying a ”unitality” condition similar to (a) above. Let us consider the algebra 𝔲^{•} itself as a regular 𝔲^{•} ⊗ (𝔲^{•})^{opp}-module. It is infinite dimensional, but is a union of finite dimensional modules, hence it may be considered as an object of the category Ind(C ⊗ C_{r}) = IndC ⊗ IndC_{r} where Ind denotes the category of Ind-objects, cf. [D4] §4. Let us denote by R the image of this object under the equivalence Id ⊗ a : IndC ⊗ IndC_{r} ⟶^{∼} IndC ⊗ IndC. Every object O ∈ C ⊗ C induces a functor F_{O} : C ⟶ C defined by

F_{O}(M) = a(aM ⊗_{C} O).

The same formula defines a functor F_{O} : IndC ⟶ IndC for O ∈ Ind(C ⊗ C). We have FR = Id_{IndC}. We can consider a version of the above formalism using semiinfinite Tor’s. An object O ∈ Ind(C ⊗ C) defines functors F_{O; \frac{∞}{2} +n} : IndC ⟶ IndC (n ∈ ℤ) defined by

F_{O; \frac{∞}{2} +n}(M) = aTor_{\frac{∞}{2} +n}^{C}(aM, O).

12.4. Theorem. (i) We have F_{R; \frac{∞}{2} +n} = Id_{IndC} if n = 0, and 0 otherwise.

(ii) Conversely, suppose we have an object Q ∈ Ind(C⊗C) together with an isomorphism of functors ϕ : F_{R; \frac{∞}{2} +•} ⟶^{∼} F_{Q; \frac{∞}{2} +•}. Then ϕ is induced by the unique isomorphism R ⟶^{∼} Q. □

12.5. Adjoint representation. For μ ∈ Y_{ℓ}, let T (μ) be a one-dimensional 𝔲^{•} ⊗ (𝔲^{•})^{opp}- module equal to L(μ) (resp. to aL(−μ)) as a 𝔲^{•}- (resp. (𝔲^{•})^{opp}-) module. Let us consider the module T_{μ}R = R⊗T (μ) ∈ Ind(C ⊗C). This object represents the same functor Id_{IndC}, hence we have a canonical isomorphism t_{μ} : R ⟶^{∼} T_{μ}R.

Let us denote by âd ∈ IndC the image of R under the tensor product ⊗ : Ind(C ⊗ C) ⟶ IndC. The isomorphisms t_{μ} above induce an action of the lattice Y_{ℓ} on âd. Set ad = âd/Y_{ℓ}. This is an object of C ⊂ IndC which is equal to the algebra 𝔲′ considered as a 𝔲^{•}-module by means of the adjoint action.

In the notations of 10.7, let us consider an object

ad_{MS} := ⊕_{μ∈Δ_{ℓ}}⟨L(μ) ⊗ L(μ)^{∗}⟩_{MS} ∈ MS,

cf. [BFM] 4.5.3.

12.6. Theorem. We have a canonical isomorphism ⟨ad⟩_{MS} = ad_{MS}. □

13. Regular sheaf

13.1. Degeneration of quadrics. The construction below is taken from [KL]II 15.2. Let us consider the quadric Q ⊂ ℙ^{1} × ℙ^{1} × 𝔸^{1} given by the equation uv = t where (u, v, t) are coordinates in the triple product. Let f : Q ⟶ 𝔸^{1} be the projection to the third coordinate; for t ∈ 𝔸^{1} denote Q_{t} := f^{−1}(t). For t ≠ 0, Q_{t} is isomorphic to ℙ^{1}; the fiber Q_{0} is a union of two projective lines clutched at a point: Q_{0} = Q_{u} ∪ Q_{v} where Q_{u} (resp. Q_{v}) is an irreducible component given (in Q_{0}) by the equation v = 0 (resp. u = 0) and is isomorphic to ℙ^{1}; their intersection being a point. We set ′Q = f^{−1}(𝔸^{1} − {0}).

We have two sections x_{1}, x_{2} : 𝔸^{1} ⟶ Q given by x_{1}(t) = (∞, 0, t), x_{2}(t) = (0, ∞, t). Consider two ”coordinate charts” at these points: the maps ϕ_{1}, ϕ_{2} : ℙ^{1} × 𝔸^{1} ⟶ Q given by

ϕ_{1}(z, t) = (\frac{tz}{z − 1}, \frac{z − 1}{z}, t); ϕ_{2}(z, t) = (\frac{z − 1}{z}, \frac{tz}{z − 1}, t).

This defines a map

(a) ϕ : 𝔸^{1} − {0} ⟶ P̃(2),

in the notations of 7.6.

13.2. For ν ∈ Y^{+}, let us consider the corresponding (relative over 𝔸^{1}) configuration scheme f^{ν} : Q_{/𝔸^{1}}^{ν} ⟶ 𝔸^{1}. For the brevity we will omit the subscript _{/𝔸^{1}} indicating that we are dealing with the relative version of configuration spaces. We denote by Q^{ν•} (resp. Q^{ν∘}) the subspace of configurations with the points distinct from x_{1}, x_{2} (resp. also pairwise distinct). We set ′Q^{ν∘} = Q^{ν∘}|_{𝔸^{1}−{0}}, etc.

The map ϕ above, composed with the canonical projection P̃(2) ⟶ P (2), induces the maps

′Q^{ν∘} ⟶ P^{ν}(2) (ν ∈ Y^{+})

(in the notations of 8.1). For ν ∈ Y^{+} and μ_{1}, μ_{2} ∈ X_{ℓ} such that μ_{1} + μ_{2} − ν = 2ρ_{ℓ}, let I_{μ_{1},μ_{2}}^{ν} denote the local system over ′Q^{ν∘} which is the inverse image of the local system H_{μ_{1},μ_{2}}^{ν} over P^{ν}(2)^{∘}. Let I_{μ_{1},μ_{2}}^{ν•} denote the perverse sheaf over ′Q^{ν•}(ℂ) which is the middle extension of I_{μ_{1},μ_{2}}^{ν}[dim Q^{ν}].

13.3. Let us take the nearby cycles and get a perverse sheaf Ψ_{f^{ν}}(I_{μ_{1},μ_{2}}^{ν•}) over Q_{0}^{ν•}(ℂ). Let us consider the space Q_{0}^{ν•} more attentively. This is a reducible scheme which is a union

Q_{0}^{ν•} = ⋃_{ν_{1}+ν_{2}=ν} 𝔸^{ν_{1}} × 𝔸^{ν_{2}},

the component 𝔸^{ν_{1}} × 𝔸^{ν_{2}} corresponding to configurations where ν_{1} (resp. ν_{2}) points are running on the affine line Q_{u} − x_{1}(0) (resp. Q_{v} − x_{2}(0)). Here we identify these affine lines with a ”standard” one using the coordinates u and v respectively. Using this decomposition we can define a closed embedding

i_{ν} : Q_{0}^{ν•} ↪ 𝔸^{ν} × 𝔸^{ν}

whose restriction to a component 𝔸^{ν_{1}} × 𝔸^{ν_{2}} sends a configuration as above, to the configuration where all remaining points are equal to zero. Let us define a perverse sheaf

R_{μ_{1},μ_{2}}^{ν,ν} = i_{ν∗}Ψ_{f^{ν}}(I_{μ_{2},μ_{1}}^{ν•}) ∈ M(𝔸^{ν}(ℂ) × 𝔸^{ν}(ℂ); S)

Let us consider the collection of sheaves {R_{μ_{1},μ_{2}}^{ν,ν}| μ_{1}, μ_{2} ∈ X_{ℓ}, ν ∈ Y^{+}, μ_{1} + μ_{2} − ν = 2ρ_{ℓ}}. One can complete this collection to an object R of the category Ind(FS ⊗ FS) where FS ⊗ FS is understood as a category of finite factorizable sheaves corresponding to the square of our initial Cartan datum, i.e. I ∐ I, etc. For a precise construction, see Part V.

13.4. Theorem. We have Φ(R) = R. □

Chapter 3. Modular

14. Heisenberg local systems

In this section we sketch a construction of certain remarkable cohesive local systems on arbitrary smooth families of compact smooth curves, to be called the Heisenberg local systems.

In the definition and construction of local sustems below we will have to assume that our base field k contains roots of unity of sufficiently high degree; the characteristic of k is assumed to be prime to this degree.

14.1. From now on until 14.11 we fix a smooth proper morphism f : C ⟶ S of relative dimension 1, S being a smooth connected scheme over ℂ. For s ∈ S, we denote C_{s} := f^{−1}(s). Let g be the genus of fibres of f.

Let S_{λ} denote the total space of the determinant line bundle λ_{C/S} = det Rf_{∗}Ω_{C/S}^{1} without the zero section. For any object (?) over S (e.g., a scheme over S, a sheaf over a scheme over S, etc.), we will denote by (?)_{λ} its base change under S_{λ} ⟶ S.

Below, if we speak about a scheme as a topological (analytic) space, we mean its set of ℂ-points with the usual topology (resp. analytic structure).

14.2. We will use the relative versions of configuration spaces; to indicate this, we will use the subscript _{/S}. Thus, if J is a finite set, C_{/S}^{J} will denote the J-fold fibered product of C with itself over S, etc.

Let C(J)_{/S} denote the subscheme of the J-fold cartesian power of the relative tangent bundle T_{C/S} consisting of J-tuples {x_{j}, τ_{j}} where x_{j} ∈ C and τ_{j} ≠ 0 in T_{C/S,x}, the points x_{j} being pairwise distinct. Let C̃(J)_{/S} denote the space of J-tuples of holomorphic embeddings ϕ_{j} : D × S ⟶ C over S with disjoint images; we have the 1-jet maps C̃(J)_{/S} ⟶ C(J)_{/S}.

An epimorphism ρ : K ⟶ J induces the maps

m_{C/S}(ρ) : ∏_{J} D̃(K_{j}) × C̃(J)_{/S} ⟶ C̃(J)_{/S}

and

m̄_{C/S}(ρ) : ∏_{J} D(K_{j}) × C̃(J)_{/S} ⟶ C(K)_{/S}

which satisfy the compatibilities as in 7.6.

14.3. We extend the function n to X_{ℓ} (see 12.1) by n(μ) = \frac{1}{2}μ · μ − μ · ρ_{ℓ} (μ ∈ X_{ℓ}). We will denote by J the X_{ℓ}-coloured local system over the operad of disks D which is defined exactly as in 7.5, with X replaced by X_{ℓ}.

14.4. A cohesive local system H of level μ ∈ X_{ℓ} over C/S is a collection of local systems H(π) over the spaces C(J)_{/S;λ} given for every map π : J ⟶ X_{ℓ} of level μ (note the base change to S_{λ}!), together with the factorization isomorphisms

ϕ_{C}(ρ) : m_{C/S}(ρ)^{∗}H(π) ⟶^{∼} × _{J}J (π_{j}) ⊠ H(ρ_{∗}π).

Here we have denoted by the same letter H(π) the lifting of H(π) to C̃_{/S;λ}. The factorization isomorphisms must satisfy the obvious analogs of properties 7.7 (a), (b).

14.5. Now we will sketch a construction of certain cohesive local system over C/S of level (2 − 2g)ρ_{ℓ}. For alternative beautiful constructions of H, see [BP].

To simplify the exposition we will assume below that g ≥ 2 (the construction for g ≤ 1 needs some modification, and we omit it here, see Part V). Let us consider the group scheme Pic(C/S) ⊗ X_{ℓ} over S. Here Pic(C/S) is the relative Picard scheme. The group of connected components π_{0}(Pic(C/S) ⊗ X_{ℓ}) is equal to X_{ℓ}. Let us denote by J ac the connected component corresponding to the element (2 − 2g)ρ_{ℓ}; this is an abelian scheme over S, due to the existence of the section S ⟶ J ac defined by Ω_{C/S}^{1} ⊗ (−ρ_{ℓ}).

For a scheme S′ over S, let H_{1}(S′/S) denote the local system of the first relative integral homology groups over S. We have H_{1}(J ac/S) = H_{1}(C/S) ⊗ X_{ℓ}. We will denote by ω the polarization of J ac (i.e. the skew symmetric form on the latter local system) equal to the tensor product of the standard form on H_{1}(C/S) and the form (μ_{1}, μ_{2}) ↦ \frac{d_{ℓ}^{g}}{ℓ} μ_{1} · μ_{2} on X_{ℓ}. Note that the assumption g ≥ 2 implies that \frac{d_{ℓ}^{g}}{ℓ} μ_{1} · μ_{2} ∈ ℤ for any μ_{1}, μ_{2} ∈ X_{ℓ}. Since the latter form is positive definite, ω is relatively ample (i.e. defines a relatively ample invertible sheaf on J ac).

14.6. Let α = ∑ n_{μ} · μ ∈ ℕ[X_{ℓ}]; set Supp(α) = {μ| n_{μ} ≠ 0}. Let us say that α is admissible if ∑ n_{μ}μ = (2 − 2g)ρ_{ℓ}. Let us denote by

aj_{α} : C_{/S}^{α} ⟶ Pic(C/S) ⊗ X_{ℓ}

the Abel-Jacobi map sending ∑ μ · x_{μ} to ∑ x_{μ} ⊗ μ. If α is admissible then the map aj_{α} lands in J ac.

Let D^{α} denote the following relative divisor on C_{/S}^{α}

D^{α} = \frac{d_{ℓ}^{g}}{ℓ} (∑_{μ≠ν} μ · νΔ_{μν} + \frac{1}{2} ∑_{μ} μ · μΔ_{μμ}).

Here Δ_{μν} (μ, ν ∈ Supp(α)) denotes the corresponding diagonal in C_{/S}^{α}. Note that all the multiplicities are integers.

Let π : J ⟶ X_{ℓ} be an unfolding of α. We will denote by D^{π} the pull-back of D^{α} to C_{/S}^{J}. Let us introduce the following line bundles

L(π) = ⊗_{j∈J}T_{j}^{⊗^{\frac{d{g \atop ℓ}}{ℓ}} n(π(j))} ⊗ O(D^{π})

on C_{/S}^{J}, and

L_{α} = L(π)/Σ_{π}

on C_{/S}^{α} (the action of Σ_{π} is an obvious one). Here T_{j} denotes the relative tangent line

bundle on C_{/S}^{J} in the direction j. Note that the numbers \frac{d_{ℓ}^{g}}{ℓ} n(μ) (μ ∈ X_{ℓ}) are integers.

14.7. Proposition. There exists a unique line bundle L on J ac such that for each admissible α, we have L_{α} = aj_{α}^{∗}(L). The first Chern class c_{1}(L) = −[ω]. □

14.8. In the sequel if L_{0} is a line bundle, let L^{•}_{0} denote the total space its with the zero section removed.

The next step is the construction of a certain local system ℌ over L^{•}_{λ}. Its dimension is equal to d_{ℓ}^{g} and the monodromy around the zero section of L (resp. of the determinant bundle) is equal to ζ^{−2ℓ/d_{ℓ}^{g}} (resp. (−1)^{rk(X)}ζ^{−12ρ_{ℓ}·ρ_{ℓ}}). The construction of ℌ is outlined below.

The previous construction assigns to a triple

(a lattice Λ, a symmetric bilinear form ( , ) : Λ × Λ ⟶ ℤ, ν ∈ Λ)

an abelian scheme J ac_{Λ} := (Pic(C/S) ⊗ Λ)_{(2g−2)ν} over S, together with a line bundle L_{Λ} on it (in the definition of L_{Λ} one should use the function n_{ν}(μ) = \frac{1}{2}μ · μ + μ · ν). We considered the case Λ = X_{ℓ}, (μ_{1}, μ_{2}) = \frac{d_{ℓ}^{g}}{ℓ} μ_{1} · μ_{2}, ν = −ρ_{ℓ}.

Now let us apply this construction to the lattice Λ = X_{ℓ} ⊕ Y_{ℓ}, the bilinear form ((μ_{1}, ν_{1}), (μ_{2}, ν_{2})) = −\frac{1}{ℓ}(ν_{1} · μ_{2} + ν_{2} · μ_{1} + ν_{1} · ν_{2}) and ν = (−ρ_{ℓ}, 0). The first projection Λ ⟶ X_{ℓ} induces the morphism

p : J ac_{Λ} ⟶ J ac

the fibers of p are abelian varieties J ac(C_{s}) ⊗ Y_{ℓ} (s ∈ S).

14.9. Theorem. (i) The line bundle L_{Λ} is relatively ample with respect to p. The direct image E := p_{∗}L_{Λ} is a locally free sheaf of rank d_{ℓ}^{g}.

(ii) We have an isomorphism

det(E) = L ⊗ λ^{d_{ℓ}^{g}(− \frac{1}{2} rk(X_{ℓ})+6 \frac{ρ_{ℓ}·ρ_{ℓ}}{ℓ})}.

□

Here λ denotes the pull-back of the determinant bundle λ_{C/S} to J ac.

14.10. Let us assume for a moment that k = ℂ and ζ = exp(−\frac{π\sqrt{−1}}{ℓ}). By the result of Beilinson-Kazhdan, [BK] 4.2, the vector bundle E carries a canonical flat projective connection. By loc. cit. 2.5, its lifting to det(E)^{•} carries a flat connection with the scalar monodromy around the zero section equal to exp(\frac{2π\sqrt{−1}}{d_{ℓ}^{g}}). We have an obvious map

m : L^{•}_{λ} ⟶ L ⊗ λ^{d_{ℓ}^{g}(− \frac{1}{2} rk(X_{ℓ})+6 \frac{ρ_{ℓ}·ρ_{ℓ}}{ℓ})}.

By definition, ℌ is the local system of horizontal sections of the pull-back of E to L^{•}_{λ}. The claim about its monodromies follows from part (ii) of the previous theorem.

This completes the construction of ℌ for k = ℂ and ζ = exp(−\frac{π\sqrt{−1}}{ℓ}). The case of arbitrary k (of sufficiently large characteristic) and ζ follows from this one.

14.11. Let us consider an obvious map q : C(J)_{/S;λ} ⟶ C_{/S;λ}^{J}. The pull-back q^{∗}L(π) has a canonical non-zero section s. Let H̃(π) be the pull-back of the local system ℌ to q^{∗}L(π). By definition, we set H(π) = s^{∗}H̃(π). For the construction of the factorization isomorphisms, see Part V.

14.12. Let 𝔤 be the simple Lie algebra connected with our Cartan datum. Assume that ζ = exp(\frac{π\sqrt{−1}}{dκ}) for some positive integer κ, cf. 10.6 (d is defined in 2.1).

We have 12ρ_{ℓ} · ρ_{ℓ} ≡ 12ρ · ρ mod l, and rk(X) ≡ dim 𝔤 mod 2. By the strange formula of Freudenthal-de Vries, we have 12ρ · ρ = dh dim 𝔤 where h is the dual Coxeter number of our Cartan datum. It follows that the monodromy of H around the zero section of the determinant line bundle is equal to exp(π\sqrt{−1}\frac{(κ−h) dim 𝔤}{κ}). This number coincides with the multiplicative central charge of the conformal field theory associated with the affine Lie algebra 𝔤̂ at level κ (see [BFM] 4.4.1, 6.1.1, 2.1.3, [TUY] 1.2.2), cf 16.2 below.

UNIVERSAL HEISENBERG SYSTEMS

14.13. Let us define a category Sew as follows (cf. [BFM] 4.3.2). Its object A is a finite set \overline{A} together with a collection N_{A} = \{n\} of non-intersecting two-element subsets n ⊂ \overline{A}. Given such an object, we set A^{1} = ⋃_{N_{A}} n, A^{0} = \overline{A} − A^{1}. A morphism f : A ⟶ B is an embedding i_{f} : \overline{B} ↪ \overline{A} and a collection N_{f} of non-intersecting two-element subsets of \overline{B} − \overline{A} such that N_{A} = N_{B} ∐ N_{f}. The composition of morphisms is obvious. (Sew coincides with the category Sets^{♯}/∅, in the notations of [BFM] 4.3.2.)

For A ∈ Sew, let us call an A-curve a data (C, {x_{a}, τ_{a}}_{A^{0}}) where C is a smooth proper (possibly disconnected) complex curve, {x_{a}, τ_{a}}_{A^{0}} is an A^{0}-tuple of distinct points x_{a} ∈ C together with non-zero tangent vectors τ_{a} at them. For such a curve, let \overline{C}_{A} denote the curve obtained from C by clutching pairwise the points x_{a′} with x_{a′′} (n = {a′, a′′}) for all sets n ∈ N_{A}. Thus, the set N_{A} is in the bijection with the set of nodes of the curve \overline{C}_{A} (\overline{C}_{A} = C if N_{A} = ∅).

Let us call an enhanced graph a pair Γ = (\overline{Γ}, g) here \overline{Γ} is a non-oriented graph and g = \{g_{v}\}_{v∈Vert(\overline{Γ})} is a ℕ- valued 0-chain of \overline{Γ}. Here Vert(\overline{Γ}) denotes the set of vertices of \overline{Γ}. Let us assign to a curve \overline{C}_{A} an enhanced graph Γ(\overline{C}_{A}) = (\overline{Γ}(\overline{C}_{A}), g(\overline{C}_{A})). By definition, \overline{Γ}(\overline{C}_{A}) is a graph with Vert(\overline{Γ}(\overline{C}_{A})) = π_{0}(C) = \{ the set of irreducible components of \overline{C}_{A}\} and the set of edges Edge(\overline{Γ}(\overline{C}_{A})) = N_{A}, an edge n = \{a′, a′′\} connecting the vertices corresponding to the components of the points x_{a′}, x_{a′′}. For v ∈ π_{0}(C), g(\overline{C}_{A})_{v} is equal to the genus of the corresponding component C_{v} ⊂ C.

14.14. Let M_{A} denote the moduli stack of A-curves (C, …) such that the curve \overline{C}_{A} is stable in the sense of [DM] (in particular connected). The stack M_{A} is smooth; we have M_{A} = ∐_{g≥0} M_{A,g} where M_{A,g} is a substack of A-curves C with \overline{C}_{A} having genus (i.e. dim H^{1}(\overline{C}_{A}, O_{\overline{C}_{A}})) equal to g. In turn, we have the decomposition into connected components

M_{A,g} = ∐_{Γ,A⃗^{0}} M_{A⃗^{0},g,Γ}

where M_{A⃗^{0},g,Γ} is the stack of A-curves (C, \{x_{a}, τ_{a}\}) as above, with Γ(\overline{C}_{A}) = Γ, A⃗^{0} = \{A_{v}^{0}\}_{v∈Vert(\overline{Γ})}, A^{0} = ∐ A_{v}^{0}, such that x_{a} lives on the connected component C_{v} for a ∈ A_{v}^{0}.

We denote by η : C_{A,g} ⟶ M_{A,g} (resp. η̄ : \overline{C}_{A,g} ⟶ M_{A,g}) the universal smooth curve (resp. stable surve). For ν ∈ Y^{+}, we have the corresponding relative configuration spaces C_{A,g}^{ν}, C_{A,g}^{ν∘}, \overline{C}_{A,g}^{ν}. For brevity we omit the relativeness subscript /M_{A,g} from these notations. The notation C_{A⃗^{0},g,Γ}^{ν} etc., will mean the restriction of these configuration spaces to the component M_{A⃗^{0},g,Γ}.

Let M_{g} be the moduli stack of smooth connected curves of genus g, and \overline{M}_{g} be its Grothendieck-Deligne-Mumford-Knudsen compactification, i.e. the moduli stack of stable curves of genus g. Let η̄ : \overline{C}_{g} ⟶ \overline{M}_{g} be the universal stable curve; let λ̄_{g} = det Rη̄_{∗}(ω_{\overline{C}_{g}/\overline{M}_{g}}) be the determinant line bundle; let \overline{M}_{g;λ} ⟶ \overline{M}_{g} be its total space with the zero section removed.

We have obvious maps M_{A,g} ⟶ \overline{M}_{g}. Let the complementary subscript (·)_{λ} denote the base change of all the above objects under \overline{M}_{g;λ} ⟶ \overline{M}_{g}.

14.15. Let us consider the configuration space C_{A,g}^{ν∘}; it is the moduli stack of ν distinct points running on A-curves (C, {x_{a}, τ_{a}}) and not equal to the marked points x_{a}. This stack decomposes into connected components as follows:

C_{A,g}^{ν∘} = ∐_{Γ,A⃗^{0},ν⃗} Cν⃗ ; A⃗^{0},g,Γ

where ν⃗ = {ν_{v}}_{v∈Vert(Γ)} and Cν⃗ ; A⃗^{0},g,Γ being the moduli stack of objects as above, with Γ(\overline{C}_{A}) = Γ and ν_{v} points running on the component C_{v}. The decomposition is taken over appropriate graphs Γ, decompositions A^{0} = ∐ A_{v}^{0} and the tuples ν⃗ with ∑ ν_{v} = ν.

Let us call an A_{0}-tuple μ⃗ = {μ_{a}} ∈ X_{ℓ}^{A^{0}} (g, ν)-good if

(a) ∑_{a∈A^{0}} μ_{a} − ν ≡ (2 − 2g)ρ_{ℓ} mod Y_{ℓ}.

Given such a tuple, we are going to define certain local system H_{μ⃗;A,g}^{ν} over C_{A,g;λ}^{ν∘}. Let us describe its restriction Hν⃗ ; μ⃗;A⃗^{0},g,Γ to a connected component Cν⃗ ; A⃗^{0},g,Γ;λ.

Let Γ′ be the first subdivision of \overline{Γ}. We have Vert(Γ′) = Vert(\overline{Γ}) ∐ Edge(\overline{Γ}) = π_{0}(C) ∐ N_{A}. The edges of Γ′ are indexed by the pairs (n, a) where n ∈ N_{A}, a ∈ n, the corresponding edge e_{n,a} having the ends a and n. Let us define an orientation of Γ′ by the requierement that a is the beginning of e_{n,a}. Consider the chain complex

C_{1}(Γ′; X_{ℓ}/Y_{ℓ}) ⟶^{d} C_{0}(Γ′; X_{ℓ}/Y_{ℓ}).

Let us define a 0-chain c = cν⃗ ; μ⃗ ∈ C_{0}(Γ′; X_{ℓ}/Y_{ℓ}) by

c(v) = ∑_{a∈A_{v}^{0}} μ_{a} + (2g_{v} − 2)ρ_{ℓ} − ν_{v} (v ∈ π_{0}(C)); c(n) = 2ρ_{ℓ} (n ∈ N_{A}).

The goodness assumption (a) ensures that c is a boundary. By definition,

Hν⃗ ; μ⃗;A⃗^{0},g,Γ = ⊕_{χ: dχ=c} H_{χ}

Note that the set {χ| dχ = c} is a torsor over the group H_{1}(Γ′; X_{ℓ}/Y_{ℓ}) = H_{1}(Γ; X_{ℓ}/Y_{ℓ}).

The local system H_{χ} is defined below, in 14.18, after a little notational preparation.

14.16. Given two finite sets J, K, let C(J; K)_{g} denote the moduli stack of objects (C, {x_{j}}, {y_{k}, τ_{k}}). Here C is a smooth proper connected curve of genus g, {x_{j}} is a J-tuple of distinct points x_{j} ∈ C and {y_{k}, τ_{k}} is a K-tuple of distinct points y_{k} ∈ C together with non-zero tangent vectors τ_{k} ∈ T_{y_{k}}C. We suppose that y_{k} ≠ x_{j} for all k, j.

We set C(J)_{g} := C(∅; J)_{g}. We have the forgetful maps C(J ∐ K)_{g} ⟶ C(J; K)_{g}.

The construction of 14.5 — 14.11 defines the Heisenberg system H(π) over the smooth stack C(J)_{g;λ} for each π : J ⟶ X_{ℓ}.

Given ν ∈ Y^{+}, choose an unfolding of ν, π : J ⟶ I, and set C^{ν}(K)_{g}^{∘} := C(J; K)/Σ_{π}. Given a K-tuple μ⃗ = {μ_{k}} ∈ X_{ℓ}^{K}, define a map π̃ : J ∐ K ⟶ X_{ℓ} by π̃(j) = −π(j) ∈ −I ⊂ X_{ℓ} (j ∈ J), π̃(k) = μ_{k} (k ∈ K). The local system H(π̃) over C(J ∐ K)_{g;λ} descends to C(J; K)_{g;λ} since ζ^{2n(−i)} = 1, and then to C^{ν}(K)_{g;λ}^{∘}, by Σ_{π}-equivariance; denote the latter local system by H̃ν ; μ⃗, and set Hν ; μ⃗ = H̃ν ; μ⃗ ⊗ Sign^{ν}, cf. 8.3.

14.17. Lemma. If μ⃗ ≡ μ′ mod Y_{ℓ}^{K} then we have canonical isomorphisms Hν ; μ⃗ = H_{μ′}^{ν}. □

Therefore, it makes sense to speak about Hν ; μ⃗ for μ⃗ ∈ (X_{ℓ}/Y_{ℓ})^{K}.

14.18. Let us return to the situation at the end of 14.15. We have Γ = (\overline{Γ}, \{g_{v}\}_{v∈Vert(\overline{Γ})}). Recall that A^{0} = ∐ A_{v}^{0}; we have also A^{1} = ∐ A_{v}^{1} where A_{v}^{1} := {a ∈ A^{1}| x_{a} ∈ C_{v}} (v ∈ Vert(\overline{Γ}). Set \overline{A}_{v} = A_{v}^{0} ∐ A_{v}^{1}, so that \overline{A} = ∐ \overline{A}_{v}. We have an obvious map

(a) M_{A⃗^{0},g,Γ} ⟶ ∏_{Vert(\overline{Γ})} C(\overline{A}_{v})_{g_{v}},

and a map

(b) Cν⃗ ; A⃗^{0},g,Γ ⟶ ∏_{Vert(\overline{Γ})} C^{ν_{v}}(\overline{A}_{v})_{g_{v}}^{∘}

over (a). For each v, define an \overline{A}_{v}-tuple μ⃗(χ; v) equal to μ_{a} at a ∈ A_{v}^{0} and χ(e_{a,n}) at a ∈ n ⊂ A^{1}. By definition, the local system H_{χ} over Cν⃗ ; A⃗^{0},g,Γ;λ is the inverse image of the product

× _{Vert(\overline{Γ})} H_{μ⃗(χ;v)}^{ν_{v}}

under the map (b) (pulled back to the determinant bundle).

This completes the definition of the local systems H_{μ⃗;A,g}^{ν}. They have a remarkable compatibility property (when the object A varies) which we are going to describe below, see theorem 14.23.

14.19. Let T̃_{A,g}^{ν} denote the fundamental groupoid π(C_{A,g;λ}^{ν∘}). We are going to show that these groupoids form a cofibered category over Sew.

14.20. A morphism f : A ⟶ B in Sew is called a sewing (resp. deleting) if A^{0} = B^{0} (resp. N_{f} = ∅). A sewing f with card(N_{f}) = 1 is called simple. Each morphism is a composition of a sewing and a deleting; each sewing is a composition of simple ones.

(a) Let f : A ⟶ B be a simple sewing. We have canonical morphisms

℘_{f} : M_{A,g;λ} ⟶ T^{•}_{∂\overline{M}_{B,g;λ}} \overline{M}_{B,g;λ}

and

℘_{f}^{(ν∘)} : C_{A,g;λ}^{ν∘} ⟶ T^{•}_{∂\overline{C_{B,g;λ}^{ν∘}}} \overline{C_{B,g;λ}^{ν∘}},

over ℘_{f}, cf. [BFM] 4.3.1. Here \overline{M}_{B,g} (resp. \overline{C_{B,g}^{ν∘}}) denotes the Grothendieck-Deligne-Mumford-Knudsen compactification of M_{B,g} (resp. of C_{B,g}^{ν∘}) and ∂\overline{M}_{B,g} (resp. ∂\overline{C_{B,g}^{ν∘}}) denotes the smooth locus of the boundary \overline{M}_{B,g}−M_{B,g} (resp. \overline{C_{B,g}^{ν∘}}−\overline{C}_{B,g}^{ν}). The subscript λ indicates the base change to the determinant bundle, as before.

Composing the specialization with the inverse image under ℘_{f}^{(ν∘)}, we get the canonical map f_{∗} : T̃_{A,g}^{ν} ⟶ T̃_{B,g}^{ν}.

(b) Let f : A ⟶ B be a deleting. It induces the obvious morphisms (denoted by the same letter)

f : M_{A,g;λ} ⟶ M_{B,g;λ}

and

f : C_{A,g;λ}^{ν∘} ⟶ C_{B,g;λ}^{ν∘}.

The last map induces f_{∗} : T̃_{A,g}^{ν} ⟶ T̃_{B,g}^{ν}.

Combining the constructions (a) and (b) above, we get a category T̃_{g}^{ν} cofibered in groupoids over Sew, with fibers T̃_{A,g}^{ν}.

14.21. Let Rep_{c;A,g}^{ν} be the category of finite dimensional representations of T̃_{A,g}^{ν} (over k) with the monodromy c ∈ k^{∗} around the zero section of the determinant bundle. The previous construction shows that these categories form a fibered category Rep_{c;g}^{ν} over Sew.

14.22. For A ∈ Sew, let us call an A^{0}-tuple μ⃗ = {μ_{a}} ∈ X_{ℓ}^{A^{0}} good if ∑_{A^{0}} μ_{a} ∈ Y.

If f : B ⟶ A is a morphism, define a B^{0}-tuple f^{∗}μ⃗ = {μ′_{b}} by μ′_{b} = μ_{i_{f}^{−1}(b)} if b ∈ i_{f}(A^{0}), and 0 otherwise. Obviously, ∑_{B^{0}} μ′_{b} = ∑_{A^{0}} μ_{a}.

Given a good μ⃗, let us pick an element ν ∈ Y^{+} such that ν ≡ ∑ μ_{a}+(2g−2)ρ_{ℓ} mod Y_{ℓ}. We can consider the local system H_{μ⃗;A,g}^{ν} as an object of Rep_{c;A,g}^{ν} where c = (−1)^{card(I)}ζ^{−12ρ·ρ}.

14.23. Theorem. For any morphism f : B ⟶ A in Sew and a g-good μ⃗ ∈ X_{ℓ}^{A^{0}}, we have the canonical isomorphism

f^{∗}H_{μ⃗;A,g}^{ν} = H_{f^{∗}μ⃗;B,g}^{ν}.

In other words, the local systems H_{μ⃗;B,g}^{ν} define a cartesian section of the fibered category Rep_{c;g}^{ν} over Sew/A. Here c = (−1)^{card(I)}ζ^{−12ρ·ρ}. □

15. Fusion structures on FS

15.1. Below we will construct a family of ”fusion structures” on the category FS (and hence, due to the equivalence Φ, on the category C) indexed by m ∈ ℤ. We should explain what a fusion structure is. This is done in 15.8 below. We will use a modification of the formalism from [BFM] 4.5.4.

15.2. Recall that we have a regular object R ∈ Ind(FS^{⊗2}), cf. Section 13. We have the canonical isomorphism t(R) = R where t : Ind(FS^{⊗2}) ⟶^{∼} Ind(FS^{⊗2}) is the permutation, hence an object R_{n} ∈ Ind(FS^{⊗2}) is well defined for any two-element set n.

For an object A ∈ Sew, we set Ã = A^{0} ∐ N_{A}. Let us call an A-collection of factorizable sheaves an Ã-tuple {Xã}_{ã∈Ã} where Xã ∈ FS_{cã} if ã ∈ A^{0} and Xã = Rã if ã ∈ N_{A}. We impose the condition that ∑_{a∈A^{0}} c_{a} = 0 ∈ X_{ℓ}/Y. We will denote such an object {X_{a}; R_{n}}_{A}. It defines an object

⊗_{A} \{X_{a}; R_{n}\} := (⊗_{a∈A^{0}} X_{a}) ⊗ (⊗_{n∈N_{A}} R_{n}) ∈ Ind(FS^{⊗\overline{A}}).

If f : B ⟶ A is a morphism in Sew, we define a B-collection f^{∗}{X_{a}; R_{n}}_{B} = {Yb̃}B̃ by Yb̃ = X_{i_{f}^{−1}(b̃)} for b̃ ∈ i_{f}(A^{0}), 1 if b̃ ∈ B^{0} − i_{f}(A^{0}) and Rb̃ if b̃ ∈ N_{B} = N_{A} ∪ N_{f}.

15.3. Given an A-collection {X_{a}, R_{n}}_{A} with λ(X_{a}) = μ_{a} such that

(a) ν := ∑_{A^{0}} μ_{a} + (2g − 2)ρ_{ℓ} ∈ Y^{+},

one constructs (following the pattern of 8.5) a perverse sheaf × _{A,g}^{(ν)} \{X_{a}; R_{n}\} over \overline{C}_{A,g;λ}^{ν}. It is obtained by planting factorizable sheaves X_{a} into the universal sections x_{a} of the stable curve \overline{C}_{A,g}, the regular sheaves R_{n} into the nodes n of this curve and pasting them together into one sheaf by the Heisenberg system H_{μ⃗;A,g}^{ν}.

15.4. Given A ∈ Sew and an A-collection {X_{a}; R_{n}}_{A}, choose elements μ_{a} ≥ λ(X_{a}) in X_{ℓ} such that (a) above holds (note that 2ρ_{ℓ} ∈ Y). Below ν will denote the element as in (a) above.

Let X′_{a} denote the factorizable sheaf isomorphic to X_{a} obtained from it by the change of the highest weight from λ(X_{a}) to μ_{a}. For each m ∈ ℤ, define a local system ⟨⊗_{A} {X_{a}; R_{n}}⟩_{g}^{(m)} over M_{A,g;λ} as follows.

Let ⟨⊗_{A} {X_{a}; R_{n}}⟩_{A⃗^{0},g,Γ}^{(m)} denote the restriction of the local system to be defined to the connected component M_{A⃗^{0},g,Γ;λ}. By definition,

⟨⊗_{A} {X_{a}; R_{n}}⟩_{A⃗^{0},g,Γ}^{(m)} := R^{m−dim M_{A⃗^{0},g,Γ;λ}}η̄_{∗}^{ν}( × _{A⃗^{0},g,Γ}^{(ν)} {X′_{a}; R_{n}}).

Here × _{A⃗^{0},g,Γ}^{(ν)} {X′_{a}; R_{n}} denotes the perverse sheaf × _{A,g}^{(ν)} {X′_{a}; R_{n}} restricted to the subspace \overline{C}_{A⃗^{0},g,Γ;λ}^{ν}. This definition does not depend on the choice of the elements μ_{a}.

15.5. Given a morphism f : B ⟶ A, we define, acting as in 14.19, 14.21, a perverse

sheaf f^{∗}( × _{A,g}^{(ν)} \{X_{a}; R_{n}\}) over \overline{C}_{B,g;λ}^{ν} and local systems f^{∗}⟨⊗_{A} \{X_{a}; R_{n}\}⟩_{g}^{(m)} over M_{B,g;λ}.

15.6. Theorem. In the above notations, we have canonical isomorphisms

f^{∗}( × _{A,g}^{(ν)} {X_{a}; R_{n}}) = × _{B,g}^{(ν)} f^{∗}{X_{a}; R_{n}}.

This is a consequence of Theorem 14.23 above and the definition of the regular sheaf R as a sheaf of nearby cycles of the braiding local system, 13.3.

15.7. Corollary. We have canonical isomorphisms of local systems

f^{∗}⟨⊗_{A} {X_{a}; R_{n}}⟩_{g}^{(m)} = ⟨⊗_{B} f^{∗}{X_{a}; R_{n}}⟩_{g}^{(m)} (m ∈ ℤ). □

15.8. The previous corollary may be expressed as follows. The various A-collections of factorizable sheaves (resp. categories R̃ep_{c;A} of finite dimensional representations of

”Teichmüller groupoids” { \atop T eich}_{A} = π(M_{A;λ}) having monodromy c around the zero section of the determinant bundle) define a fibered category FS^{♯} (resp. R̃ep_{c}) over Sew.

For any m ∈ ℤ, the collection of local systems ⟨⊗_{A} {X_{a}; R_{n}}⟩_{g}^{(m)} and the canonical isomorphisms of the previous theorem define a cartesian functor

⟨ ⟩^{(m)} : FS^{♯} ⟶ R̃ep_{c}

where c = (−1)^{card(I)}ζ^{−12ρ·ρ}. We call such a functor a fusion structure of multiplicative central charge c on the category FS. The category FS with this fusion structure will be denoted by FS^{(m)}.

The difference from the definition of a fusion structure given in [BFM] 4.5.4 is that our fibered categories live over Sew = Sets^{♯}/∅ and not over Sets^{♯}, as in op. cit.

15.9. Example. For A ∈ Sew, let us consider an A-curve P = (ℙ^{1}, {x_{a}, τ_{a}}); it defines a geometric point Q of the stack M_{A,g} where g = card(N_{A}) and hence a geometric point P = (Q, 1) of the stack M_{A,g;λ} since the determinant bundle is canonically trivialized at Q.

15.9.1. Theorem. For an A-collection {X_{a}; R_{n}}, the stalk of the local system ⟨⊗_{A} X_{a}⟩_{g}^{(m)} at a point P is isomorphic to

Tor_{\frac{∞}{2} −m}^{C}(1_{r}, (⊗_{A^{0}} Φ(X_{a})) ⊗ ad^{⊗g}).

To prove this, one should apply theorem 11.2 and the following remark. Degenerating all nodes into cusps, one can include the nodal curve \overline{P}_{A} into a one-parameter family whose special fiber is ℙ^{1}, with card(A^{0})+g marked points. The nearby cycles of the sheaf × _{A} {X_{a}; R_{n}} will be the sheaf obtained by the gluing of the sheaves X_{a} and g copies of the sheaf Φ^{−1}(ad) into these marked points.

15.10. Note that an arbitrary A-curve may be degenerated into a curve considered in the previous example. Due to 15.7, this determines the stalks of all our local systems (up to a non canonical isomorphism).

16. Conformal blocks (higher genus)

In this section we assume that k = ℂ.

16.1. Let us make the assumptions of 10.6, 14.12. Consider the full subcategory MS ⊂ C. Let us define the regular object R_{MS} by

R_{MS} = ⊕_{μ∈Δ_{ℓ}} (𝔏(μ) ⊗ 𝔏(μ)^{∗}) ∈ MS^{⊗2}.

As in the previous section, we have a notion of A-collection {L_{a}; R_{MSn}}_{A} (A ∈ Sew, {L_{a}} ∈ MS^{A^{0}}). The classical fusion structure on MS, [TUY], defines for each A-collection as above, a local system

⟨⊗_{A} {L_{a}; R_{MSn}}⟩_{MS}

on the moduli stack M_{A;λ}.

We have A = (\overline{A}, N_{A}); let us define another object A′ = (\overline{A} ∪ \{∗\}, N_{A}). We have an obvious deleting f_{A} : A′ ⟶ A. Given an A-collection as above, define an A′-collection in {L_{a}; L(2ρ_{ℓ}); R_{n}}_{A′} in the category C. Using the equivalence Φ, we transfer to C the fusion structures defined in the previous section on FS; we denote them ⟨ ⟩_{C}^{(m)}.

The following theorem generalizes Theorem 11.5 to higher genus.

16.2. Theorem. For each A ∈ Sew, the local system f_{A}^{∗}⟨⊗_{A} {L_{a}; R_{MSn}}⟩_{MS} on M_{A′;λ} is a canonical subquotient of the local system ⟨⊗_{A′} {L_{a}; L(2ρ_{ℓ}); R_{n}}⟩_{C}^{(0)}. □

16.3. Let us consider the special case A = (A^{0}, ∅); for an A-collection {L_{a}}_{A^{0}} in MS, we have the classical local systems of conformal blocks ⟨⊗_{A^{0}} {L_{a}}⟩_{MS} on M_{A;λ}.

16.4. Corollary. The local systems ⟨⊗_{A^{0}} {L_{a}}⟩_{MS} are semisimple. They carry a canonical non-degenerate Hermitian form.

In fact, the local system f_{A}^{∗}⟨⊗_{A^{0}} {L_{a}}⟩_{MS} is semisimple by the previous theorem and by Beilinson-Bernstein-Deligne-Gabber, [BBD] 6.2.5. The map of fundamental groupoids f_{A∗} : { \atop T eich}_{A′} ⟶ { \atop T eich}_{A} is surjective; therefore the initial local system is semisimple.

The Hermitian form is defined in the same manner as in genus zero, cf. 11.7.

Part I. INTERSECTION COHOMOLOGY

OF REAL ARRANGEMENTS

1. Introduction

1.1. Let 𝔸 be a complex affine space, H a finite collection of hyperplanes in 𝔸. We suppose that all H ∈ H are given by real equations, and denote by H_{ℝ} ⊂ H the subspace of real points. An arrangement H induces naturally a stratification of 𝔸 denoted by S_{H} (for precise definitions see section 2 below). The main goal of this paper is the study of the category M(𝔸; S_{H}) of perverse sheaves (of vector spaces over a fixed ground field) over 𝔸 which are smooth along S_{H}.

In section 1 we collect topological notations and known facts we will need in the sequel.

Section 2. To each object M ∈ M(𝔸; S_{H}) we assign ”a linear algebra data”. Namely, for each facet of H — i.e. a connected component of an intersection of some H_{ℝ}’s — we define a vector space Φ_{F} (M); if F is contained and has codimension 1 in the closure of another facet E, we define two linear operators between Φ_{F} (M) and Φ_{E}(M) acting in opposite directions. These data contain all information about M we need.

In fact, it is natural to expect that the sheaf M may be uniquely reconstructed from the linear algebra data above (one can check this in the simplest cases).

The spaces Φ_{F} (M) are defined using certain relative cohomology. They are similar to a construction by R. MacPherson (see [B2]). The spaces Φ_{F} (M) are analogous to functors of vanishing cycles.

The main technical properties of functors M ↦ Φ_{F} (M) are (a) commutation with Verdier duality, and (b) exactness. In fact, (b) follows from (a) without difficulty; the proof of (a) is the principal geometrical result of this paper (see Theorem 3.5). Actually, we prove a more general statement concerning all complexes smooth along S_{H}. This is a result of primary importance for us.

Using these linear algebra data, we define an exact functor from M(𝔸; S_{H}) to complexes of finite dimensional vector spaces computing cohomology RΓ(𝔸; •), see Theorem 3.14. This is the main result we will need below. The idea of constructing such functorial complexes on categories of perverse sheaves (quite analogous to the usual computation of singular cohomology using cell decompositions) is due to A. Beilinson, [B1]. It was an important source of inspiration for us.

A similar problem was considered in [C].

In section 3 we present explicit computations for standard extensions of an arbitrary one-dimensional local system over an open stratum. The main result is the computation of intersection cohomology, see Theorem 4.17. In our computations we use some simple geometric ideas from M. Salvetti’s work, [Sa]. However, we do not use a more difficult main theorem of this paper. On the contrary, maybe our considerations shed some new light on it.

1.2. The idea of this work appeared several years ago, in some discussions with Hélène Esnault and Eckart Viehweg. We use this occasion to express to them our sincere gratitude. We are also very grateful to Paul Bressler for a stimulating discussion. We thank Kari Vilonen for pointing out an error in an earlier version of the manuscript.

2. Topological preliminaries

In this section we introduce our notations and recall some basic facts from topology. Main references are [KS], [BBD].

2.1. Throughout the paper, all our topological spaces will be locally compact, in particular Hausdorff. {pt} will denote a one-point space. For a topological space X, a_{X} : X ⟶ {pt} will be the unique map. If Y ⊂ X is a subspace, Ȳ will denote the closure of Y.

Throughout this chapter we fix an arbitrary ground field B. A vector space will mean a vector space over B. For a finite dimensional vector space V, V^{∗} will denote a dual space. Vect will denote a category of vector spaces.

A sheaf will mean a sheaf of vector spaces. For a topological space X, Sh(X) will denote a category of sheaves over X, D^{∗}(X) will denote the derived category of Sh(X), ∗ = +, −, b or ∅ will have the usual meaning.

For p ∈ ℤ, D^{≤p}(X) ⊂ D(X) will denote a full subcategory of complexes K with H^{i}(K) = 0 for all i > p.

If A, B are abelian categories, we will say that a left exact functor F : D^{∗}(A) ⟶ D(B) (∗ = +, −, b or ∅) has cohomological dimension ≤ r, and write cd(F) ≤ r, if H^{i}(F (A)) = 0 for any A ∈ A and i > r. (Left exactness here means that H^{i}(F (A)) = 0 for i < 0).

2.2. Let f : X ⟶ Y be a continuous map. We will use the following notations for standard functors.

f^{∗} : D(Y) ⟶ D(X) — the inverse image; f_{∗} : D^{+}(X) ⟶ D^{+}(Y) — (the right derived of) the direct image; f_{!} : D^{+}(X) ⟶ D^{+}(Y) — (the right derived of) the direct image with proper supports; f^{!} : D^{+}(Y) ⟶ D^{+}(X) — the right adjoint to f_{!} (defined when f_{!} has finite cohomological dimension), see [KS], 3.1.

We will denote the corresponding functors on sheaves as follows: f^{∗} : Sh(Y) ⟶ Sh(X); R^{0}f_{∗} : Sh(X) ⟶ Sh(Y); R^{0}f_{!} : Sh(X) ⟶ Sh(Y).

We will denote RΓ(X; ·) := p_{X∗}; RΓ_{c}(X; ·) := p_{X!}. For K ∈ D^{+}(X), i ∈ ℤ, we set H^{i}(X; K) := H^{i}(RΓ(X; K)); H_{c}^{i}(X; K) := H^{i}(RΓ_{c}(X; K)).

For V ∈ Vect = Sh({pt}) we denote by V_{X} the constant sheaf p_{X}^{∗}V.

If Y is a subspace of X, we will use a notation i_{Y,X}, or simply i_{Y} if X is understood, for the embedding Y ↪ X. If Y is open in X, we will also write j_{Y} instead of i_{Y}.

For K ∈ D(X), we will use notations K|_{Y} := i_{Y,X}^{∗}K; RΓ(Y; K) := RΓ(Y; K|_{Y}), H^{i}(Y; K) := H^{i}(Y; K_{Y}), etc.

2.3. We have functors

RHom : D^{−}(X)^{opp} × D^{+}(X) ⟶ D^{+}(X),  (1)
⊗ : D^{−}(X) × D^{∗}(X) ⟶ D^{∗}(X)  (2)

where ∗ =, −, b or ∅;

RHom = RΓ ∘ RHom : D^{−}(X)^{opp} × D^{+}(X) ⟶ D^{+}(Vect)  (3)

For K, L ∈ D^{b}(X) we have

Hom_{D(X)}(K, L) = H^{0}(RHom(K, L))  (4)

We denote K^{∗} := RHom(K, B_{X}).

2.4. Let X be a topological space, j : U ⟶ X an embedding of an open subspace, i : Y ⟶ X an embedding of the complement. In this case i_{∗} = i_{!} and j^{!} = j^{∗}. R^{0}i_{∗} and R^{0}j_{!} are exact, so we omit R^{0} from their notation. j_{!} is the functor of extension by zero. i^{!} is the right derived of the functor R^{0}i^{!} : Sh(X) ⟶ Sh(Y) of the subsheaf of sections with supports in Y (in notations of [KS], R^{0}i^{!}(F) = Γ_{Y} (F)).

Let K ∈ D^{+}(X). We will use the following notations for relative cohomology: RΓ(X, Y; K) := RΓ(X; j_{!}K|_{U}); H^{i}(X, Y; K) := H^{i}(RΓ(X, Y; K)).

If Z ↪ Y is a closed subspace, we have a canonical exact triangle

i_{X−Y!}K|_{X−Y} ⟶ i_{X−Z!}K|_{X−Z} ⟶ i_{Y −Z!}K|_{Y −Z} ⟶ i_{X−Y!}K|_{X−Y} [1]  (5)

(of course, i_{X−Y} = j) inducing a long cohomology sequence

… ⟶ H^{i}(X, Y; K) ⟶ H^{i}(X, Z; K) ⟶ H^{i}(Y, Z; K) ⟶^{∂} H^{i+1}(X, Y; K) ⟶ … (6)

2.5. Let X be a topological space such that the functor RΓ_{c}(X; ·) has a finite cohomological dimension. We define a dualizing complex:

ω_{X} := a_{X}^{!}B ∈ D^{b}(X)  (7)

For K ∈ D^{b}(X) we set D_{X}K = RHom(K, ω_{X}) ∈ D^{b}(X). If there is no risk of confusion we denote DK instead of D_{X}K. We get a functor

D : D^{b}(X)^{opp} ⟶ D^{b}(X)  (8)

which comes together with a natural transformation

Id_{D^{b}(X)} ⟶ D ∘ D  (9)

2.6. Orientations. (Cf. [KS], 3.3.) Let X be an n-dimensional C^{0}-manifold. We define an orientation sheaf Or_{X} as the sheaf associated to a presheaf U ↦ Hom_{ℤ}(H_{c}^{n}(U; ℤ), ℤ). It is a locally constant sheaf of abelian groups of rank 1. Its tensor square is constant. By definition, orientation of X is an isomorphism Or_{X} ⟶^{∼} ℤ_{X}.

We have a canonical isomorphism

ω_{X} ⟶^{∼} Or_{X} ⊗_{ℤ} B[n]  (10)

2.7. Homology. Sometimes it is quite convenient to use homological notations. Let Y ⊂ X be a closed subspace of a topological space X (Y may be empty), K ∈ D^{b}(X). We define homology groups as

H_{i}(X, Y; F) := H^{i}(X, Y; F^{∗})^{∗}.

These groups behave covariantly with respect to continuous mappings.

Let σ be a relative singular n-cell, i.e. a continuous mapping

σ : (D^{n}, ∂D^{n}) ⟶ (X, Y)

where, D^{n} denotes a standard closed unit ball in ℝ^{n}. We supply D^{n} with the standard orientation. Let D^{∘}^{n} denote the interior of D^{n}.

Suppose that σ^{∗}K|_{D^{∘}}^{n} is constant. Then by Poincaré duality we have isomorphisms

H_{n}(D^{n}, ∂D^{n}; σ^{∗}K) = H_{c}^{n}(D^{∘}^{n}; σ^{∗}K^{∗})^{∗} ⟶^{∼} H^{0}(D^{∘}^{n}; σ^{∗}K).

(recall that D^{∘}^{n} is oriented). Thus, given an element s ∈ H^{0}(D^{∘}^{n}; K), we can define a homology class

cl(σ; s) := σ_{∗}(s) ∈ H_{n}(X, Y; K).

We will call the couple (σ; s) a relative singular n-cell for K.

These classes enjoy the following properties.

2.7.1. Homotopy. Let us call to cells (σ_{0}; s_{0}) and (σ_{1}, s_{1}) homotopic if there exists a map

σ : (D^{n} × I, ∂D^{n} × I) ⟶ (X, Y)

(where I denotes a unit interval) such that σ^{∗}K|_{D^{∘}^{n}×I} is constant, and a section

s ∈ H^{0}(D^{∘}^{n} × I; σ^{∗}K)

such that (σ, s) restricted to D^{n} × {i} is equal to (σ_{i}; s_{i}), i = 0, 1.

We have

(H) if (σ_{0}; s_{0}) is homotopic to (σ_{1}; s_{1}) then cl(σ_{0}; s_{0}) = cl(σ_{1}; s_{1}).

2.7.2. Additivity. Suppose D^{n} is represnted as a union of its upper and lower half-balls D^{n} = D_{1}^{n} ∪ D_{2}^{n} where D_{1}^{n} = {(x_{1}, …, x_{n}) ∈ D^{n}| x_{n} ≥ 0} and D_{2}^{n} = {(x_{1}, …, x_{n}) ∈ D^{n}| x_{n} ≤ 0}. Let us supply D_{i}^{n} with the induced orientation.

Suppose we are given a relative n-cell (σ; s) such that σ(D_{1}^{n}∩D_{2}^{n}) ⊂ Y. Then its restriction to D_{i}^{n} gives us two relative n-cells (σ_{i}; s_{i}), i = 1, 2. We have

(A) cl(σ; s) = cl(σ_{1}; s_{1}) + cl(σ_{2}; s_{2}).

2.8. We will call a local system a locally constant sheaf of finite rank.

Let X be a topological space, i : Y ↪ X a subspace. We will say that K ∈ D(X) is smooth along Y if all cohomology sheaves H^{p}(i^{∗}K), p ∈ ℤ, are local systems.

We will call a stratification of X a partition X = ⋃_{S∈S} S into a finite disjoint of locally closed subspaces — strata — such that the closure of each stratum is the union of strata.

We say that K ∈ D(X) is smooth along S if it is smooth along each stratum. We will denote by D^{∗}(X; S) the full subcategory of D^{∗}(X) (∗ = +, −, b or ∅) consisting of complexes smooth along S.

2.9. Let us call a stratification S of a topological space X good if the following conditions from [BBD], 2.1.13 hold.

(b) All strata are equidimensional topological varieties. If a stratum S lies in the closure of a stratum T, dim S < dim T.

(c) If i : S ↪ X is a stratum, the functor i_{∗} : D^{+}(S) ⟶ D^{+}(X) has a finite cohomological dimension. If F ∈ Sh(S) is a local system, i_{∗}(F) is smooth along S.

Let S be a good stratification such that all strata have even dimension. We will denote by M(X; S) ⊂ D^{b}(X; S) the category of smooth along S perverse sheaves corresponding to the middle perversity p(S) = − dim S/2, cf. loc. cit.

2.10. Let X be a complex algebraic variety. Let us call its stratification S algebraic if all strata are algebraic subvarieties. Following [BBD], 2.2.1, let us call a sheaf F ∈ Sh(X) constructible if it is smooth along some algebraic stratification. (According to Verdier, every algebraic stratification admits a good refinement.) We denote by D_{c}^{b}(X) a full subcategory of D^{b}(X) consisting of complexes with constructible cohomology.

We denote by M(X) ⊂ D_{c}^{b}(X) the category of perverse sheaves corresponding to the middle perversity. It is a filtered union of categories M(X; S) over all good algebraic stratifications S, cf. loc.cit.

2.11. Let X be a complex manifold, f : X ⟶ ℂ a holomorphic map. Set Y = f^{−1}(0), U = X − Y, let i : Y ↪ X, j : U ↪ X denote the embeddings.

We define a functor of nearby cycles

Ψ_{f} : D^{b}(U) ⟶ D^{b}(Y)  (11)

as Ψ_{f}(K) = ψ_{f}j_{∗}(K)[−1] where ψ_{f} is defined in [KS], 8.6.1.

We define a functor of vanishing cycles

Φ_{f} : D^{b}(X) ⟶ D^{b}(Y)  (12)

as ϕ_{f} from loc. cit.

2.12. Lemma. Let X be topological space, Y ↪ X a closed subspace, F ∈ Sh(X). Then natural maps

lim_{→} H^{i}(U; F) ⟶ H^{i}(Y; F), i ∈ ℤ,

where U ranges through all the open neighbourhoods of U, are isomorphisms.

Proof. See [KS], 2.5.1, 2.6.9. □

2.13. Conic sheaves. (Cf. [KS], 3.7.) Let ℝ^{∗+} denote the multiplicative group of positive real numbers. Let X be a topological space endowed with an ℝ^{∗+}-action.

Following loc.cit., we will call a sheaf F over X conic (with respect to the given ℝ^{∗+}- action) if its restriction to every ℝ^{∗+}-orbit is constant. We will call a complex K ∈ D(X) conic if all its cohomology sheaves are conic.

We will denote by Sh_{ℝ^{∗+}}(X) ⊂ Sh(X), D_{ℝ^{∗+}}^{∗}(X) ⊂ D^{∗}(X), ∗ = b, +, − or ∅, the full subcategories of conic objects.

2.13.1. Lemma. Let U ↪ X be an open subset. Suppose that for every ℝ^{∗+}-orbit O ⊂ X, O ∩ U is contractible (hence, non-empty). Then for every conic K ∈ D^{+}(X) the restriction morphism

RΓ(X; K) ⟶ RΓ(U; K)

is an isomorphism.

Proof. See loc. cit., 3.7.3. □

3. Vanishing cycles functors

3.1. Arrangements. Below we use some terminology from [Sa]. Let 𝔸_{ℝ} be a real affine space 𝔸_{ℝ} of dimension N, and H_{ℝ} = {H_{ℝ}} a finite set of distinct real affine hyperplanes in 𝔸_{ℝ}. Such a set is called a real arrangement. We pick once and for all a square root i = \sqrt{−1} ∈ ℂ.

Let 𝔸 = 𝔸_{ℝ}⊗_{ℝ}ℂ denote the complexification of 𝔸_{ℝ}, and H = {H} where H := H_{ℝ}⊗_{ℝ}ℂ. (A finite set of complex hyperplanes in a complex affine space will be called a complex arrangement).

We will say that H is central if ⋂_{H∈H} H consists of one point.

For a subset K ⊂ H denote

H_{K} = ⋂_{H∈K} H; _{K}H = ⋃_{H∈K} H,

and

H_{ℝ,K} = ⋂_{H∈K} H_{ℝ}, _{K}H_{ℝ} = ⋃_{H∈K} H_{ℝ}.

The nonempty subspaces H_{K} and H_{K,ℝ} are called complex and real edges respectively. Set

H^{∘}_{K} = H_{K} − ∪L,

the union over all the complex edges L ⊂ H_{K}, L ≠ H_{K}. We set H^{∘}_{ℝ,K} = H_{ℝ,K} ∩ 𝔸_{ℝ}.

Connected components of H^{∘}_{ℝ,K} are called facets of H_{ℝ}. Facets of codimension 0 (resp.,

1) are called chambers (resp., faces).

Let us denote by S_{H} a stratification of 𝔸 whose strata are all non-empty H^{∘}_{K}. We will denote by 𝔸^{∘}_{H} a unique open stratum

𝔸^{∘}_{H} = H^{∘}_{∅} = 𝔸 −_{H} H.

In this Section we will study categories of sheaves D(𝔸; S_{H}) and M(𝔸; S_{H}).

We will denote by S_{H_{ℝ}} a stratification of 𝔸_{ℝ} whose strata are all facets. We set

𝔸^{∘}_{H,ℝ} = 𝔸_{ℝ} −_{H} H_{ℝ}.

It is a union of all chambers.

3.2. Dual cells. (cf. [Sa]). Let us fix a point ^{F} w on each facet F. We will call this set of points w = {^{F} w} marking of our arrangement.

For two facets F, E let us write F < E if F ⊂ \overline{E} and dim F < dim E. We will say that E is adjacent to F. We will denote by Ch(F) the set of all chambers adjacent to F.

Let us call a flag a sequence of q − p + 1 facets F = (F_{p} < F_{p+1} < … < F_{q}) with dim F_{i} = i. We say that F_{p} is the beginning and F_{q} the end of F.

Let us denote by FΔ a closed (q − p)-symplex with vertices ^{F_{p}}w, …,^{F_{q}} w. Evidently, FΔ ⊂ \overline{F}_{q}.

Suppose we are given two facets F_{p} < F_{q}, dim F_{i} = i. We will denote

D_{F_{p}<F_{q}} = ⋃ FΔ,

the sum over all flags beginning at F_{p} and ending at F_{q}. This is a (q − p)-dimensional cell contained in \overline{F}_{q}.

For a facet F let us denote

D_{F} = ⋃_{C∈Ch(F)} D_{F <C}

We set

S_{F} := ⋃_{E,C:F <E<C;} D_{E<C},

the union over all facets E and chambers C. The space S_{F} is contained in D_{F} (in fact, in the defintion of S_{F} it is enough to take the union over all facets E such that dim E = dim F + 1). If q = codim F, then D_{F} is homeomorphic to a q-dimensional disc

and S_{F} — to a (q −1)- dimensional sphere, cf. [Sa], Lemma 6. We denote D^{∘}_{F} := D_{F} −S_{F}. We will call D_{F} a dual cell corresponding to F.

We set D^{∘}_{F <C} := D_{F <C} ∩ D^{∘}_{F}.

3.3. Generalized vanishing cycles. Let K ∈ D^{b}(𝔸; S_{H}), F a p-dimensional facet. Let us introduce a complex

Φ_{F} (K) := RΓ(D_{F} , S_{F}; K)[−p] ∈ D^{b}(pt)  (13)

This complex will be called a complex of generalized vanishing cycles of K at a facet F.

Formally, the definition of functor Φ_{F} depends upon the choice of a marking w. However, functors defined using two different markings are canonically isomorphic. This is evident. Because of this, we omit markings from the notations.

3.4. Transversal slices. Let F be a facet of dimension p which is a connected component of a real edge M_{ℝ} with the complexification M. Let us choose a real affine subspace L_{ℝ} of codimension p transversal to F and passing through ^{F} w. Let L be its complexification.

Let us consider a small disk L_{ϵ} ⊂ L with the centrum at ^{F} w = L ∩ M. We identify L_{ϵ} with an affine space by dilatation. Our arrangement induces a central arrangement H_{L} in L_{ϵ}. Given K ∈ D^{b}(𝔸; S_{H}), define K_{L} := i_{L_{ϵ}}^{∗}K[−p].

3.4.1. Lemma. We have a natural isomorphism

D(K_{L}) ⟶^{∼} (DK)_{L}  (14)

Proof. Consider an embedding of smooth complex manifolds i_{L} : L ⟶ 𝔸. Let us consider the following complexes: ω_{L/𝔸} := i_{L}^{!}B_{𝔸} (cf. [KS], 3.1.16 (i)) and Or_{L/𝔸} := Or_{L} ⊗_{ℤ} Or_{𝔸}. We have canonical isomorphism

ω_{L/𝔸} ⟶^{∼} Or_{L/𝔸}[−2p] ⊗ B.

The chosen orientation of ℂ enables us to identify Or_{𝔸} and Or_{L}, and hence Or_{L/𝔸} with constant sheaves; consequently, we get an isomorphism

ω_{L/𝔸} ⟶^{∼} B[−2p].

The canonical map

i_{L}^{!}K ⟶ i^{∗}K ⊗ ω_{L/𝔸}  (15)

(cf. [KS], (3.1.6)) is an isomorphism since singularities of K are transversal to L (at least in the neighbourhood of ^{F} w). Consequently we get an isomorphism i_{L}^{!}K ⟶^{∼} i^{∗}K_{L}[−2p].

Now we can compute:

D(K_{L}) = D(i_{L}^{∗}K[−p]) ⟶^{∼} D(i_{L}^{∗}K)[2p] ⟶^{∼} i_{L}^{!}DK[p] ⟶^{∼} i_{L}^{∗}DK[−p] = (DK)_{L},

QED. □

3.4.2. Lemma. We have a natural isomorphism

Φ_{F} (K) ≅ Φ_{{^{F} w}}(K_{L})  (16)

This follows directly from the definition of functors Φ.

This remark is often useful for reducing the study of functors Φ_{F} to the case of a central arrangement.

3.4.3. Lemma. If M ∈ M(𝔸; S_{H}) then M_{L} ∈ M(L_{ϵ}; S_{H_{Lϵ}}).

Proof follows from transversality of L_{ϵ} to singularities of M. □

3.5. Duality. Theorem. Functor Φ_{F} commutes with Verdier duality. More precisely, for every K ∈ D^{b}(𝔸, S_{H}) there exists a natural isomorphism

Φ_{F} (DK) ⟶^{∼} DΦ_{F} (K)  (17)

This is the basic property of our functors.

3.6. 1-dimensional case. To start with the proof, let us treat first the simplest case. Consider an arrangement consisting of one point — origin — in a one-dimensional space 𝔸. It has one 0-face F and two 1-faces E_{±} — real rays ℝ_{>0} and ℝ_{<0}. A marking consists of two points w_{±} ∈ E_{±} and F. We set w_{±} = ±1 (we pick a coordinate on 𝔸).

For a positive r denote 𝔸_{≤r} := {z ∈ 𝔸| |z| ≤ r}, S_{r} := ∂𝔸_{≤r}; 𝔸_{<r}, 𝔸_{≥r}, etc. have an evident meaning. We also set 𝔸_{(r′,r′′)} := 𝔸_{>r′} ∩ 𝔸_{<r′′}. A subscript ℝ will denote an intersection of these subsets with 𝔸_{ℝ}. Evidently, D_{F} = 𝔸_{≤1,ℝ}, S_{F} = S_{1,ℝ}. Define D_{F}^{opp} := 𝔸_{≥1,ℝ}, Y := i · D_{F}^{opp}.

One sees easily (cf. infra, Lemma 3.7) that one has isomorphisms

Φ_{F} (K) ≅ RΓ(𝔸, S_{F}; K) ⟶^{∼} RΓ(𝔸, D_{F}^{opp}; K).  (18)

Let us choose real numbers ϵ, r′, r′′ such that 0 < ϵ < r′ < 1 < r′′. Set Y := ϵi · D_{F}^{opp}. Denote j := j_{𝔸−S_{F}}. We have natural isomorphisms

DΦ_{F} (K) ≅ DRΓ(𝔸, S_{F}; K) ≅ RΓ_{c}(𝔸; j_{∗}j^{∗}DK)

(Poincaré duality)

≅ RΓ(𝔸, 𝔸_{≥r′′}; j_{∗}j^{∗}DK) ≅ RΓ(𝔸, Y ∪ 𝔸_{≥r′′}; j_{∗}j^{∗}DK)

(homotopy). Consider the restriction map

res : RΓ(𝔸, Y ∪ 𝔸_{≥r′′}; j_{∗}j^{∗}DK) ⟶ RΓ(𝔸_{≤r′}, Y ∩ 𝔸_{≤r′}; DK)  (19)

We claim that res is an isomorphism. In fact, Cone(res) is isomorphic to

RΓ(𝔸, 𝔸_{≤r′} ∪ 𝔸_{≥r′′} ∪ Y; j_{∗}j^{∗}DK) = RΓ_{c}(𝔸_{<r′′}, 𝔸_{≤r′} ∪ Y; j_{∗}j^{∗}DK) ≅
≅ DRΓ(𝔸_{<r′′} − (𝔸_{≤r′} ∪ Y); j_{!}j^{∗}K)

We have by definition

RΓ(𝔸_{<r′′} − (𝔸_{≤r′} ∪ Y); j_{!}j^{∗}K) = RΓ(𝔸_{(r′,r′′)} − Y, S_{F}; K)

On the other hand, evidently K is smooth over 𝔸_{(r′,r′′)}, and we have an evident retraction of 𝔸_{(r′,r′′)} on S_{F} (see Fig. 1). Therefore, RΓ(𝔸_{(r′,r′′)} − Y, S_{F}; K) = 0 which proves the claim.

[figure: S_{r′′}; Y; iϵw_{+}; w_{+}; w_{−}; S_{r′}; z; Fig. 1]

A clockwise rotation by π/2 induces an isomorphism

RΓ(𝔸_{≤r′}, Y ∩ 𝔸_{≤r′}; DK) ≅ RΓ(𝔸_{≤r′}, ϵ · D_{F}^{opp}; DK),

and the last complex is isomorphic to Φ_{F} (DK) by dilatation and (18). This proves the theorem for Φ_{F}. The statement for functors Φ_{E_{±}} is evident.

Let us return to the case of an arbitrary arrangement.

3.7. Lemma. Suppose that H is central. Let F be the unique 0- dimensional facet. The evident restriction maps induce canonical isomorphisms

Φ_{F} (K) ≅^{(1)} RΓ(𝔸_{ℝ}, S_{F}; K) ≅^{(2)} RΓ(𝔸, S_{F}; K) ≅^{(3)} RΓ(𝔸, D_{F}^{opp}; K) ≅^{(4)} RΓ(𝔸_{ℝ}, D_{F}^{opp}; K)

where D_{F}^{opp} = 𝔸_{ℝ} − D^{∘}_{F}.

Proof. Let us fix a coordinate system with the origin at F, and hence a metric on 𝔸. For ϵ > 0 let U_{ϵ} ⊂ 𝔸_{ℝ} denote the set of points x ∈ 𝔸_{ℝ} having distance < ϵ from D_{F}.

It follows from 2.12 that

Φ_{F} (K) = RΓ(D_{F} , S_{F}; K) ≅ lim_{→}_{ϵ} RΓ(U_{ϵ}, S_{F}; K).

On the other hand, from 2.13.1 it follows that restriction maps

RΓ(𝔸_{ℝ}, S_{F}; K) ⟶ RΓ(U_{ϵ}, S_{F}; K)

are isomorphisms. This establishes an isomorphism (1). To prove (3), one remarks that its cone is acyclic.

The other isomorphisms are proven by the similar arguments. We leave the proof to the reader. □

3.8. Proof of 3.5. First let us suppose that H is central and F is its 0-dimensional facet. We fix a coordinate system in 𝔸 as in the proof of the lemma above. We have a decomposition 𝔸 = 𝔸_{ℝ}⊕i·𝔸_{ℝ}. We will denote by ℜ, ℑ : 𝔸 ⟶ 𝔸_{ℝ} the evident projections.

Let 𝔸_{≤r}, etc. have the meaning similar to the one-dimensional case above. Let us denote j := j_{𝔸−S_{F}}. We proceed as in one-dimensional case.

Let us choose positive numbers r′, r′′, ϵ, such that

ϵD_{F} ⊂ 𝔸_{<r′} ⊂ D^{∘}_{F} ⊂ D_{F} ⊂ 𝔸_{<r′′}

Let us introduce a subspace

Y = ϵi · D_{F}^{opp}

We have isomorphisms

DΦ_{F} (K) ≅ DRΓ(𝔸, S_{F}; K)

(by Lemma 3.7)

≅ RΓ_{c}(𝔸; j_{∗}j^{∗}DK) ≅ RΓ(𝔸, 𝔸_{≥r′′}; j_{∗}j^{∗}DK) ≅ RΓ(𝔸, Y ∪ 𝔸_{≥r′′}; j_{∗}j^{∗}DK)

(homotopy). Consider the restriction map

res : RΓ(𝔸, Y ∪ 𝔸_{≥r′′}; j_{∗}j^{∗}DK) ⟶ RΓ(𝔸_{≤r′}, Y ∩ 𝔸_{≤r′}; DK)  (20)

Cone(res) is isomorphic to

RΓ(𝔸, 𝔸_{≤r′} ∪ 𝔸_{≥r′′} ∪ Y; j_{∗}j^{∗}DK) = RΓ_{c}(𝔸_{<r′′}, 𝔸_{≤r′} ∪ Y; j_{∗}j^{∗}DK) ≅
≅ DRΓ(𝔸_{<r′′} − (𝔸_{≤r′} ∪ Y); j_{!}j^{∗}K)

We have by definition RΓ(𝔸_{<r′′} − (𝔸_{≤r′} ∪ Y); j_{!}j^{∗}K) = RΓ(𝔸_{(r′,r′′)} − Y, S_{F}; K)

3.8.1. Lemma. Set 𝔸′_{ℝ} := 𝔸_{ℝ} − {0}. We have RΓ(𝔸 − i · 𝔸_{ℝ}, 𝔸′_{ℝ}; K) = 0. Proof. We have to prove that the restriction map

RΓ(𝔸 − i · 𝔸_{ℝ}; K) ⟶ RΓ(𝔸′_{ℝ}; K)  (21)

is an isomorphism. Consider projection to the real part

π : 𝔸 − i · 𝔸_{ℝ} ⟶ 𝔸′_{ℝ}.

Evidently,

RΓ(𝔸 − i · 𝔸_{ℝ}; K) = RΓ(𝔸′_{ℝ}, π_{∗}K).

On the other hand, since K is conic along fibers of π, we have

π_{∗}K ⟶^{∼} e^{∗}K

where e : 𝔸′_{ℝ} ⟶ 𝔸 − i · 𝔸_{ℝ} denotes the embedding (cf. 2.13). This implies our lemma. □ It follows easily that Cone(res) is acyclic, therefore the map res is an isomorphism. In other words, we have constructed an isomorphism

DΦ_{F} (K) ⟶^{∼} RΓ(𝔸_{≤r′}, Y ∩ 𝔸_{≤r′}; DK).

A clockwise rotation by π/2 induces an isomorphism

RΓ(𝔸_{≤r′}, Y ∩ 𝔸_{≤r′}; DK) ≅ RΓ(𝔸_{≤r′}, ϵ · D_{F}^{opp}; DK),

and the last complex is isomorphic to Φ_{F} (DK) by dilatation and 3.7 (3). This proves the theorem for Φ_{F}. Note that we have constructed an explicit isomorphism.

3.8.2. If F is an arbitrary facet, we consider a transversal slice L as in 3.4. By the results of loc.cit., and the above proven case, we have natural isomorphisms

DΦ_{F} (K) ⟶^{∼} DΦ_{{^{F} w}}(K_{L}) ⟶^{∼} Φ_{{^{F} w}}(DK_{L}) ⟶^{∼} Φ_{{^{F} w}}((DK)_{L}) ⟶^{∼} Φ_{F} (DK).

This proves the theorem. □

3.9. Theorem. For every M ∈ M(𝔸; S_{H}) and every facet F we have H^{i}(Φ_{F} (M)) = 0 for all i ≠ 0. In other words, functors Φ_{F} are t-exact with respect to the middle perversity. Proof. First let us suppose that H is central and F is its 0-dimensional facet. Let us prove that Φ_{F} is right exact, that is, H^{i}(Φ_{F} (M)) = 0 for i > 0 and every M as above. In fact, we know that 3.9.1. if S is any stratum of S_{H} of dimension p then M|_{S} ∈ D^{≤−p}(S) by the condition of perversity. In particular, RΓ(𝔸; M) = i_{F}^{∗} M ∈ D^{≤0}({pt}). On the other hand, one deduces from 3.9.1 that RΓ(S_{F}; M) ∈ D^{≤−1}({pt}). In fact, by definition S_{F} is a union of certain simplices Δ, all of whose edges lie in strata of positive dimension. This implies that RΓ(Δ; M|_{Δ}) ∈ D^{≤−1}({pt}), and one concludes by Mayer-Vietoris argument, using similar estimates for intersections of simplices. Consequently we have

Φ_{F} (M) ≅ RΓ(𝔸, S_{F}; M) ∈ D^{≤0}({pt}),

as was claimed. On the other hand by Duality theorem 3.5 we have an opposite inequality, which proves that Φ_{F} is exact in our case. The case of an arbitrary facet is reduced immediately to the central one by noting that an operation of the restriction to a transversal slice composed with a shift by its codimension is t-exact and using (16). The theorem is proved. □

3.10. By the above theorem, the restriction of functors Φ_{F} to the abelian subcategory M(𝔸, S_{H}) lands in subcategory Vect ⊂ D({pt}). In other words, we get exact functors

Φ_{F} : M(𝔸, S_{H}) ⟶ Vect  (22)

These functors commute with Verdier duality. We will also use the notation M_{F} for Φ_{F} (M).

3.11. Canonical and variation maps. Suppose we have a facet E. Let us denote by F⊣⌋^{1}(E) the set of all facets F such that E < F, dim F = dim E + 1. We have

S_{E} = ⋃_{F ∈F⊣⌋^{1}(E)} D_{F}  (23)

Suppose we have K ∈ D(𝔸, S_{H}).

3.11.1. Lemma. We have a natural isomorphism

RΓ(S_{E}, ⋃_{F ∈F⊣⌋^{1}(E)} S_{F}; K) ≅ ⊕_{F ∈F⊣⌋^{1}E}RΓ(D_{F} , S_{F}; K)

Proof. Note that S_{E}−⋃_{F ∈F⊣⌋^{1}(E)} S_{F} = ⋃_{F ∈F⊣⌋^{1}(E)} D^{∘}_{F} (disjoint union). The claim follows now from the Poincaré duality. □

Therefore, for any F ∈ F⊣⌋^{1}(E) we get a natural inclusion map

i_{E}^{F} : RΓ(D_{F} , S_{F}; K) ↪ RΓ(S_{E}, ⋃_{F′∈F⊣⌋^{1}(E)} S_{F′}; K)  (24)

Let us define a map

u_{E}^{F}(K) : Φ_{F} (K) ⟶ Φ_{E}(K)

as a composition

RΓ(D_{F} , S_{F}; K)[−p] ⟶^{i_{E}^{F}} RΓ(S_{E}, ⋃_{F′∈F⊣⌋^{1}(E)} S_{F′}; K)[−p] ⟶ RΓ(S_{E}; K)[−p] ⟶ RΓ(D_{E}, S_{E})[−p+1]

where the last arrow is the coboundary map for the couple (S_{E}, D_{E}), and the second one is evident.

This way we get a natural transormation

u_{E}^{F} : Φ_{F} ⟶ Φ_{E}  (25)

which will be called a canonical map.

We define a variation map

v_{F}^{E} : Φ_{E} ⟶ Φ_{F}  (26)

as follows. By definition, v_{F}^{E}(K) is the map dual to the composition

DΦ_{F} (K) ⟶^{∼} Φ_{F} (DK) ⟶^{u_{E}^{F}(DK)} Φ_{E}(DK) ⟶^{∼} DΦ_{E}(K).

3.12. Lemma. Suppose we have 4 facets A, B_{1}, B_{2}, C such that A < B_{1} < C, A < B_{2} < C and dim A = dim B_{i} − 1 = dim C − 2 (see Fig. 2). Then

u_{A}^{B_{1}} ∘ u_{B_{1}}^{C} = −u_{A}^{B_{2}} ∘ u_{B_{2}}^{C}

and

v_{C}^{B_{1}} ∘ v_{B_{1}}^{A} = −v_{C}^{B_{2}} ∘ v_{B_{2}}^{A}.

For a proof, see below, 3.13.2.

[figure: B_{2}; B_{1}; A; C; Fig. 2]

3.13. Cochain complexes. For each integer p, 0 ≤ p ≤ N, and M ∈ M(𝔸, S_{H}) introduce vector spaces

C_{H}^{−p}(𝔸; M) = ⊕_{F:dim F =p} M_{F}  (27)

For i > 0 or i < −N set C_{H}^{i}(𝔸; M) = 0.

Define operators

d : C_{H}^{−p}(𝔸; M) ⟶ C_{H}^{−p+1}(𝔸; M)

having components u_{E}^{F}.

3.13.1. Lemma. d^{2} = 0.

Proof. Let us denote X_{p} := ⋃_{F:dim F =p} D_{F}. We have X_{p} ⊃ X_{p+1}. Evident embeddings of couples (D_{F} , S_{F}) ↪ (X_{p}, X_{p+1}) induce maps

RΓ(X_{p}, X_{p+1}; M) ⟶ ⊕_{F:dim F =p} RΓ(D_{F} , S_{F}; M)

which are easily seen to be isomorphisms. Thus, we can identify C^{−p}(𝔸; M) with RΓ(X_{p}, X_{p+1}; M). In these terms, d is a boundary homomorphism for the triple X_{p} ⊃ X_{p+1} ⊃ X_{p+2}. After this description, the equality d^{2} = 0 is a general fact from homological algebra. □

3.13.2. Proof of 3.12. The above lemma is equivalent to the statement of 3.12 about maps u, which is thus proven. The claim for variation maps follows by duality. □

This way we get a complex C_{H}^{•}(𝔸; M) lying in degrees from −N to 0. It will be called the cochain complex of our arrangement H_{ℝ} with coefficients in M.

3.14. Theorem. (i) A functor

M ↦ C_{H}^{•}(𝔸; M)

is an exact functor from M(𝔸; S_{H}) to the category of complexes of vector spaces.

(ii) We have a canonical natural isomorphism in D({pt})

C_{H}^{•}(𝔸; M) ⟶^{∼} RΓ(𝔸; M)

Proof. (i) is obvious from the exactness of functors Φ_{F}, cf. Thm. 3.9. To prove (ii), let us consider the filtration

𝔸 ⊃ X_{0} ⊃ X_{1} ⊃ … X_{N} ⊃ 0.

It follows easily from homotopy argument (cf. 2.12, 2.13) that the restriction

RΓ(𝔸; M) ⟶ RΓ(X_{0}; M)

is an isomorphism. On the other hand, a ”Cousin” interpretation of C_{H}^{•}(𝔸; M) given in the proof of Lemma 3.13.1, shows that one has a canonical isomorphism RΓ(X_{0}; M) ⟶^{∼} C_{H}^{•}(𝔸; M). □

4. Computations for standard sheaves

4.1. Suppose we have a connected locally simply connected topological space X and a subspace Y ⊂ X such that each connected component of Y is simply connected. Recall that a groupoid is a category all of whose morphisms are isomorphisms. Let us define a Poincaré groupoid π_{1}(X; Y) as follows.

We set Obπ_{1}(X; Y) = π_{0}(Y). To define morphisms, let us choose a point y_{i} on each connected component Y_{i} ⊂ Y. By definition, for two connected components Y_{i} and Y_{j}, the set of homomorphisms Hom_{π_{1}(X,Y)}(Y_{i}, Y_{j}) is the set of all homotopy classes of paths in X starting at y_{i} and ending at y_{j}.

A different choice of points y_{i} gives a canonically isomorphic groupoid. If Y is reduced to one point we come back to a usual definition of the fundamental group.

Given a local system L on X, we may assign to it a ”fiber” functor

F_{L} : π_{1}(X; Y) ⟶ Vect,

carrying Y_{i} to the fiber L_{y_{i}}. This way we get an equivalence of the category of local systems on X and the category of functors π_{1}(X, Y) ⟶ Vect.

4.2. Return to the situation of the previous section. It is known (cf. [Br]) that the

homology group H_{1}(𝔸^{∘}_{H}; ℤ) is a free abelian group with a basis consisting of classes of small loops around hyperplanes H ∈ H. Consequently, for each map

q : H ⟶ ℂ^{∗}  (28)

there exists a one-dimensional local system L(q) whose monodromy around H ∈ H is equal to q(H). Such a local system is unique up to a non-unique isomorphism.

Let us construct such a local system explicitely, using a language of the previous subsection.

4.3. From now on we fix a real equation for each H ∈ H, i.e. a linear function f_{H} : 𝔸_{ℝ} ⟶ ℝ such that H_{ℝ} = f^{−1}(0). We will denote also by f_{H} the induced function 𝔸 ⟶ ℂ.

The hyperplane H_{ℝ} divides 𝔸_{ℝ} into two halfspaces: 𝔸_{ℝ,H}^{+} = {x ∈ 𝔸_{ℝ}|f_{H}(x) > 0} and 𝔸_{ℝ,H}^{−} = {x ∈ 𝔸_{ℝ}|f_{H}(x) < 0}.

Let F ⊂ H_{ℝ} be a facet of dimension N − 1. We have two chambers F_{±} adjacent to F, where F_{±} ⊂ 𝔸_{ℝ,H}^{±}. Pick a point w ∈ F. Let us choose a real affine line l_{ℝ} ⊂ 𝔸_{ℝ} transversal to H_{ℝ} and passing through w. Let l denote its complexification.

The function f_{H} induces isomorphism l ⟶^{∼} ℂ, and f_{H}^{−1}(ℝ) ∩ l = l_{ℝ}. Let us pick a real ϵ > 0 such that two points f_{H}^{−1}(±ϵ) ∩ l_{ℝ} lie in F_{±} respectively. Denote these points by w_{±}.

[figure: H_{ℝ}; F; w; w_{+}; w_{−}; F_{−}; F_{+}; Fig. 3]

Let us denote by τ^{+} (resp., τ^{−}) a counterclockwise (resp., clockwise) path in the upper (resp., lower) halfplane connecting ϵ with −ϵ. Let us denote

τ_{F}^{±} = f_{H}^{−1}(τ^{±})

This way we get two well-defined homotopy classes of paths connecting chambers F_{+} and F_{−}. The argument arg f_{H} increases by ∓\frac{π}{2} along τ_{F}^{±}.

Note that if H′ is any other hyperplane of our arrangement then arg f_{H′} gets no increase along τ_{F}^{±}.

4.4. Now suppose we have q as in (28). Note that all connected components of 𝔸^{∘}_{H,ℝ} — chambers of our arrangement — are contractible. Let us define a functor

F (q^{2}) : π_{1}(𝔸^{∘}_{H}, 𝔸^{∘}_{H,ℝ}) ⟶ Vect_{ℂ}

as follows. For each chamber C we set F (q^{2})(C) = ℂ. For each facet F of codimension 1 which lies in a hyperplane H we set

F (q^{2})(τ_{F}^{±}) = q(H)^{±1}

It follows from the above remark on the structure of H_{1}(𝔸^{∘}_{H}) that we get a correctly defined functor.

The corresponding abelian local system over 𝔸_{H} will be denoted L(q^{2}); it has a monodromy q(H)^{2} around H ∈ H. If all numbers q(H) belong to some subfield B ⊂ ℂ then the same construction gives a local system of B-vector spaces.

4.5. From now on untill the end of this section we fix a map

q : H ⟶ B^{∗},

B being a subfield of ℂ, and denote by L the local system of B-vector spaces L(q^{2}) constructed above. We denote by L^{−1} the dual local system L(q^{−2}).

Let j : 𝔸^{∘}_{H} ⟶ 𝔸 denote an open embedding. For ? equal to one of the symbols !, ∗ or !∗, let us consider perverse sheaves L_{?} := j_{?}L[N]. They belong to M(𝔸; S_{H}); these sheaves will be called standard extensions of L, or simply standard sheaves. Note that

DL_{!} ≅ L_{∗}^{−1}; DL_{∗} ≅ L_{!}^{−1}  (29)

We have a canonical map

m : L_{!} ⟶ L_{∗},  (30)

and by definition L_{!∗} coincides with its image.

Our aim in this section will be to compute explicitely the cochain complexes of standard sheaves.

4.6. Orientations. Let F be a facet which is a connected component of a real edge L_{F}. Consider a linear space L_{F}^{⊥} = 𝔸_{ℝ}/L_{F}. Let us define

λ_{F} := H^{0}(L_{F}^{⊥}; Or_{L_{F}^{⊥}}),

it is a free abelian group of rank 1. To choose an orientation of L_{F}^{⊥} (as a real vector space) is the same as to choose a basis vector in λ_{F}. We will call an orientation of L_{F}^{⊥} a coorientation of F.

We have an evident piecewise linear homeomorphism of D_{F} = ⋃ D_{F <C} onto a closed disk in L_{F}^{⊥}; thus, a coorientation of F is the same as an orientation of D_{F} (as a C^{0}- manifold); it defines orientations of all cells D_{F <C}.

4.6.1. From now on untill the end of the section, let us fix coorientations of all facets. Suppose we have a pair E < F, dim E = dim F − 1. The cell D_{E} is a part of the boundary of D_{F}. Let us define the sign sgn(F, E) = ±1 as follows. Complete an orienting basis of D_{E} by a vector directed outside D_{F}; if we get the given orientation of D_{F}, set sgn(F, E) = 1, otherwise set sgn(F, E) = −1.

4.7. Basis in Φ_{F} (L_{!})^{∗}. Let F be a facet of dimension p. We have by definition

Φ_{F} (L_{!}) = H^{−p}(D_{F} , S_{F}; L_{!}) = H^{N−p}(D_{F} , S_{F}; j_{!}L) ≅ H^{N−p}(D_{F} , S_{F} ∪ (_{H}H_{ℝ} ∩ D_{F}); j_{!}L).

By Poincaré duality,

H^{N−p}(D_{F} , S_{F} ∪ (_{H}H_{ℝ} ∩ D_{F}); j_{!}L)^{∗} ≅ H^{0}(D_{F} − (S_{F} ∪_{H} H_{ℝ}); L^{−1})

(recall that we have fixed an orientation of D_{F}). The space D_{F} − (S_{F} ∪_{H} H_{ℝ}) is a disjoint union

D_{F} − (S_{F} ∪_{H} H_{ℝ}) = ⋃_{C∈Ch(F)} D^{∘}_{F <C}.

Consequently,

H^{0}(D_{F} − (S_{F} ∪_{H} H_{ℝ}); L^{−1}) ≅ ⊕_{C∈Ch(F)}H^{0}(D^{∘}_{F <C}; L^{−1}).

By definition of L, we have canonical identifications H^{0}(D^{∘}_{F <C}; L^{−1}) = B. We will denote

by c(L_{!})_{F <C} ∈ Φ_{F} (L)^{∗} the image of 1 ∈ H^{0}(D^{∘}_{F <C}; L^{−1}) with respect to the embedding

H^{0}(D^{∘}_{F <C}; L^{−1}) ↪ Φ_{F} (L_{!})^{∗}

following from the above.

Thus, classes c(L_{!})_{F <C}, C ∈ Ch(F), form a basis of Φ_{F} (L_{!})^{∗}.

4.8. Let us describe canonical maps for L_{!}. If F < E, dim E = dim F + 1, let u^{∗} : Φ_{F} (L_{!})^{∗} ⟶ Φ_{E}(L_{!})^{∗} denote the map dual to u_{F}^{E}(L_{!}). Let C be a chamber adjacent to F. Then

u^{∗}(c(L_{!})_{F <C}) = \{ \begin{matrix}sgn(F, E)c(L_{!})_{E<C} & if E < C \\ 0 & otherwise \end{matrix}  (31)

4.9. Basis in Φ_{F} (L_{∗})^{∗}. We have isomorphisms

Φ_{F} (L_{∗}) ≅ Φ_{F} (DL_{!}^{−1}) ≅ Φ_{F} (L_{!}^{−1})^{∗}  (32)

Hence, the defined above basis {c(L_{!}^{−1})_{F <C}}_{C∈Ch(F)} of Φ_{F} (L_{!}^{−1})^{∗}, gives a basis in Φ_{F} (L_{∗}). We will denote by {c(L_{∗})_{F <C}}_{C∈Ch(F)} the dual basis of Φ_{F} (L_{∗}).

4.10. Example. Let us describe our chains explicitely in the simplest one-dimensional case, in the setup 3.6. We choose a natural orientation on 𝔸_{ℝ}. A local system L = L(q^{2}) is uniquely determined by one nonzero complex number q. By definition, the upper (resp., lower) halfplane halfmonodromy from w_{+} to w_{−} is equal to q (resp., q^{−1}).

4.10.1. Basis in Φ_{F} (L_{!})^{∗}. The space Φ_{F} (L_{!})^{∗} admits a basis consisting of two chains c_{±} = c(L_{!})_{F <E_{±}} shown below, see Fig. 3(a). By definition, a homology class is represented by a cell together with a section of a local system L^{−1} over it. The section of L^{−1} over c_{+} (resp., c_{−}) takes value 1 over w_{+} (resp., w_{−}).

[figure: S_{r′′}; w_{+}; w_{−}; c_{−}; c_{+}; Dc_{−}; Dc_{+}; (a); S_{r′′}; Y; w_{+}; w_{−}; c_{−}; c_{+}; D̃c_{+}; D̃c_{−}; (b); Fig. 3]

4.10.2. Basis in Φ_{F} (L_{∗})^{∗}. Let us adopt notations of 3.8, with K = L_{!}^{−1}. It is easy to find the basis {Dc_{+}, Dc_{−}} of the dual space

Φ_{F} (L_{!}^{−1}) ≅ H^{0}(𝔸, 𝔸_{≥r′′}; j^{∗}j_{∗}DL_{!}^{−1})^{∗} = H^{1}(𝔸 − {w_{+}, w_{−}}, 𝔸_{≥r′′}; L)^{∗}

dual to {c(L^{−1})_{F <E_{+}}, c(L^{−1})_{F <E_{−}}}. Namely, Dc_{±} is represented by the relative 1-chain

\{±\frac{1}{2} + y · i| − \sqrt{(r′′)^{2} − \frac{1}{4}} ≤ y ≤ \sqrt{(r′′)^{2} − \frac{1}{4}}\},

with evident sections of L^{−1} over them, see Fig. 3(a).

Next, one has to deform these chains to chains D̃c_{±} with their ends on Y, as in Fig. 3(b). Finally, one has to make a clockwise rotation of the picture by π/2. As a result, we arrive at the following two chains c_{±}^{∗} forming a basis of Φ_{F} (L_{∗})^{∗}:

[figure: w_{+}; w_{−}; c_{−}^{∗}; c_{+}^{∗}; c_{−}; c_{+}; Fig. 4]

The section of L^{−1} over c_{+}^{∗} (resp., c_{−}) has value 1 at w_{+} (resp., w_{−}). It follows from this description that the natural map

m : Φ_{F} (L_{∗})^{∗} ⟶ Φ_{F} (L_{!})^{∗}  (33)

is given by the formulas

m(c_{+}^{∗}) = c_{+} + qc_{−}; m(c_{−}^{∗}) = qc_{+} + c_{−}  (34)

By definition, spaces Φ_{E_{±}}(L_{?})^{∗} may be identified with fibers L_{w_{±}} respectively, for both ? =! and ? = ∗, and hence with B. Let us denote by c_{w_{±}} and c_{w_{±}}^{∗} the generators corresponding to 1 ∈ B. It follows from the above description that the canonical maps u^{∗} are given by the formulas

u^{∗}(c_{+}) = c_{w_{+}}; u^{∗}(c_{−}) = −c_{w_{−}};  (35)

and

u^{∗}(c_{+}^{∗}) = c_{w_{+}}^{∗} − qc_{w_{−}}^{∗}; u^{∗}(c_{−}^{∗}) = qc_{w_{+}}^{∗} − c_{w_{−}}^{∗};  (36)

Let us compute variation maps. To get them for the sheaf L_{!}, we should by definition replace q by q^{−1} in (36) and take the conjugate map:

v^{∗}(c_{w_{+}}) = c_{+} + q^{−1}c_{−}; v^{∗}(c_{w_{−}}) = −q^{−1}c_{+} − c_{−}  (37)

To compute v^{∗} for L_{∗}, note that the basis in

H^{0}(𝔸, {w_{+}, w_{−}}; L_{!}^{−1})^{∗} = H_{1}(𝔸, {w_{+}, w_{−}, 0}; L)

dual to {c_{+}^{∗}, c_{−}^{∗}}, is {q^{−1}c̃_{−}, q^{−1}c̃_{+}} (sic!) where c̃_{±} denote the chains defined in the same way as c_{±}, with L replaced by L^{−1}. From this remark it follows that

v^{∗}(c_{w_{+}}^{∗}) = q^{−1}c_{−}^{∗}; v^{∗}(c_{w_{−}}^{∗}) = −q^{−1}c_{+}^{∗}  (38)

4.11. Let us return to the case of an arbitrary arrangement. Let us say that a hyperplane H ∈ H separates two chambers C, C′ if they lie in different halfspaces with respect to H_{ℝ}. Let us define numbers

q(C, C′) = ∏ q(H),  (39)

the product over all hyperplanes H ∈ H separating C and C′. In particular, q(C, C) = 1.

4.12. Lemma. Let F be a face, C ∈ Ch(F). The canonical mapping

m : Φ_{F} (L_{∗})^{∗} ⟶ Φ_{F} (L_{!})^{∗}

is given by the formula

m(c(L_{∗})_{F <C}) = ∑_{C′∈Ch(F)} q(C, C′)c(L_{!})_{F <C′}  (40)

4.12.1. Since Φ_{F} (L_{∗}) is dual to Φ_{F} (L_{!}), we may view m as a bilinear form on Φ_{F} (L_{!}). By (40) it is symmetric.

Proof of lemma. We generalize the argument of the previous example. First consider the case of zero-dimensional F. Given a chain c(L_{!}^{−1})_{F <C}, the corresponding dual chain may be taken as

Dc_{F <C} = ϵ · ^{C}w ⊕ i · 𝔸_{ℝ},

were ϵ is a sufficiently small positive real. Next, to get the dual chain c(L_{∗})_{F <C}, we should make a deformation similar to the above one, and a rotation by \frac{π}{2}. It is convenient to make the rotation first. After the rotation, we get a chain 𝔸_{ℝ} − ϵi · ^{C}w. The value of m(c(L_{∗})_{F <C}) is given by the projection of this chain to 𝔸_{ℝ}.

The coefficient at c(L_{!})_{F <C′} is given by the monodromy of L^{−1} along the following path from C to C′. First, go ”down” from ^{C}w to −ϵi · ^{C}w; next, travel in 𝔸_{ℝ} − ϵi · ^{C}w along the straight line from −ϵi · ^{C}w to −ϵi ·^{C′} w, and then go ”up” to ^{C′}w. Each time we are passing under a hyperplane H_{ℝ} separating C and C′, we gain a factor q(H). This gives desired coefficient for the case dim F = 0.

For an arbitrary F we use the same argument by considering the intersection of our picture with a transversal slice. □

4.13. Let E be a facet which is a component of a real edge L_{E,ℝ}; as usually L will denote the complexification. Let H_{L} ⊂ H be a subset consisting of all hyperplanes containg L. If we assign to a chamber C ∈ Ch(E) a unique chamber of the subarrangement H_{L} comtaining C, we get a bijection of Ch(E) with the set of all chambers of H_{L}.

Let F < E be another facet. Each chamber C ∈ Ch(F) is contained in a unique chamber of H_{L}. Taking into account a previous bijection, we get a mapping

π_{E}^{F} : Ch(F) ⟶ Ch(E)  (41)

4.14. Lemma. Let F be a facet, C ∈ Ch(F). We have

u^{∗}(c(L_{∗})_{F <C}) = ∑ sgn(F, E)q(C, π_{E}^{F}(C))c(L_{∗})_{E<π_{E}^{F}(C)},  (42)

the summation over all facets E such that F < E and dim E = dim F + 1.

(Signs sgn(F, E) have been defined in 4.6.1.)

Proof. Again, the crucial case is dim F = 0 — the case of arbitrary dimension is treated using a transversal slice. So, let us suppose that F is zero-dimensional. In order to compute the coefficient of u^{∗}(c(L_{∗})_{F <C}) at c(L^{∗})_{E<C′} where E is a one-dimensional facet adjacent to F and C′ ∈ Ch(E), we have to do the following.

Consider the intersection of a real affine subspace 𝔸_{ℝ} − ϵ · i · ^{C}w (as in the proof of the previous lemma) with a complex hyperplane M_{E} passing through ^{E}w and transversal to E. The intersection will be homotopic to a certain chain c(L_{∗})_{E<C′′} where C′′ is easily seen to be equal to π_{E}^{F}(C), and the coefficient is obtained by the same rule as described in the previous proof. The sign will appear in accordance with compatibility of orientations of D_{F} and D_{E}. □

4.15. Let us collect our results. Let us denote by {b(L_{?})_{F <C}}_{C∈Ch(F)} the basis in Φ_{F} (L_{?}) dual to {c(L_{?})_{F <C}}, where ? =! or ∗.

4.16. Theorem. (i) The complex C_{H}^{•}(𝔸; L_{!}) is described as follows. For each p, 0 ≤ p ≤ N, the space C_{H}^{−p}(𝔸; L_{!}) admits a basis consisting of all cochains b(L_{!})_{F <C} where F runs through all facets of H_{ℝ} of dimension p, and C through Ch(F). The differential

d : C_{H}^{−p}(𝔸; L_{!}) ⟶ C_{H}^{−p+1}(𝔸; L_{!})

is given by the formula

d(b(L_{!})_{F <C}) = ∑_{E:E<F, dim E=dim F −1} sgn(E, F)b(L_{!})_{E<C}  (43)

(ii) The complex C_{H}^{•}(𝔸; L_{∗}) is described as follows. For each p, 0 ≤ p ≤ N, the space C_{H}^{−p}(𝔸; L_{∗}) admits a basis consisting of all cochains b(L_{∗})_{F <C} where F runs through all facets of H_{ℝ} of dimension p, and C through Ch(F). The differential

d : C_{H}^{−p}(𝔸; L_{∗}) ⟶ C_{H}^{−p+1}(𝔸; L_{∗})

is given by the formula

d(b(L_{∗})_{F <C}) = ∑ sgn(E, F)q(C, C′)b(L_{∗})_{E<C′},  (44)

the summation over all facets E < F such that dim E = dim F − 1 and all chambers C′ ∈ Ch(E) such that π_{F}^{E}(C′) = C.

(iii) The natural map of complexes

m : C_{H}^{•}(𝔸; L_{!}) ⟶ C_{H}^{•}(𝔸; L_{∗})  (45)

induced by the canonical map L_{!} ⟶ L_{∗}, is given by the formula

m(b(L_{!})_{F <C}) = ∑_{C′∈Ch(F)} q(C, C′)b(L_{∗})_{F <C′}  (46)

All statements have already been proven.

4.16.1. Corollary. The complexes C_{H}^{•}(𝔸; L_{!}) and C_{H}^{•}(𝔸; L_{∗}) described explicitely in the above theorem, compute the relative cohomology H^{•}(𝔸, _{H}H; L) and the cohomology of

the open stratum H^{•}(𝔸^{∘}, L) respectively, and the map m induces the canonical map in cohomology.

Proof. This follows immediately from 3.14. □

This corollary was proven in [V], Sec. 2, by a different argument.

4.17. Theorem. The complex C_{H}^{•}(𝔸; L_{!∗}) is canonically isomorphic to the image of (45).

Proof. This follows from the previous theorem and the exactness of the functor M ↦ C_{H}^{•}(𝔸; M), cf. Thm. 3.14 (i). □

The above description of cohomology is analogous to [SV1], p. I, whose results may be considered as a ”quasiclassical” version of the above computations.

Part II. CONFIGURATION SPACES

AND QUANTUM GROUPS

1. Introduction

1.1. We are starting here the geometric study of the tensor category C associated with a quantum group (corresponding to a Cartan matrix of finite type) at a root of unity (see [AJS], 1.3 and the present paper, 11.3 for the precise definitions).

The main results of this paper are Theorems 8.18, 8.23, 12.7 and 12.8 which

— establish isomorphisms between homogeneous components of irreducible objects in C and spaces of vanishing cycles at the origin of certain Goresky-MacPherson sheaves on configuration spaces;

— establish isomorphisms of the stalks at the origin of the above GM sheaves with certain Hochschild complexes (which compute the Hochschild homology of a certain ”triangular” subalgebra of our quantum group with coefficients in the coresponding irreducible representation);

— establish the analogous results for tensor products of irreducibles. In geometry, the tensor product of representations corresponds to a ”fusion” of sheaves on configuration spaces — operation defined using the functor of nearby cycles, see Section 10.

We must mention that the assumption that we are dealing with a Cartan matrix of finite type and a root of unity appears only at the very end (see Chapter 4). We need these assumptions in order to compare our representations with the conventional definition of the category C. All previous results are valid in more general assumptions. In particular a Cartan matrix could be arbitrary and a deformation parameter ζ not necessarily a root of unity.

1.2. A part of the results of this paper constitutes the description of the cohomology of certain ”standard” local systems over configuration spaces in terms of quantum groups. These results, due to Varchenko and one of us, were announced several years ago in [SV2]. The proofs may be found in [V]. Our proof of these results uses completely different approach. Some results close to this paper were discussed in [S].

Certain results of a similar geometric spirit are discussed in [FW].

1.3. We are grateful to A.Shen who made our communication during the writing of this paper possible.

1.4. Notations. We will use all the notations from the part I. References to loc. cit. will look like I.1.1. If a, b are two integers, we will denote by [a, b] the set of all integers c such that a ≤ c ≤ b; [1, a] will be denoted by [a]. ℕ will denote the set of non-negative integers. For r ∈ ℕ, Σ_{r} will denote the group of all bijections [r] ⟶^{∼} [r]. We suppose that our ground field B has characteristic 0, and fix an element ζ ∈ B, ζ ≠ 0. For a ∈ ℤ we will use the notation

[a]_{ζ} = 1 − ζ^{−2a}  (47)

The word ”t-exact” will allways mean t-exactness with respect to the middle perversity.

Chapter 1. Algebraic discussion

2. Free algebras and bilinear forms

Most definitions of this section follow [L1] and [SV2] (with slight modifications). We also add some new definitions and computations important for the sequel. Cf. also [V], Section 4.

2.1. Untill the end of this paper, let us fix a finite set I and a symmetric ℤ-valued bilinear form ν, ν′ ↦ ν · ν′ on the free abelian group ℤ[I] (cf. [L1], 1.1). We will denote by X the dual abelian group Hom(ℤ[I], ℤ). Its elements will be called weights. Given ν ∈ ℤ[I], we will denote by λ_{ν} ∈ X the functional i ↦ i · ν. Thus we have

⟨λ_{ν}, μ⟩ = ν · μ  (48)

for all ν, μ ∈ ℕ[I].

2.2. Let 𝔉 denote a free associative B-algebra with 1 with generators θ_{i}, i ∈ I. Let ℕ[I] be a submonoid of ℤ[I] consisting of all linear combinations of elements of I with coefficients in ℕ. For ν = ∑ ν_{i}i ∈ ℕ[I] we denote by 𝔉_{ν} the B-subspace of 𝔉 spanned by all monomials θ_{i_{1}}θ_{i_{2}} · . . . · θ_{i_{p}} such that for any i ∈ I, the number of occurences of i in the sequence i_{1}, …, i_{p} is equal to ν_{i}.

We have a direct sum decomposition 𝔉 = ⊕_{ν∈ℕ[I]}𝔉_{ν}, all spaces 𝔉_{ν} are finite dimensional, and we have 𝔉_{0} = B · 1, 𝔉_{ν} · 𝔉_{ν′} ⊂ 𝔉_{ν+ν′}.

Let ϵ : 𝔉 ⟶ B denote the augmentation — a unique B-algebra map such that ϵ(1) = 1 and ϵ(θ_{i}) = 0 for all i. Set 𝔉^{+} := Ker(ϵ). We have 𝔉^{+} = ⊕_{ν≠0}𝔉_{ν}.

An element x ∈ 𝔉 is called homogeneous if it belongs to 𝔉_{ν} for some ν. We then set |x| = ν. We will use the notation depth(x) for the number ∑_{i} ν_{i} if ν = ∑_{i} ν_{i}i; it will be called the depth of x.

2.3. Given a sequence K⃗ = (i_{1}, …, i_{N}), i_{j} ∈ I, let us denote by θK⃗ the monomial θ_{i_{1}} · . . . · θ_{i_{N}}. For an empty sequence we set θ_{∅} = 1.

For τ ∈ Σ_{N} let us introduce the number

ζ(K⃗; τ) = ∏ ζ^{i_{a}·i_{b}},  (49)

the product over all a, b such that 1 ≤ a < b ≤ N and τ(a) > τ(b).

We will call this number the twisting number of the sequence K⃗ with respect to the permutation τ.

We will use the notation

τ(K⃗) = (i_{τ(1)}, i_{τ(2)}, …, i_{τ(N)})  (50)

2.4. Let us regard the tensor product 𝔉⊗𝔉 (in the sequel ⊗ will mean ⊗_{B} unless specified otherwise) as a B-algebra with multiplication

(x_{1} ⊗ x_{2}) · (x′_{1} ⊗ x′_{2}) = ζ^{|x_{2}|·|x′_{1}|}x_{1}x′_{1} ⊗ x_{2}x′_{2}  (51)

for homogeneous x_{2}, x′_{1}. Let us define a map

Δ : 𝔉 ⟶ 𝔉 ⊗ 𝔉  (52)

as a unique algebra homomorphism carrying θ_{i} to θ_{i} ⊗ 1 + 1 ⊗ θ_{i}.

2.5. Let us define a coalgebra structure on 𝔉⊗𝔉 as follows. Let us introduce the braiding isomorphism

r : 𝔉 ⊗ 𝔉 ⟶^{∼} 𝔉 ⊗ 𝔉  (53)

by the rule

r(x ⊗ y) = ζ^{|x|·|y|}y ⊗ x  (54)

for homogeneous x, y. By definition,

Δ_{𝔉⊗𝔉} : 𝔉 ⊗ 𝔉 ⟶ (𝔉 ⊗ 𝔉) ⊗ (𝔉 ⊗ 𝔉)  (55)

coincides with the composition (1_{𝔉} ⊗ r ⊗ 1_{𝔉}) ∘ (Δ_{𝔉} ⊗ Δ_{𝔉}).

The multiplication

𝔉 ⊗ 𝔉 ⟶ 𝔉  (56)

is a coalgebra morphism.

2.6. Let us describe Δ more explicitely. Suppose a sequence K⃗ = (i_{1}, …, i_{N}), i_{j} ∈ I, is given. For a subset A = {j_{1}, …, j_{a}} ⊂ [N], j_{1} < … < j_{a}, let A′ = [N] − A = {k_{1}, …, k_{N−a}}, k_{1} < … < k_{N−a}. Define a permutation τ_{A} by the formula

(τ(1), …, τ(N)) = (j_{1}, j_{2}, …, j_{a}, k_{1}, k_{2}, …, k_{N−a})  (57)

Set K⃗_{A} := (i_{j_{1}}, i_{j_{2}}, …, i_{j_{a}}), K⃗_{A′} := (i_{k_{1}}, i_{k_{2}}, …, i_{k_{N−a}}).

2.6.1. Lemma.

Δ(θK⃗) = ∑_{A⊂K} ζ(K⃗; τ_{A})θK⃗_{A} ⊗ θK⃗_{A′},

the summation ranging over all subsets A ⊂ [N].

Proof follows immediately from the definitions. □

2.7. Let us denote by

Δ^{(N)} : 𝔉 ⟶ 𝔉^{⊗N}  (58)

iterated coproducts; by the coassociativity they are well defined.

Let us define a structure of an algebra on 𝔉^{⊗N} as follows:

(x_{1} ⊗ … ⊗ x_{N}) · (y_{1} ⊗ … ⊗ y_{N}) = ζ^{∑_{j<i} |x_{i}|·|y_{j}|}x_{1}y_{1} ⊗ … ⊗ x_{N}y_{N}  (59)

for homogeneous x_{1}, …, x_{N}; y_{1}, …, y_{N}. The map Δ^{(N)} is an algebra morphism.

2.8. Suppose we have a sequence K⃗ = (i_{1}, …, i_{N}). Let us consider an element Δ^{(N)}(θK⃗); let Δ^{(N)}(θK⃗)^{+} denote its projection to the subspace 𝔉^{+⊗N}.

2.8.1. Lemma.

Δ^{(N)}(θK⃗)^{+} = ∑_{τ∈Σ_{N}} ζ(K⃗; τ)θ_{i_{τ(1)}} ⊗ … ⊗ θ_{i_{τ(N)}}

Proof follows from 2.6.1 by induction on N. □

2.9. For each component 𝔉_{ν} consider the dual B-space 𝔉_{ν}^{∗}, and set 𝔉^{∗} := ⊕𝔉_{ν}^{∗}. Graded components Δ_{ν,ν′} : 𝔉_{ν+ν′} ⟶ 𝔉_{ν} ⊗ 𝔉_{ν′} define dual maps 𝔉_{ν}^{∗} ⊗ 𝔉_{ν′}^{∗} ⟶ 𝔉_{ν+ν′}^{∗} which give rise to a multiplication

𝔉^{∗} ⊗ 𝔉^{∗} ⟶ 𝔉^{∗}  (60)

making 𝔉^{∗} a graded associative algebra with 1 (dual to the augmentation of 𝔉). This follows from the coassociativity of Δ, cf. [L1], 1.2.2.

Here and in the sequel, we will use identifications (V ⊗ W)^{∗} = V^{∗} ⊗ W^{∗} (for finite dimensional spaces V, W) by the rule ⟨ϕ ⊗ ψ, x ⊗ y⟩ = ⟨ϕ, x⟩ · ⟨ψ, y⟩.

The dual to (56) defines a comultiplication

δ : 𝔉^{∗} ⟶ 𝔉^{∗} ⊗ 𝔉^{∗}  (61)

It makes 𝔉^{∗} a graded coassociative coalgebra with a counit.

The constructions dual to 2.4 and 2.5 equip 𝔉^{∗} ⊗ 𝔉^{∗} with a structure of a coalgebra and an algebra. It follows from loc. cit that (60) is a coalgebra morphism, and δ is an algebra morphism.

By iterating δ we get maps

δ^{(N)} : 𝔉^{∗} ⟶ 𝔉^{∗⊗N}  (62)

If we regard 𝔉^{∗⊗N} as an algebra by the same construction as in (59), δ^{(N)} is an algebra morphism.

2.10. Lemma. There exists a unique bilinear form

S( , ) : 𝔉 ⊗ 𝔉 ⟶ B

such that

(a) S(1, 1) = 1 and (θ_{i}, θ_{j}) = δ_{i,j} for all i, j ∈ I;

(b) S(x, y′y′′) = S(Δ(x), y′ ⊗ y′′) for all x, y′, y′′ ∈ 𝔉;

(c) S(xx′, y′′) = S(x ⊗ x′, Δ(y′′)) for all x, x′, y′′ ∈ 𝔉.

(The bilinear form

(𝔉 ⊗ 𝔉) ⊗ (𝔉 ⊗ 𝔉) ⟶ B

given by

(x_{1} ⊗ x_{2}) ⊗ (y_{1} ⊗ y_{2}) ↦ S(x_{1}, y_{1})S(x_{2}, y_{2})

is denoted again by S( , ).)

The bilinear form S( , ) on 𝔉 is symmetric. The different homogeneous components 𝔉_{ν} are mutually orthogonal.

Proof. See [L1], 1.2.3. Cf. also [SV2], (1.8)-(1.11). □

2.11. Following [L1], 1.2.13 and [SV2], (1.10)-(1.11), let us introduce operators δ_{i} : 𝔉 ⟶ 𝔉, i ∈ I, as unique linear mappings satisfying

δ_{i}(1) = 0; δ_{i}(θ_{j}) = δ_{i,j}, j ∈ I; δ_{i}(xy) = δ_{i}(x)y + ζ^{|x|·i}xδ_{i}(y)  (63)

for homogeneous x.

It follows from 2.10 (c) that

S(θ_{i}x, y) = S(x, δ_{i}(y))  (64)

for all i ∈ I, x, y ∈ 𝔉, and obviously S is determined uniquely by this property, together with the requirement S(1, 1) = 1.

2.12. Lemma. For any two sequences K⃗, K⃗′ of N elements from I we have

S(θK⃗, θK⃗′) = ∑_{τ∈Σ_{N}: τ(K⃗)=K⃗′} ζ(K⃗; τ).

Proof follows from (64) by induction on N, or else from 2.8.1. □

2.13. Let us define elements θ_{i}^{∗} ∈ 𝔉_{i}^{∗} by the rule < θ_{i}^{∗}, θ_{i} >= 1. The form S defines a homomomorphism of graded algebras

S : 𝔉 ⟶ 𝔉^{∗}  (65)

carrying θ_{i} to θ_{i}^{∗}. S is determined uniquely by this property.

2.14. Lemma. The map S is a morphism of coalgebras. Proof. This follows from the symmetry of S. □

VERMA MODULES

2.15. Let us pick a weight Λ. Our aim now will be to define certain X-graded vector space V (Λ) equipped with the following structures.

(i) A structure of left 𝔉-module 𝔉 ⊗ V (Λ) ⟶ V (Λ); (ii) a structure of left 𝔉-comodule V (Λ) ⟶ 𝔉 ⊗ V (Λ); (iii) a symmetric bilinear form S_{Λ} on V (Λ). As a vector space, we set V (Λ) = 𝔉. We will define on V (Λ) two gradings. The first one, ℕ[I]-grading coincides with the grading on 𝔉. If x ∈ V (Λ) is a homogeneous element, we will denote by depth(x) its depth as an element of 𝔉. The second grading — X-grading — is defined as follows. By definition, we set

V (Λ)_{λ} = ⊕_{ν∈ℕ[I]|Λ−λ_{ν}=λ}𝔉_{ν}

for λ ∈ X. In particular, V (Λ)_{Λ} = 𝔉_{0} = B · 1. We will denote the element 1 in V (Λ) by v_{Λ}. By definition, multiplication

𝔉 ⊗ V (Λ) ⟶ V (Λ)  (66)

coincides with the multiplication in 𝔉. Let us define an X-grading in 𝔉 by setting

𝔉_{λ} = ⊕_{ν∈ℕ[I]|−λ_{ν}=λ}𝔉_{ν}

for λ ∈ X. The map (66) is compatible with both ℕ[I] and X-gradings (we define gradings on the tensor product as usually as a sum of gradings of factors).

2.16. The form S_{Λ}. Let us define linear operators ϵ_{i} : V (Λ) ⟶ V (Λ), i ∈ I, as unique operators such that ϵ_{i}(v_{Λ}) = 0 and

ϵ_{i}(θ_{j}x) = [⟨β, i⟩]_{ζ}δ_{i,j}x + ζ^{i·j}θ_{j}ϵ_{i}(x)  (67)

for j ∈ I, x ∈ V (Λ)_{β}. We define S_{Λ} : V (Λ) ⊗ V (Λ) ⟶ B as a unique linear map such that S_{Λ}(v_{Λ}, v_{Λ}) = 1, and

S_{Λ}(θ_{i}x, y) = S_{Λ}(x, ϵ_{i}(y))  (68)

for all x, y ∈ V (Λ), i ∈ I. Let us list elementary properties of S_{Λ}.

2.16.1. Different graded components V (Λ)_{ν}, ν ∈ ℕ[I], are orthogonal with respect to S_{Λ}. This follows directly from the definition.

2.16.2. The form S_{Λ} is symmetric. This is an immediate corollary of the formula

S_{Λ}(ϵ_{i}(y), x) = S_{Λ}(y, θ_{i}x)  (69)

which in turn is proved by an easy induction on depth(x).

2.16.3. ”Quasiclassical” limit. Let us consider restriction of our form to the homogeneous component V (Λ)_{λ} of depth N. If we divide our form by (ζ − 1)^{N} and formally pass to the limit ζ ⟶ 1, we get the ”Shapovalov” contravariant form as defined in [SV1], 6.4.1. The next lemma is similar to 2.12.

2.17. Lemma. For any K⃗, K⃗′ as in 2.12 we have

S_{Λ}(θK⃗v_{Λ}, θK⃗′v_{Λ}) = ∑_{τ∈Σ_{N}: τ(K⃗)=K⃗′} ζ(K⃗; τ)A(K⃗, Λ; τ)

where

A(K⃗, Λ; τ) = ∏_{a=1}^{N}[⟨Λ − ∑_{b: b<a,τ(b)<τ(a)} λ_{i_{b}}, i_{a}⟩]_{ζ}.

Proof. Induction on N, using definition of S_{Λ}. □

2.18. Coaction. Let us define a linear map

Δ_{Λ} : V (Λ) ⟶ 𝔉 ⊗ V (Λ)  (70)

as follows. Let us introduce linear operators t_{i} : 𝔉^{+} ⊗ V (Λ) ⟶ 𝔉^{+} ⊗ V (Λ), i ∈ I, by the formula

t_{i}(x ⊗ y) = θ_{i}x ⊗ y − ζ^{i·ν−2⟨λ,i⟩} · xθ_{i} ⊗ y + ζ^{i·ν}x ⊗ θ_{i}y  (71)

for x ∈ 𝔉_{ν} and y ∈ V (Λ)_{λ}. By definition,

Δ_{Λ}(θ_{i_{N}} · . . . · θ_{i_{1}}v_{Λ}) = 1 ⊗ θ_{i_{N}} · . . . · θ_{i_{1}}v_{Λ} (72)
+[⟨Λ − λ_{i_{1}} − … − λ_{i_{N−1}}, i_{N}⟩]_{ζ} · θ_{i_{N}} ⊗ θ_{i_{N−1}} · . . . · θ_{i_{1}}v_{Λ}

+ ∑_{j=1}^{N−1}[⟨Λ − λ_{i_{1}} − … − λ_{i_{j−1}}, i_{j}⟩]_{ζ} · t_{i_{N}} ∘ t_{i_{N−1}} ∘ … ∘ t_{i_{j+1}}(θ_{i_{j}} ⊗ θ_{i_{j−1}} · . . . · θ_{i_{1}}v_{Λ})

2.19. Let us define linear operators

ad_{θ_{i},λ} : 𝔉 ⟶ 𝔉, i ∈ I, λ ∈ X  (73)

by the formula

ad_{θ_{i},λ}(x) = θ_{i}x − ζ^{i·ν−2⟨λ,i⟩} · xθ_{i}  (74)

for x ∈ 𝔉_{ν}. Let us note the following relation

(δ_{i} ∘ ad_{θ_{j},λ} − ζ^{i·j} · ad_{θ_{j},λ} ∘ δ_{i})(x) = [⟨λ − λ_{ν}, i⟩]_{ζ}δ_{ij}x  (75)

for x ∈ 𝔉_{ν}, where δ_{i} are operators defined in 2.11, and δ_{ij} the Kronecker symbol.

2.20. Formula for coaction. Let us pick a sequence I⃗ = (i_{N}, i_{N−1}, …, i_{1}). To shorten the notations, we set

ad_{j,λ} := ad_{θ_{i_{j}} ,λ}, j = 1, …, N  (76)

2.20.1. Quantum commutators. For any non-empty subset Q ⊂ [N], set θ_{I⃗,Q} := θI⃗_{Q} where I⃗_{Q} denotes the sequence obtained from I⃗ by omitting all entries i_{j}, j ∈ Q. We will denote 𝔉_{Q} = 𝔉_{ν_{Q}} where ν_{Q} := ∑_{j∈Q} i_{j}. Let us define an element [θ_{I⃗,Q,Λ}] ∈ 𝔉_{Q} as follows. Set

[θ_{I⃗,{j},Λ}] = ζ^{i_{j}·(∑_{k>j} i_{k})}θ_{i_{j}}  (77)

for all j ∈ [N]. Suppose now that card(Q) = l + 1 ≥ 2. Let Q = {j_{0}, j_{1}, …, j_{l}}, j_{0} < j_{1} < … < j_{l}. Define the weights

λ_{a} = Λ − λ_{∑ i_{k}}, a = 1, …, l,

where the summation is over k from 1 to j_{a} − 1, k ≠ j_{1}, j_{2}, …, j_{a−1}. Let us define sequences N⃗ := (N, N − 1, …, 1), Q⃗ = (j_{l}, j_{l−1}, …, j_{0}) and N⃗_{Q} obtained from N⃗ by omitting all entries j ∈ Q. Define the permutation τ_{Q} ∈ Σ_{N} by the requirement

τ_{Q}(N⃗) = Q⃗||N⃗_{Q}

where || denotes concatenation. Set by definition

[θ_{I⃗,Q,Λ}] = ζ(I⃗, τ_{Q}) · ad_{j_{l},λ_{l}} ∘ ad_{j_{l−1},λ_{l−1}} ∘ … ∘ ad_{j_{1},λ_{1}}(θ_{i_{j_{0}}})  (78)

2.20.2. Lemma. We have

Δ_{Λ}(θI⃗v_{Λ}) = 1 ⊗ θI⃗v_{Λ} + ∑_{Q}[⟨Λ − λ_{i_{1}} − λ_{i_{2}} − … − λ_{i_{j(Q)−1}}, i_{j(Q)}⟩]_{ζ} · [θ_{I⃗,Q,Λ}] ⊗ θ_{I⃗,Q}v_{Λ},(79)

the summation over all non-empty subsets Q ⊂ [N], j(Q) denotes the minimal element of Q.

Proof. The statement of the lemma follows at once from the inspection of definition (72), after rearranging the summands. □

Several remarks are in order.

2.20.3. Formula (79) as similar to [S], 2.5.4.

2.20.4. If all elements i_{j} are distinct then the part of the sum in the rhs of (79) corresponding to one-element subsets Q is equal to ∑_{j=1}^{N} θ_{i_{j}} ⊗ ϵ_{i_{j}}(θI⃗v_{Λ}).

2.20.5. ”Quasiclassical” limit. It follows from the definition of quantum commutators that if we divide the rhs of (79) by (ζ − 1)^{N}) and formally pass to the limit ζ ⟶ 1, we get the expression for the coaction obtained in [SV1], 6.15.3.2.

2.21. Let us define the space V (Λ)^{∗} as the direct sum ⊕_{ν}V (Λ)_{ν}^{∗}. We define an ℕ[I]- grading on it as (V (Λ)^{∗})_{ν} = V (Λ)_{ν}^{∗}, and an X-grading as V (Λ)_{λ}^{∗} = ⊕_{ν: Λ−λ_{ν}=λ}V (Λ)_{ν}^{∗}.

The form S_{Λ} induces the map

S_{Λ} : V (Λ) ⟶ V (Λ)^{∗}  (80)

compatible with both gradings.

2.22. Tensor products. Suppose we are given n weights Λ_{0}, …, Λ_{n−1}.

2.22.1. For every m ∈ ℕ we introduce a bilinear form S = S_{m;Λ_{0},... ,Λ_{n−1}} on the tensor product 𝔉^{⊗m} ⊗ V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1}) by the formula

S(x_{1}⊗…⊗x_{m}⊗y_{0}⊗…⊗y_{n−1}, x′_{1}⊗…⊗x′_{m}⊗y′_{0}⊗…⊗y′_{n−1}) = ∏_{i=1}^{m} S(x_{i}, x′_{i}) ∏_{j=0}^{n−1} S_{Λ_{j}}(y_{j}, y′_{j})

(in the evident notations). This form defines mappings

S : 𝔉^{⊗m} ⊗ V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1}) ⟶ 𝔉^{∗⊗m} ⊗ V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗} (81)

2.22.2. We will regard V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1}) as an 𝔉^{⊗n}-module with an action

(u_{0} ⊗ … ⊗ u_{n−1}) · (x_{1} ⊗ … ⊗ x_{n−1}) = ζ^{− ∑_{j<i}⟨λ_{j},ν_{i}⟩}u_{0}x_{0} ⊗ … ⊗ u_{n−1}x_{n−1} (82)

for u_{i} ∈ 𝔉_{ν_{i}}, x_{j} ∈ V (Λ_{j})_{λ_{j}}, cf. (59). Here we regard 𝔉^{⊗n} as an algebra according to the rule of loc. cit.; one checks easily using (48) that we really get a module structure.

Using the iterated comultiplication Δ^{(n)}, we get a structure of an 𝔉-module on V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1}).

2.23. Theorem. We have an identity

S_{Λ}(xy, z) = S_{1;Λ}(x ⊗ y, Δ_{Λ}(z))  (83)

for any x ∈ 𝔉, y, z ∈ V (Λ) and any weight Λ.

2.24. Proof. We may suppose that x, y and z are monomials. Let z = θI⃗v_{Λ} where I⃗ = (i_{N}, …, i_{1}).

(a) Let us suppose first that all indices i_{j} are distinct. We will use the notations and computations from 2.20. The sides of (83) are non-zero only if y is equal to z_{Q} := θ_{I⃗,Q}v_{Λ} for some subset Q ⊂ [N].

Therefore, it follows from Lemma 2.20.2 that it is enough to prove

2.24.1. Lemma. For every non-empty Q ⊂ [N] and x ∈ 𝔉_{Q} we have

S_{Λ}(xz_{Q}, z) = [⟨Λ, i_{j(Q)}⟩ − μ_{Q} · i_{j(Q)}]_{ζ} · S(x, [θ_{I⃗,Q,Λ}]) · S_{Λ}(z_{Q}, z_{Q})  (84)

where j(Q) denotes the minimal element of Q, and

μ_{Q} := ∑_{a=1}^{j(Q)−1} i_{a}.

Proof. If card(Q) = 1 the statement follows from the definiton (77). The proof will proceed by the simultaneous induction by l and N. Suppose that x = θ_{i_{p}} · x′, so x′ ∈ 𝔉_{Q′} where Q′ = Q − {i_{p}}, p = j_{a} for some a ∈ [0, l]. Let us set I⃗′ = I⃗ − {i_{p}}, z′ = z_{{i_{p}}}, so that z_{Q} = z′_{Q′}.

We have

S_{Λ}(θ_{i_{p}}x′ · z_{Q}, z) = S_{Λ}(x′ · z_{Q}, ϵ_{i_{p}}(z)) =  (85)

= [⟨Λ, i_{p}⟩ − (∑_{k<p} i_{k}) · i_{p}]_{ζ} · ζ^{(∑_{k>p} i_{k})·i_{p}} · S(x′ · z_{Q}, z′) =

= [⟨Λ, i_{p}⟩ − (∑_{k<p} i_{k}) · i_{p}]_{ζ} · [⟨Λ, i_{j(Q′)}⟩ − μ_{Q′} · i_{j(Q′)}]_{ζ} · ζ^{(∑_{k>p} i_{k})·i_{p}} ·

·S(x, [θ_{I⃗,Q,Λ}]) · S_{Λ}(z′_{Q′}, z′_{Q′})

by induction hypothesis. On the other hand,

S(θ_{i_{p}} · x′, [θ_{I⃗,Q,Λ}]) = S(x′, δ_{i_{p}}([θ_{I⃗,Q,Λ}])).

Therefore, to complete the induction step it is enough to prove that

[⟨Λ, i_{j(Q)}⟩ − μ_{Q} · i_{j(Q)}]_{ζ} · δ_{i_{p}}([θ_{I⃗,Q,Λ}]) =  (86)

= [⟨Λ, i_{p}⟩ − (∑_{k<p} i_{k}) · i_{p}]_{ζ} · [⟨Λ, i_{j(Q′)}⟩ − μ_{Q′} · i_{j(Q′)}]_{ζ} · ζ^{(∑_{k>p} i_{k})·i_{p}}[θ_{I⃗′,Q′,Λ}]

This formula follows directly from the definition of quantum commutators (78) and formula (75). One has to treat separately two cases: a > 0, in which case j(Q) = j(Q′) = j_{0} and a = 0, in which case j(Q) = j_{0}, j(Q′) = j_{1}. Lemma is proven. □

This completes the proof of case (a).

(b) There are repeating indices in the sequence I⃗. Suppose that θI⃗ ∈ 𝔉_{ν}. At this point we will use symmetrization constructions (and simple facts) from Section 4 below. The reader will readily see that there is no vicious circle. So, this part of the proof must be read after loc.cit.

There exists a finite set J and a map π : J ⟶ I such that ν = ν_{π}. Using compatibility of the coaction and the forms S with symmetrization — cf. Lemmata 4.5 and 4.8 below — our claim is immediately reduced to the analogous claim for the algebra ^{π}𝔉, the module V (^{π}Λ) and homogeneous weight χ_{J} which does not contain multiple indices and therefore follows from (a) above.

This completes the proof of the theorem. □

2.25. Let us pick a weight Λ. We can consider numbers q_{ij} := ζ^{i·j} and r_{i} := ⟨Λ, i⟩, i, j ∈ I as parameters of our bilinear forms.

More precisely, for a given ν ∈ ℕ[I] the matrix elements of the form S (resp., S_{Λ}) on 𝔉_{ν} (resp., on V (Λ)_{Λ−λ_{ν}}) in the standard bases of these spaces are certain universal polynomials of q_{ij} (resp., q_{ij} and r_{i}). Let us denote their determinants by det(S_{ν})(q) and det(S_{Λ,ν})(q; r) respectively. These determinants are polynomials of corresponding variables with integer coefficients.

2.25.1. Lemma. Polynomials det(S_{ν})(q) and det(S_{Λ,ν})(q; r) are not identically zero.

In other words, bilinear forms S and S_{Λ} are non-degenerate for generic values of parameters — ”Cartan matrix” (q_{ij}) and ”weight” (r_{i}).

Proof. Let us consider the form S_{Λ} first. The specialization of the matrix of S_{Λ,ν} at ζ = 1 is the identity matrix. It follows easily that det(S_{V,ν})(q; r) ≠ 0.

Similarly, the matrix of S_{ν} becomes identity at ζ = 0, which implies the generic nondegeneracy. □

2.26. Theorem. Coaction Δ_{Λ} is coassociative, i.e.

(1_{𝔉} ⊗ Δ_{Λ}) ∘ Δ_{Λ} = (Δ ⊗ 1_{V (Λ)}) ∘ Δ_{Λ}.  (87)

Proof. The equality (87) is a polynomial identity depending on parameters q_{ij} and r_{i} of the preceding subsection. For generic values of these parameters it is true due to associativity of the action of 𝔉 an V (Λ), Theorem 2.23 and Lemma 2.25.1. Therefore it is true for all values of parameters. □

2.27. The results Chapter 2 below provide a different, geometric proof of Theorems 2.23 and 2.26. Namely, the results of Section 8 summarized in Theorem 8.21 provide an isomorphism of our algebraic picture with a geometric one, and in the geometrical language the above theorems are obvious: they are nothing but the naturality of the canonical morphism between the extension by zero and the extension by star, and the claim that a Cousin complex is a complex. Lemma 2.25.1 also follows from geometric considerations: the extensions by zero and by star coincide for generic values of monodromy.

2.28. By Theorem 2.26 the dual maps

Δ_{Λ}^{∗} : 𝔉 ⊗ V (Λ)^{∗} ⟶ V (Λ)^{∗}  (88)

give rise to a structure of a 𝔉^{∗}-module on V (Λ)^{∗}.

More generally, suppose we are given n modules V (Λ_{0}), …, V (Λ_{n−1}). We regard the tensor product V (Λ_{0})^{∗}⊗…⊗V (Λ_{n−1})^{∗} as a 𝔉^{∗⊗n}-module according to the ”sign” rule (82). Using iterated comultiplication (62) we get a structure of a 𝔉^{∗}-module on V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗}.

2.28.1. The square

𝔉 ⊗ V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1}) ⟶ V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})
S ↓  ↓ S
𝔉^{∗} ⊗ V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗} ⟶ V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗}

commutes.

This follows from 2.23 and 2.14.

3. Hochschild complexes

3.1. If A is an augmented B-algebra, A^{+} — the kernel of the augmentation, M an Amodule, let C_{A}^{•}(M) denote the following complex. By definition, C_{A}^{•}(M) is concentrated in non-positive degrees. For r ≥ 0

C_{A}^{−r}(M) = A^{+⊗r} ⊗ M.

We will use a notation a_{r}| … |a_{1}|m for a_{r} ⊗ … a_{1} ⊗ m.

The differential d : C_{A}^{−r}(M) ⟶ C_{A}^{−r+1}(M) acts as

d(a_{r}| … |a_{1}|m) = ∑_{p=1}^{r−1}(−1)^{p}a_{r}| … |a_{p+1}a_{p}| … a_{1}|m + a_{r}| … a_{2}|a_{1}m.

We have canonically H^{−r}(C_{A}^{•}(M)) ≅ Tor_{r}^{A}(B, M) where B is considered as an A-module by means of the augmentation, cf. [M], Ch. X, §2.

We will be interested in the algebras 𝔉 and 𝔉^{∗}. We define the augmentation 𝔉 ⟶ B as being zero on all 𝔉_{ν}, ν ∈ ℕ[I], ν ≠ 0, and identity on 𝔉_{0}; in the same way it is defined on 𝔉^{∗}.

3.2. Let M be a ℕ[I]-graded 𝔉-module. Each term C_{𝔉}^{−r}(M) is ℕ[I]-graded by the sum of gradings of tensor factors. We will denote _{ν}C_{𝔉}^{−r}(M) the weight ν component.

For ν = (ν_{0}, …, ν_{r}) ∈ ℕ[I]^{r+1} we set

νC_{𝔉}^{−r}(M) = _{ν_{r},... ,ν_{0}}C_{𝔉}^{−r}(M) = 𝔉_{ν_{r}} ⊗ … ⊗ 𝔉_{ν_{1}} ⊗ M_{ν_{0}}.

Thus,

νC_{𝔉}^{−r}(M) = ⊕_{ν_{0}+...ν_{r}=ν} _{ν_{r},... ,ν_{0}}C_{𝔉}^{−r}(M).

Note that all ν_{p} must be > 0 for p > 0 since tensor factors lie in 𝔉^{+}.

The differential d clearly respects the ℕ[I]-grading; thus the whole complex is ℕ[I]-graded:

C_{𝔉}^{•}(M) = ⊕_{ν∈ℕ[I]} _{ν}C_{𝔉}^{•}(M).

The same discussion applies to ℕ[I]-graded 𝔉^{∗}-modules.

3.3. Let us fix weights Λ_{0}, …, Λ_{n−1}, n ≥ 1. We will consider the Hochschild complex C_{𝔉}^{•}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})) where the structure of an 𝔉-module on V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1}) has been introduced in 2.22.

3.3.1. In the sequel we will use the following notation. If K ⊂ I is a subset, we will denote by χ_{K} := ∑_{i∈K} i ∈ ℕ[I].

3.3.2. Suppose we have a map

ϱ : I ⟶ [−n + 1, r]  (89)

where r is some non-negative integer. Let us introduce the elements

ν_{a}(ϱ) = χ_{ϱ^{−1}(a)},  (90)

a ∈ [−n + 1, r]. Let us denote by P_{r}(I; n) the set of all maps (89) such that ϱ^{−1}(a) ≠ ∅ for all a ∈ [r]. It is easy to see that this set is not empty iff 0 ≤ r ≤ N.

Let us assign to such a ϱ the space

ϱC_{𝔉}^{−r}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})) := 𝔉_{ν_{r}(ϱ)} ⊗ … ⊗ 𝔉_{ν_{1}(ϱ)} ⊗ V (Λ_{0})_{ν_{0}(ϱ)} ⊗ … ⊗ V (Λ_{n−1})_{ν_{−n+1}(ϱ)} (91)

For each ϱ ∈ P_{r}(I; n) this space is non-zero, and we have

χ_{I}C_{𝔉}^{−r}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})) = ⊕_{ϱ∈P_{r}(I;n)} _{ϱ}C_{𝔉}^{−r}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})) (92)

3.4. Bases. Let us consider the set P_{N}(I; n). Obviously, if ϱ ∈ P_{N}(I; n) then ϱ(I) = [N], and the induced map I ⟶ [N] is a bijection; this way we get an isomorphism between P_{N}(I; n) and the set of all bijections I ⟶^{∼} [N] or, to put it differently, with the set of all total orders on I.

For an arbitrary r, let ϱ ∈ P_{r}(I; n) and τ ∈ P_{N}(I; n). Let us say that τ is a refinement of ϱ, and write ϱ ≤ τ, if ϱ(i) < ϱ(j) implies τ(i) < τ(j) for each i, j ∈ I. The map τ induces total orders on all subsets ϱ^{−1}(a). We will denote by Ord(ϱ) the set of all refinements of a given ϱ.

Given ϱ ≤ τ as above, and a ∈ [−n + 1, r], suppose that ϱ^{−1}(a) = {i_{1}, …, i_{p}} and τ(i_{1}) < τ(i_{2}) < … < τ(i_{p}). Let us define a monomial

θ_{ϱ≤τ;a} = θ_{i_{p}}θ_{i_{p−1}} · . . . · θ_{i_{1}} ∈ 𝔉_{ν_{a}(ϱ)}

If ϱ^{−1}(a) = ∅, we set θ_{ϱ≤τ;a} = 1. This defines a monomial

θ_{ϱ≤τ} = θ_{ϱ≤τ;r} ⊗ … ⊗ θ_{ϱ≤τ;1} ⊗ θ_{ϱ≤τ;0}v_{Λ_{0}} ⊗ … ⊗ θ_{ϱ≤τ;−n+1}v_{Λ_{n−1}} ∈ _{ϱ}C_{𝔉}^{−r}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})) (93)

3.4.1. Lemma. The set {θ_{ϱ≤τ}|τ ∈ Ord(ϱ)} forms a basis of the space _{ϱ}C_{𝔉}^{−r}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})).

Proof is obvious. □

3.4.2. Corollary. The set {θ_{ϱ≤τ}|ϱ ∈ P_{r}(I; n), τ ∈ Ord(ϱ)} forms a basis of the space χ_{I}C_{𝔉}^{−r}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})). □

3.5. We will also consider dual Hochschild complexes C_{𝔉^{∗}}^{•}(V (Λ_{0})^{∗}⊗…⊗V (Λ_{n−1})^{∗}) where V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗} is regarded as an 𝔉^{∗}-module as in 2.28.1.

We have obvious isomorphisms

C_{𝔉^{∗}}^{−r}(V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗}) ≅ C_{𝔉}^{−r}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1}))^{∗}

We define graded components

ϱC_{𝔉^{∗}}^{−r}(V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗}), ϱ ∈ P_{r}(I; n),

as duals to _{ϱ}C_{𝔉}^{−r}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})).

We will denote by {θ_{ϱ≤τ}^{∗}|ϱ ∈ P_{r}(I; n), τ ∈ Ord(ϱ)} the basis of _{χ_{I}}C_{𝔉^{∗}}^{−r}(V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗}) dual to the basis {θ_{ϱ≤τ}|ϱ ∈ P_{r}(I; n), τ ∈ Ord(ϱ)}, 3.4.2.

3.6. The maps S_{r;Λ_{0},... ,Λ_{n−1}}, cf. (81), for different r are compatible with differentials in Hochschild complexes, and therefore induce morphism of complexes

S : C_{𝔉}^{•}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})) ⟶ C_{𝔉^{∗}}^{•}(V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗})  (94)

This follows from 2.28.1 and 2.10 (b).

4. Symmetrization

4.1. Let us fix a finite set J and a map π : J ⟶ I. We set ν_{π} := ∑_{i} N_{i}i ∈ ℤ[I] where N_{i} := card(π^{−1}(i)). The map π induces a map ℤ[J] ⟶ ℤ[I] also to be denoted by π. We will use the notation χ_{K} := ∑_{j∈K} j ∈ ℕ[J] for K ⊂ J. Thus, π(χ_{J}) = ν_{π}.

We will denote also by μ, μ′ ↦ μ · μ′ := π(μ) · π(μ′) the bilinear form on ℤ[J] induced by the form on ℤ[I].

We will denote by Σ_{π} the group of all bijections σ : J ⟶ J preserving fibers of π.

Let ^{π}𝔉 be a free associative B-algebra with 1 with generators θ̃_{j}, j ∈ J. It is evidently ℕ[J]-graded. For ν ∈ ℕ[J] the corresponding homogeneous component will be denoted π𝔉_{ν}. The degree of a homogeneous element x ∈ ^{π}𝔉 will be denoted by |x| ∈ ℕ[J]. The group Σ_{π} acts on algebras ^{π}𝔉, ^{π}𝔉^{∗} by permutation of generators.

4.2. In the sequel, if G is a group and M is a G-module, M^{G} will denote the subset of G-invariants in M.

Let us define a B-linear ”averaging” mapping

πa : 𝔉_{ν_{π}} ⟶ (^{π}𝔉_{χ_{J}})^{Σ_{π}}  (95)

by the rule

πa(θ_{i_{1}} · . . . · θ_{i_{N}}) = ∑ θ̃_{j_{1}} · . . . · θ̃_{j_{N}} ,  (96)

the sum being taken over the set of all sequences (j_{1}, …, j_{N}) such that π(j_{p}) = i_{p} for any p. Note that this set is naturally a Σ_{π}-torsor. Alternatively, ^{π}a may be defined as follows. Pick some sequence (j_{1}, …, j_{N}) as above, and consider an element

∑_{σ∈Σ_{π}} σ(θ̃_{j_{1}} · . . . · θ̃_{j_{N}});

this element obviously lies in (^{π}𝔉_{χ_{J}})^{Σ_{π}} and is equal to ^{π}a(θ_{i_{1}} · . . . · θ_{i_{N}}).

The map π induces the map between homogeneous components

π : ^{π}𝔉_{χ_{J}} ⟶ 𝔉_{ν_{π}}.  (97)

It is clear that the composition π ∘ ^{π}a is equal to the multiplication by card(Σ_{π}), and πa ∘ π — to the action of operator ∑_{σ∈Σ_{π}} σ. As a consequence, we get

4.2.1. Lemma, [SV1], 5.11. The map ^{π}a is an isomorphism. □

4.3. Let us consider the dual to the map (97): 𝔉_{ν_{π}}^{∗} ⟶ ^{π}𝔉_{χ_{J}}^{∗}; it is obvious that it lands in the subspace of Σ_{π}-invariant functionals. Let us consider the induced map

πa^{∗} : 𝔉_{ν_{π}}^{∗} ⟶^{∼} (^{π}𝔉_{χ_{J}}^{∗})^{Σ_{π}}  (98)

It follows from the above discussion that ^{π}a^{∗} is an isomorphism.

4.4. Given a weight Λ ∈ X = Hom(ℤ[I], ℤ), we will denote by ^{π}Λ the composition ℤ[J] ⟶^{π} ℤ[I] ⟶^{Λ} ℤ, and by V (^{π}Λ) the corresponding Verma module over ^{π}𝔉.

Suppose we are given n weights Λ_{0}, …, Λ_{n−1}. Let us consider the Hochschild complex C_{^{π}𝔉}^{•}(V (^{π}Λ_{0})⊗…⊗V (^{π}Λ_{n−1})). By definition, its (−r)-th term coincides with the tensor power^{π}𝔉^{⊗n+r}. Therefore we can identify the homogeneous component_{χ_{J}} C_{^{π}𝔉}^{−r}(V (^{π}Λ_{0})⊗ … ⊗ V (^{π}Λ_{n−1})) with (^{π}𝔉^{⊗n+r})_{χ_{J}} which in turn is isomorphic to ^{π}𝔉_{χ_{J}}, by means of the multiplication map ^{π}𝔉^{⊗n+r} ⟶ ^{π}𝔉. This defines a map

χ_{J} C_{^{π}𝔉}^{−r}(V (^{π}Λ_{0}) ⊗ … ⊗ V (^{π}Λ_{n−1})) ⟶ ^{π}𝔉_{χ_{J}}  (99)

which is an embedding when restricted to polygraded components. The Σ_{π}-action on 𝔉 induces the Σ_{π}-action on _{χ_{J}} C_{^{π}𝔉}^{−r}(V (^{π}Λ_{0}) ⊗ … ⊗ V (^{π}Λ_{n−1})).

In the same manner we define a map

ν_{π}C_{𝔉}^{−r}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})) ⟶ 𝔉_{ν_{π}}  (100)

Let us define an averaging map

πa : _{ν_{π}}C_{𝔉}^{−r}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})) ⟶ _{χ_{J}} C_{^{π}𝔉}^{−r}(V (^{π}Λ_{0}) ⊗ … ⊗ V (^{π}Λ_{n−1}))^{Σ_{π}} (101)

as the map induced by (95). It follows at once that this map is is an isomorphism.

These maps for different r are by definition compatible with differentials in Hochschild complexes. Therefore we get

4.4.1. Lemma. The maps (101) induce isomorphism of complexes

πa : _{ν_{π}}C_{𝔉}^{•}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})) ⟶^{∼} _{χ_{J}} C_{^{π}𝔉}^{•}(V (^{π}Λ_{0}) ⊗ … ⊗ V (^{π}Λ_{n−1}))^{Σ_{π}}. □(102)

4.5. Lemma. The averaging is compatible with coaction. In other words, for any Λ ∈ X the square

V (Λ) ⟶^{Δ_{Λ}} 𝔉 ⊗ V (Λ)
πa ↓  ↓ ^{π}a
V (^{π}Λ) ⟶^{Δ_{^{π}Λ}} ^{π}𝔉 ⊗ V (^{π}Λ)

commutes.

Proof follows at once by inspection of the definition (72). □

4.6. Consider the dual Hochschild complexes. We have an obvious isomorphism

χ_{J} C_{^{π}𝔉^{∗}}^{−r}(V (^{π}Λ_{0})^{∗} ⊗ … ⊗ V (^{π}Λ_{n−1})^{∗}) ≅ _{χ_{J}} C_{^{π}𝔉}^{−r}(V (^{π}Λ_{0}) ⊗ … ⊗ V (^{π}Λ_{n−1}))^{∗};

using it, we define the isomorphism

χ_{J} C_{^{π}𝔉^{∗}}^{−r}(V (^{π}Λ_{0})^{∗} ⊗ … ⊗ V (^{π}Λ_{n−1})^{∗}) ⟶^{∼} ^{π}𝔉_{χ_{J}}^{∗}

as the dual to (99). The Σ_{π}-action on the target induces the action on _{χ_{J}} C_{^{π}𝔉^{∗}}^{−r}(V (^{π}Λ_{0})^{∗} ⊗ … ⊗ V (^{π}Λ_{n−1})^{∗}). Similarly, the isomorphism

ν_{π}C_{𝔉^{∗}}^{−r}(V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗}) ⟶^{∼} 𝔉_{ν_{π}}^{∗}

is defined. We define the averaging map

πa^{∗} :_{ν_{π}} C_{𝔉^{∗}}^{−r}(V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗}) ⟶_{χ_{J}} C_{^{π}𝔉^{∗}}^{−r}(V (^{π}Λ_{0})^{∗} ⊗ … ⊗ V (^{π}Λ_{n−1})^{∗})^{Σ_{π}}(103)

as the map which coincides with (98) modulo the above identifications. Again, this map is an isomorphism.

Due to Lemma 4.5 these maps for different r are compatible with the differentials in Hochschild complexes. Therefore we get

4.6.1. Lemma. The maps (103) induce isomorphism of complexes

πa^{∗} :_{ν_{π}} C_{𝔉^{∗}}^{•}(V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗}) ⟶^{∼} _{χ_{J}} C_{^{π}𝔉^{∗}}^{•}(V (^{π}Λ_{0})^{∗} ⊗ … ⊗ V (^{π}Λ_{n−1})^{∗})^{Σ_{π}}. □ (104)

BILINEAR FORMS

4.7. Using the bilinear form on ℤ[J] introduced above, we define the symmetric bilinear form S( , ) on ^{π}𝔉 exactly in the same way as the form S on 𝔉. Similarly, given Λ ∈ X, we define the bilinear form S_{^{π}Λ} on V (^{π}Λ) as in 2.16, with I replaced by J.

4.7.1. Lemma. (i) The square

𝔉_{ν_{π}}  ⟶^{S} 𝔉_{ν_{π}}^{∗}
πa ↓  ↓ ^{π}a^{∗}
π𝔉_{χ_{J}} ⟶^{S} ^{π}𝔉_{χ_{J}}^{∗}

commutes.

(ii) For any Λ ∈ X the square

V (Λ)_{ν_{π}}  ⟶^{S_{Λ}} V (Λ)_{ν_{π}}^{∗}
πa ↓  ↓ ^{π}a^{∗}
V (^{π}Λ)_{χ_{J}} ⟶^{S_{^{π}Λ}} V (^{π}Λ)_{χ_{J}}^{∗}

commutes.

Proof. (i) Let us consider an element θI⃗ = θ_{i_{1}} · . . . · θ_{i_{N}} ∈ 𝔉_{ν_{π}} (we assume that N = card(J)). The functional ^{π}a^{∗} ∘ S(θI⃗) carries a monomial θ̃_{j_{1}} · . . . · θ̃_{j_{N}} to

S(θ_{i_{1}} · . . . · θ_{i_{N}} , θ_{π(j_{1})} · . . . · θ_{π(j_{N})}).

On the other hand,

S ∘ ^{π}a(θI⃗)(θ̃_{j_{1}} · . . . · θ̃_{j_{N}}) = ∑ S(θ̃_{k_{1}} · . . . · θ̃_{k_{N}} , θ̃_{j_{1}} · . . . · θ̃_{j_{N}}),

the summation ranging over all sequences K⃗ = (k_{1}, …, k_{N}) such that π(K⃗) = I⃗. It follows from Lemma 2.12 that both expressions are equal.

(ii) The same argument as in (i), using Lemma 2.17 instead of 2.12. □

More generally, we have

4.8. Lemma. For every m ≥ 0 and weights Λ_{0}, …, Λ_{n−1} ∈ X the square

(𝔉^{⊗m} ⊗ V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1}))_{ν_{π}}  ⟶^{S_{m;Λ_{0},... ,Λ_{n−1}}}  (𝔉^{∗⊗m} ⊗ V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗})_{ν_{π}}
πa ↓  ↓ ^{π}a^{∗}
(^{π}𝔉^{⊗m} ⊗ V (^{π}Λ_{0}) ⊗ … ⊗ V (^{π}Λ_{n−1}))_{χ_{J}} ⟶^{S_{m;^{π}Λ_{0},... ,^{π}Λ_{n−1}}} (^{π}𝔉^{∗⊗m} ⊗ V (^{π}Λ_{0})^{∗} ⊗ … ⊗ V (^{π}Λ_{n−1})^{∗})_{χ_{J}}

commutes.

Proof is quite similar to the proof of the previous lemma. We leave it to the reader. □

5. Quotient algebras

5.1. Let us consider the map (65) S : 𝔉 ⟶ 𝔉^{∗}. Let us consider its kernel Ker(S). It follows at once from (64) that Ker(S) is a left ideal in 𝔉. In the same manner, it is easy to see that it is also a right ideal, cf. [L1], 1.2.4.

We will denote by 𝔣 the quotient algebra 𝔉/Ker(S). It inherits the ℕ[I]-grading and the coalgebra structure from 𝔉, cf. loc.cit. 1.2.5, 1.2.6.

5.2. In the same manner, given a weight Λ, consider the kernel of S_{Λ} : V (Λ) ⟶ V (Λ)^{∗}. Let us denote by L(Λ) the quotient space V (Λ)/Ker(S_{Λ}). It inherits ℕ[I]- and X-gradings from V (Λ). Due to Theorem 2.23 the structure of 𝔉-module on V (Λ) induces the structure of 𝔣-module on L(Λ).

More generally, due to the structure of a coalgebra on 𝔣, all tensor products L(Λ_{0}) ⊗ … ⊗ L(Λ_{n−1}) become 𝔣-modules (one should take into account the ”sign rule” (82)).

5.3. We can consider Hochschild complexes C_{𝔣}^{•}(L(Λ_{0}) ⊗ … ⊗ L(Λ_{n−1})). 5.3.1. Lemma. The map

S : C_{𝔉}^{•}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{n−1})) ⟶ C_{𝔉^{∗}}^{•}(V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{n−1})^{∗})

factors through the isomorphism

Im(S) ⟶^{∼} C_{𝔣}^{•}(L(Λ_{0}) ⊗ … ⊗ L(Λ_{n−1}))  (105)

Proof. This follows at once from the definitions. □

Chapter 2. Geometric discussion

6. Diagonal stratification and related algebras

6.1. Let us adopt notations of 4.1. We set N := card(J). Let ^{π}𝔸_{ℝ} denote a real affine space with coordinates t_{j}, j ∈ J, and ^{π}𝔸 its complexification. Let us consider an arrangement H_{∅} consisting of all diagonals Δ_{ij}, i, j ∈ J. Let us denote by S_{∅} the corresponding stratification; S_{∅,ℝ} will denote the corresponding real stratification of 𝔸_{ℝ}.

The stratification S_{∅} has a unique minimal stratum

Δ = ⋂ Δ_{ij}  (106)

— main diagonal; it is one-dimensional. We will denote by ^{π}𝔸^{∘}_{∅} (resp., ^{π}𝔸^{∘}_{∅,ℝ}) the open stratum of S_{∅} (resp., of S_{∅,ℝ}).

6.2. Let us describe the chambers of S_{∅,ℝ}. If C is a chamber and x = (x_{j}) ∈ C, i.e. the embedding J ↪ ℝ, j ↦ x_{j}, it induces an obvious total order on J, i.e. a bijection

τ_{C} : J ⟶^{∼} [N]  (107)

Namely, τ_{C} is determined uniquely by the requirement τ_{C}(i) < τ_{C}(j) iff x_{i} < x_{j}; it does not depend on the choice of x. This way we get a one-to-one correspondence between the set of chambers of S_{∅} and the set of all bijections (107). We will denote by C_{τ} the chamber corresponding to τ.

Given C and x as above, suppose that we have i, j ∈ J such that x_{i} < x_{j} and there is no k ∈ J such that x_{i} < x_{k} < x_{j}. We will say that i, j are neighbours in C, more precisely that i is a left neighbour of j.

Let x′ = (x′_{j}) be a point with x′_{p} = x_{p} for all p ≠ j, and x′_{j} equal to some number smaller than x_{i} but greater than any x_{k} such that x_{k} < x_{i}. Let ^{ji}C denote the chamber containing x′. Let us introduce a homotopy class of paths ^{C}γ_{ij} connecting x and x′ as shown on Fig. 1 below.

[figure: i; j; ^{C}γ_{ij}; j′; Fig. 1.]

We can apply the discussion I.4.1 and consider the groupoid π_{1}(^{π}𝔸^{∘}_{∅},^{π} 𝔸^{∘}_{∅,ℝ}). It has as the set of objects the set of all chambers. The set of morphisms is generated by all morphisms ^{C}γ_{ij} subject to certain evident braiding relations. We will need only the following particular case.

To define a one-dimensional local system L over ^{π}𝔸^{∘}_{∅} is the same as to give a set of onedimensional vector spaces L_{C}, C ∈ π_{0}(^{π}𝔸^{∘}_{∅,ℝ}), together with arbitrary invertible linear operators

CT_{ij} : L_{C} ⟶ L_{^{ji}C}  (108)

(”half-monodromies”) defined for chambers having i as a left neighbour of j.

6.3. We define a one-dimensional local system ^{π}I over ^{π}𝔸^{∘}_{∅} as follows. Its fibers ^{π}I_{C} are one-dimensional linear spaces with fixed basis vectors; they will be identified with B.

Half-monodromies are defined as

CT_{ij} = ζ^{i·j}, i, j ∈ J

6.4. Let j : ^{π}𝔸^{∘}_{∅} ⟶ ^{π}𝔸 denote an open embedding. We will study the following objects of M(^{π}𝔸; S_{∅}):

πI_{?} = j_{?}^{π}I[N],

where ? =!, ∗. We have a canonical map

m : ^{π}I_{!} ⟶ ^{π}I_{∗}  (109)

and by definition ^{π}I_{!∗} is its image, cf. I.4.5.

6.5. For an integer r let us denote by P_{r}(J) the set of all surjective mappings J ⟶ [r]. It is evident that P_{r}(J) ≠ ∅ if and only if 1 ≤ r ≤ N. To each ρ ∈ P_{r}(J) let us assign a point w_{ρ} = (ρ(j)) ∈ ^{π}𝔸_{ℝ}. Let F_{ρ} denote the facet containing w_{ρ}. This way we get a bijection between P_{r}(J) and the set of r-dimensional facets. For r = N we get the bijection from 6.2.

At the same time we have defined a marking of H_{∅}: by definition, ^{F_{ρ}}w = w_{ρ}. This defines cells D_{F} , S_{F}.

6.6. The main diagonal Δ is a unique 1-facet; it corresponds to the unique element ρ_{0} ∈ P_{1}(J).

We will denote by Ch the set of all chambers; it is the same as Ch(Δ) in notations of Part I. Let C_{τ} be a chamber. The order τ identifies C with an open cone in the standard coordinate space ℝ^{N}; we provide C with the orientation induced from ℝ^{N}.

6.7. Basis in Φ_{Δ}(^{π}I_{∗}). The construction I.4.7 gives us the basis {c_{Δ<C}} in Φ_{Δ}(^{π}I_{!})^{∗} indexed by C ∈ Ch. We will use notation c_{τ,!} := c_{Δ<C_{τ}}.

A chain c_{τ,!} looks as follows.

[figure: j_{1}; j_{2}; …; j_{N}; Fig. 2. A chain c_{τ,!}.]

Here τ(j_{i}) = i. We will denote by {b_{τ,!}} the dual basis in Φ_{Δ}(I_{!}).

6.8. Basis in Φ_{Δ}(I_{∗}). Similarly, the definition I.4.9 gives us the basis {c_{Δ<C}}, C ∈ Ch in Φ_{Δ}(I_{∗})^{∗}. We will use the notations c_{τ,∗} := c_{Δ<C_{τ}}.

If we specify the definition I.4.9 and its explanation I.4.12 to our arrangement, we get the following picture for a dual chain c_{τ,∗}.

[figure: j_{1}; j_{2}; j_{3}; …; j_{N}; Fig. 3. A chain c_{τ,∗} .]

This chain is represented by the section of a local system I^{−1} over the cell in ^{π}𝔸^{∘}_{∅} shown above, which takes value 1 at the point corresponding to the end of the travel (direction of travel is shown by arrows).

To understand what is going on, it is instructive to treat the case N = 2 first, which essentially coincides with the Example I.4.10.

We will denote by {b_{τ,∗}} the dual basis in Φ_{Δ}(I_{∗}).

6.9. Obviously, all maps τ : J ⟶ [N] from P_{N}(J) are bijections. Given two such maps τ_{1}, τ_{2}, define the sign sgn(τ_{1}, τ_{2}) = ±1 as the sign of the permutation τ_{1}τ_{2}^{−1} ∈ Σ_{N}.

For any τ ∈ P_{N}(J) let us denote by J⃗_{τ} the sequence (τ^{−1}(N), τ^{−1}(N − 1), …, τ^{−1}(1)).

6.10. Let us pick η ∈ P_{N}(J). Let us define the following maps:

πϕ_{Δ,!}^{(η)} : Φ_{Δ}(^{π}I_{!}) ⟶ ^{π}𝔉_{χ_{J}}  (110)

which carries b_{τ,!} to sgn(τ, η) · θJ⃗_{τ}, and

πϕ_{Δ,∗}^{(η)} : Φ_{Δ}(^{π}I_{∗}) ⟶ ^{π}𝔉_{χ_{J}}^{∗}  (111)

which carries b_{τ,∗} to sgn(τ, η) · θ∗ ; J⃗_{τ}.

6.11. Theorem. (i) The maps ^{π}ϕ_{Δ,!}^{(η)} and ^{π}ϕ_{Δ,∗}^{(η)} are isomorphisms. The square

Φ_{Δ}(^{π}I_{!}) ⟶^{∼}^{^{π}ϕ_{Δ,!}^{(η)}}  ^{π}𝔉_{χ_{J}}
m ↓  ↓ S
Φ_{Δ}(^{π}I_{∗}) ⟶^{∼}^{^{π}ϕ_{Δ,∗}^{(η)}}  ^{π}𝔉_{χ_{J}}^{∗}

commutes.

(ii) The map ^{π}ϕ_{Δ,!}^{(η)} induces an isomorphism

πϕ_{Δ,!∗}^{(η)} : Φ_{Δ}(^{π}I_{!∗}) ⟶^{∼} ^{π}𝔣_{χ_{J}} □  (112)

Proof. This theorem is particular case of I.14.16, I.4.17. The claim about isomorphisms in (i) is clear. To prove the commutativity of the square, we have to compute the action of the canonical map m on our standard chains. The claim follows at once from their geometric description given above. Note that here the sign in the definition of morphisms ϕ is essential, due to orientations of our chains. (ii) is a direct corollary of (i) □

SYMMETRIZED CONFIGURATIONAL SPACES

6.12. Colored configuration spaces. Let us fix ν = ∑ ν_{i}i ∈ ℕ[I], ∑_{i} ν_{i} = N. There exists a finite set J and a morphism π : J ⟶ I such that card(π^{−1}(i)) = ν_{i} for all i ∈ I. Let us call such π an unfolding of ν. It is unique up to a non-unique isomorphism; the automorphism group of π is precisely Σ_{π}, and ν = ν_{π} in our previous notations.

Let us pick an unfolding π. As in the above discussion, we define ^{π}𝔸 as a complex affine space with coordinates t_{j}, j ∈ J. Thus, dim ^{π}𝔸 = N. The group Σ_{π} acts on the space π𝔸 by permutations of coordinates.

Let us denote by A_{ν} the quotient manifold ^{π}𝔸/Σ_{π}. As an algebraic manifold, A_{ν} is also a complex N-dimensional affine space. We have a canonical projection

π : ^{π}𝔸 ⟶ A_{ν}  (113)

The space A_{ν} does not depend on the choice of an unfolding π. It will be called the configuration space of ν-colored points on the affine line 𝔸^{1}.

We will consider the stratification on A_{ν} whose strata are π(S), S ∈ S_{∅}; we will denote this stratification also by S_{∅}; this definition does not depend on the choice of π. We will study the category M(A_{ν}; S_{∅}).

We will denote by A^{∘}_{ν,∅} the open stratum. It is clear that π^{−1}(A^{∘}_{ν,∅}) = ^{π}𝔸^{∘}_{∅}. The morphism

π is unramified over A^{∘}_{ν,∅}.

The action of Σ_{π} on ^{π}𝔸 may be extended in the evident way to the local system ^{π}I, hence all our spaces of geometric origin — like Φ_{Δ}(^{π}I_{!}), etc. — get an action of Σ_{π}.

6.12.1. If M is an object with a Σ_{π}-action (for example a vector space or a sheaf), we will denote by M^{Σ_{π},−} the subobject {x ∈ M| for every σ ∈ Σ_{π} σx = sgn(σ)x} where sgn(σ) = ±1 is the sign of a permutation.

A morphism f : M ⟶ N between two objects with Σ_{π}-action will be called skew (Σ_{π})- equivariant if for any x ∈ M, σ ∈ Σ_{π}, f(σx) = sgn(σ)σf(x).

Let us define a local system over A_{ν}

I_{ν} = (π_{∗} ^{π}I)^{Σ_{π},−}  (114)

6.13. Let j : ^{π}𝔸^{∘}_{∅} ↪ ^{π}𝔸, j_{A} : A^{∘}_{ν,∅} ↪ A_{ν} be the open embeddings. Let us define the following objects of M(A_{ν}; S_{∅}):

I_{ν?} := j_{A?}I_{ν}[N]  (115)

where ? =!, ∗ or !∗. We have by definition

I_{ν!} = (π_{∗} ^{π}I_{!})^{Σ_{π},−}  (116)

The morphism π is finite; consequently π_{∗} is t-exact (see [BBD], 4.1.3) and commutes with the Verdier duality. Therefore,

I_{ν∗} = (π_{∗} ^{π}I_{∗})^{Σ_{π},−}; I_{ν!∗} = (π_{∗} ^{π}I_{!∗})^{Σ_{π},−}  (117)

6.14. Let us define vector spaces

Φ_{Δ}(I_{ν?}) := (Φ_{Δ}(^{π}I_{?}))^{Σ_{π},−}  (118)

where ? =!, ∗ or !∗.

Let us pick a Σ_{π}-equivariant marking of H_{∅}, for example the one from 6.5; consider the corresponding cells D_{Δ}, S_{Δ}. It follows from (116) and (117) that

Φ_{Δ}(I_{ν?}) = RΓ(π(D_{Δ}), π(S_{Δ}); I_{ν?})[−1]  (119)

where ? =!, ∗ or !∗, cf. I.3.3.

6.15. The group Σ_{π} is acting on on ^{π}𝔉. Let us pick η ∈ P_{N}(J). It follows from the definitions that the isomorphisms^{π}ϕ_{Δ,!}^{(η)}, ^{π}ϕ_{Δ,∗}^{(η)} are skew Σ_{π}-equivariant. Therefore, passing to invariants in Theorem 6.11 we get

6.16. Theorem. The maps ^{π}ϕ_{Δ,!}^{(η)},^{π} ϕ_{Δ,∗}^{(η)} induce isomorphisms included into a commutative square

Φ_{Δ}(I_{ν!}) ⟶^{∼}^{ϕ_{ν,!}^{(η)}} 𝔉_{ν}
m ↓  ↓ S
Φ_{Δ}(I_{ν∗}) ⟶^{∼}^{ϕ_{ν,∗}^{(η)}} 𝔉_{ν}^{∗}

and

ϕ_{ν,!∗}^{(η)} : Φ_{Δ}(I_{ν!∗}) ⟶^{∼} 𝔣_{ν} □  (120)

7. Principal stratification

The contents of this section is parallel to I, Section 3. However, we present here certain modification of general constructions from loc. cit.

7.1. Let us fix a finite set J of cardinality N. In this section we will denote by 𝔸_{ℝ} a real affine space with fixed coordinates t_{j} : 𝔸_{ℝ} ⟶ ℝ, j ∈ J, and by 𝔸 its complexification. For z ∈ ℂ, i, j ∈ J denote by H_{j}(z) ⊂ 𝔸 a hyperplane t_{j} = z, and by Δ_{ij} a hyperplane t_{i} = t_{j}.

Let us consider an arrangement H in 𝔸 consisting of hyperplanes H_{i}(0) and Δ_{ij}, i, j ∈ J, i ≠ j. It is a complexification of an evident real arrangement H_{ℝ} in 𝔸_{ℝ}. As usual, the subscript _{ℝ} will denote real points.

Denote by S the corresponding stratification of 𝔸. To distinguish this stratification from the diagonal stratification of the previous section, we will call it the principal stratification. To shorten the notation, we will denote in this paper by D(A, S) a category which would be denoted D^{b}(𝔸; S) in I. In this section we will study the category M(𝔸; S).

7.1.1. Let us consider a positive cone

𝔸_{ℝ}^{+} = {(t_{j})| all t_{j} ≥ 0} ⊂ 𝔸_{ℝ}

A facet will be called positive if it lies inside 𝔸_{ℝ}^{+}.

A flag F is called positive if all its facets are positive.

7.2. Let us fix a marking w = { ^{F} w} of H_{ℝ} (cf. I.3.2). For a positive facet F define

D_{F}^{+} = D_{F} ∩ 𝔸_{ℝ}^{+}; S_{F}^{+} = S_{F} ∩ 𝔸_{ℝ}^{+}; D^{∘}_{F}^{+} = D_{F}^{+} − S_{F}^{+}.

Note that D_{F}^{+} coincides with the union of FΔ over all positive flags beginning at F, and S_{F}^{+} coincides with the union of FΔ as above with dim FΔ < codim F. It follows that only marking points ^{F} w for positive facets F take part in the definition of cells D_{F}^{+}, S_{F}^{+}.

7.3. Let K be an object of D(𝔸; S), F a positive facet of dimension p. Let us introduce a notation

Φ_{F}^{+}(K) = Γ(D_{F}^{+}, S_{F}^{+}; K)[−p].

This way we get a functor

Φ_{F}^{+} : D(𝔸; S) ⟶ D^{b}(pt)  (121)

7.4. Theorem. Functors Φ_{F}^{+} commute with Verdier duality. More precisely, we have canonical natural isomorphisms

DΦ_{F}^{+}(K) ⟶^{∼} Φ_{F}^{+}(DK).  (122)

Proof goes along the same lines as the proof of Theorem I.3.5.

7.5. First let us consider the case N = 1, cf. I.3.6. We will adopt notations from there and from I, Fig. 1. Our arrangement has one positive 1-dimensional facet E = ℝ_{>0}, let w ∈ E be a marking.

[figure: F; S_{r′′}; E; Y; iϵw; w; S_{r′}; Fig. 4]

We have an isomorphism

Φ_{F}^{+}(K) ⟶^{∼} RΓ(𝔸, {w}; K)  (123)

Denote j := j_{𝔸−{w}}. We have by Poincaré duality

DΦ_{F}^{+}(K) ⟶^{∼} RΓ_{c}(𝔸; j_{∗}j^{∗}DK) ⟶^{∼} RΓ(𝔸, 𝔸_{≥r′′}; j_{∗}j^{∗}DK)  (124)

Let us denote D_{F}^{+opp} := ℝ_{≥w}, and Y = ϵi · D_{F}^{+opp}. We have

RΓ(𝔸, 𝔸_{≥r′′}; j_{∗}j^{∗}DK) ⟶^{∼} RΓ(𝔸, Y ∪ 𝔸_{≥r′′}; j_{∗}j^{∗}DK)  (125)

by homotopy. Consider the restriction map

res : RΓ(𝔸, Y ∪ 𝔸_{≥r′′}; j_{∗}j^{∗}DK) ⟶ RΓ(𝔸_{≤r′}, Y ∩ 𝔸_{≤r′}; DK)  (126)

7.5.1. Claim. res is an isomorphism.

In fact, Cone(res) is isomorphic to

RΓ(𝔸, 𝔸_{≤r′} ∪ 𝔸_{≥r′′} ∪ Y; j_{∗}j^{∗}DK) = RΓ_{c}(𝔸_{<r′′}, 𝔸_{≤r′} ∪ Y; j_{∗}j^{∗}DK) ≅
≅ DRΓ(𝔸_{<r′′} − (𝔸_{≤r′} ∪ Y); j_{!}j^{∗}K)

But

RΓ(𝔸_{<r′′} − (𝔸_{≤r′} ∪ Y); j_{!}j^{∗}K) = RΓ(𝔸_{<r′′} − (𝔸_{≤r′} ∪ Y), {w}; K) = 0  (127)

evidently. This proves the claim. □ A clockwise rotation by π/2 induces an isomorphism

RΓ(𝔸_{≤r′}, Y ∩ 𝔸_{≤r′}; DK) ≅ RΓ(𝔸_{≤r′}, ϵ · D_{F}^{opp}; DK),

and the last complex is isomorphic to Φ_{F}^{+}(DK). This proves the theorem for N = 1.

7.6. Now let us return to an arbitrary J. Let us prove the theorem for F equal to the unique 0-dimensional facet. Let us introduce the following subspaces of 𝔸_{ℝ} (as usually, a circle on the top will denote the interior).

D_{F}^{opp} := 𝔸_{ℝ} − D^{∘}_{F}; D_{F}^{+opp} := 𝔸_{ℝ}^{+} − D^{∘}_{F}^{+}; for each cell D_{F <C}, C ∈ Ch(F) define D_{F <C}^{opp} :=

C − D^{∘}_{F <C}.

It is easy to see that the restriction induces isomorphism

Φ_{F}^{+}(K) ⟶^{∼} RΓ(𝔸, D_{F}^{+opp}; K).

We use the notations of I.3.8. Let us choose positive numbers r′ < r′′, ϵ, such that

ϵD_{F} ⊂ 𝔸_{<r′} ⊂ D^{∘}_{F} ⊂ D_{F} ⊂ 𝔸_{<r′′}

Define the subspace

Y^{+} = ϵi · D_{F}^{+opp};

denote j := j_{𝔸−S_{F}^{+}}. We have isomorphisms

DΦ_{F}^{+}(K) ≅ DRΓ(𝔸, S_{F}^{+}; K) ≅ RΓ_{c}(𝔸; j_{∗}j^{∗}DK) ≅  (128)
≅ RΓ(𝔸, 𝔸_{≥r′′}; j_{∗}j^{∗}DK) ≅ RΓ(𝔸, Y^{+} ∪ 𝔸_{≥r′′}; j_{∗}j^{∗}DK)

Consider the restriction map

res : RΓ(𝔸, Y^{+} ∪ 𝔸_{≥r′′}; j_{∗}j^{∗}DK) ⟶ RΓ(𝔸_{≤r′}, Y^{+} ∩ 𝔸_{≤r′}; DK)  (129)

Cone(res) is isomorphic to

RΓ(𝔸, 𝔸_{≤r′} ∪ 𝔸_{≥r′′} ∪ Y^{+}; j_{∗}j^{∗}DK) = RΓ_{c}(𝔸_{<r′′}, 𝔸_{≤r′} ∪ Y^{+}; j_{∗}j^{∗}DK) ≅

≅ DRΓ(𝔸_{<r′′} − (𝔸_{≤r′} ∪ Y^{+}); j_{!}j^{∗}K) = RΓ(𝔸_{(r′,r′′)} − Y^{+}, S_{F}^{+}; K)

7.6.1. Lemma. (Cf. I.3.8.1.) RΓ(𝔸_{(r′,r′′)} − Y^{+}, S_{F}^{+}; K) = 0.

Proof. Let us define the following subspaces of 𝔸.

A := {(t_{j})| for all j |t_{j}| < 1; there exists j : t_{j} ≠ 0}; A_{ℝ}^{+} := A ∩ 𝔸_{ℝ}^{+}. Note that A_{ℝ}^{+} ∩ i · A_{ℝ}^{+} = ∅. Due to monodromicity, it is easy to see that

RΓ(𝔸_{(r′,r′′)} − Y^{+}, S_{F}^{+}; K) ≅ RΓ(A − i · A_{ℝ}^{+}, A_{ℝ}^{+}; K).

Therefore, it is enough to prove the following

7.6.2. Claim. The restriction map

RΓ(A − i · A_{ℝ}^{+}; K) ⟶ RΓ(A_{ℝ}^{+}; K)  (130)

is an isomorphism.

Proof of the Claim. Let us introduce for each k ∈ J open subspaces

A_{k} = {(t_{j}) ∈ A|t_{k} ∉ i · ℝ_{≥0}} ⊂ A − i · A_{ℝ}^{+}

and

A′_{k} = {(t_{j}) ∈ A_{ℝ}^{+}|t_{k} > 0} ⊂ A_{ℝ}^{+}

Obviously A′_{k} = A_{k} ∩ A_{ℝ}^{+}. For each subset M ⊂ J set A_{M} := ⋂_{k∈M} A_{k}; A′_{M} := ⋂_{k∈M} A′_{k}.

For each non-empty M define the spaces B_{M} := {(t_{j})_{j∈M}| for all j |t_{j}| < 1, t_{j} ∉ i · ℝ_{≥0}} and B′_{M} := {(t_{j})_{j∈M}| for all j t_{j} ∈ ℝ, 0 < t_{j} < 1}. We have obvious projections f_{M} : A_{M} ⟶ B_{M}, f′_{M} : A′_{M} ⟶ B′_{M}.

Let us look at fibers of f_{M} and f′_{M}. Given b = (t_{j})_{j∈M} ∈ B_{M}, the fiber f_{M}^{−1}(b) is by definition {(t_{k})_{k∈J−M}||t_{k}| < 1}, the possible singularities of our sheaf K are at the hyperplanes t_{k} = t_{j} and t_{k} = 0. It is easy to see that f_{M} is ”lisse” with respect to K which means in particular that a stalk (f_{M∗}K)_{b} is equal to RΓ(f_{M}^{−1}(b); K). The same considerations apply to f′_{M}. Moreover, it follows from I.2.12 that the restriction maps

RΓ(f_{M}^{−1}(b); K) ⟶ RΓ((f′_{M})^{−1}(b); K)

are isomorphisms for every b ∈ B′_{M}. This implies that f′_{M∗}K is equal to the restriction of f_{M∗}K to B′_{M}.

The sheaf f_{M∗}K is smooth along the diagonal stratification. For a small δ > 0 let U_{δ} ⊂ B_{M} denote an open subset {(t_{j}) ∈ B_{M}| | arg(t_{j})| < δ for all j}. The restriction maps RΓ(B_{M}; f_{M∗}K) ⟶ RΓ(U_{δ}; f_{M∗}K) are isomorphisms. We have B′_{M} = ⋂_{δ} U_{δ}, therefore by I.2.12 the restriction RΓ(B_{M}; f_{M∗}K) ⟶ RΓ(B′_{M}; f_{M∗}K) is an isomorphism. This implies, by Leray, that restriction maps

RΓ(A_{M}; K) ⟶ RΓ(A′_{M}; K)

are isomorphisms for every non-empty M.

Obviously A − i · A_{ℝ}^{+} = ⋃_{k∈J} A_{k} and A_{ℝ}^{+} = ⋃_{k∈J} A′_{k}. Therefore, by Mayer-Vietoris the map (130) is an isomorphism. This proves the claim, together with the lemma. □

A clockwise rotation by π/2 induces an isomorphism

RΓ(𝔸_{≤r′}, Y^{+} ∩ 𝔸_{≤r′}; DK) ≅ RΓ(𝔸_{≤r′}, ϵ · D_{F}^{+opp}; DK) ≅ RΓ(𝔸_{≤r′}, ϵ · S_{F}^{+}; DK)

and the last complex is isomorphic to Φ_{F}^{+}(DK) in view of 7.6. This proves the theorem for the case of the 0-facet F.

7.7. Suppose that F is an arbitrary positive facet. From the description of positive facets (see infra, 8.4) one sees that the cell D_{F}^{+} is homeomorphic to a cartesian product of the form

D_{F_{0}}^{+} × D_{F_{1}}^{+} × … × D_{F_{a}}^{+}

where F_{0} is a 0-facet of the principal stratification in some affine space of smaller dimension, and D_{F_{i}}^{+} are the cells of the diagonal stratification discussed in the previous section.

Using this remark, we apply a combination of the arguments of the previous subsection (to the first factor) and of I.3.8 (to the remaining factors) to get the required isomorphism. We leave details to the reader.

The theorem is proved. □

7.8. Theorem. All functors Φ_{F}^{+} are t-exact. In other words, for all positive facets F,

Φ_{F}^{+}(M(𝔸; S)) ⊂ Vect ⊂ D^{b}(pt).

Proof. The same as that of I.3.9. □.

7.9. Thus we get exact functors

Φ^{+} : M(A; S) ⟶ Vect  (131)

commuting with Verdier duality.

We will also denote vector spaces Φ_{F}^{+}(M) by M_{F}^{+}.

7.10. Canonical and variation maps. Suppose we have a positive facet E. Let us denote by^{+}F⊣⌋^{1}(E) the set of all positive facets F such that E < F, dim F = dim E +1. We have

S_{E}^{+} = ⋃_{F ∈ ^{+}F⊣⌋^{1}(E)} D_{F}^{+}  (132)

Suppose we have K ∈ D(𝔸, S).

7.10.1. Lemma. We have a natural isomorphism

RΓ(S_{E}^{+}, ⋃_{F ∈ ^{+}F⊣⌋^{1}(E)} S_{F}^{+}; K) ≅ ⊕_{F ∈ ^{+}F⊣⌋^{1}(E)} RΓ(D_{F}^{+}, S_{F}^{+}; K)

Proof. Note that S_{E}^{+} − ⋃_{F ∈ ^{+}F⊣⌋^{1}(E)} S_{F}^{+} = ⋃_{F ∈ ^{+}F⊣⌋^{1}(E)} D^{∘}_{F}^{+} (disjoint union). The claim follows now from the Poincaré duality. □ Therefore, for any F ∈ ^{+}F⊣⌋^{1}(E) we get a natural inclusion map

i_{E}^{F} : RΓ(D_{F}^{+}, S_{F}^{+}; K) ↪ RΓ(S_{E}^{+}, ⋃_{F′∈F⊣⌋^{1}(E)} S_{F′}^{+}; K)  (133)

Let us define a map

u_{E}^{F}(K) : Φ_{F}^{+}(K) ⟶ Φ_{E}^{+}(K)

as a composition

RΓ(D_{F}^{+}, S_{F}^{+}; K)[−p] ⟶^{i_{E}^{F}} RΓ(S_{E}^{+}, ⋃_{F′∈ ^{+}F⊣⌋^{1}(E)} S_{F′}^{+}; K)[−p] ⟶  (134)
RΓ(S_{E}^{+}; K)[−p] ⟶ RΓ(D_{E}^{+}, S_{E}^{+})[−p + 1]

where the last arrow is the coboundary map for the couple (S_{E}^{+}, D_{E}^{+}), and the second one is evident. This way we get a natural transormation

+u_{E}^{F} : Φ_{F}^{+} ⟶ Φ_{E}^{+}  (135)

which will be called a canonical map. We define a variation morphism

+v_{F}^{E} : Φ_{E}^{+} ⟶ Φ_{F}^{+}  (136)

as follows. By definition, ^{+}v_{F}^{E}(K) is the map dual to the composition

DΦ_{F}^{+}(K) ⟶^{∼} Φ_{F}^{+}(DK) ⟶^{^{+}u_{E}^{F}(DK)} Φ_{E}^{+}(DK) ⟶^{∼} DΦ_{E}^{+}(K).

7.11. Cochain complexes. For each r ∈ [0, N] and M ∈ M(𝔸, S) introduce vector spaces

+C^{−r}(𝔸; M) = ⊕_{F:F positive, dim F =r} M_{F}^{+}  (137)

For r > 0 or r < −N set ^{+}C^{r}(𝔸; M) = 0. Define operators

d : ^{+}C^{−r}(𝔸; M) ⟶ ^{+}C^{−r+1}(𝔸; M)

having components ^{+}u_{E}^{F}.

7.11.1. Lemma. d^{2} = 0.

Proof. The same as that of I.3.13.1. □

This way we get a complex ^{+}C^{•}(𝔸; M) lying in degrees from −N to 0. It will be called the complex of positive cochains of the sheaf M.

7.12. Theorem. (i) A functor

M ↦ ^{+}C^{•}(𝔸; M)

is an exact functor from M(𝔸; S) to the category of complexes of vector spaces.

(ii) We have a canonical natural isomorphism in D({pt})

+C^{•}(𝔸; M) ⟶^{∼} RΓ(𝔸; M)

Proof. One sees easily that restriction maps

RΓ(𝔸; M) ⟶ RΓ(𝔸_{ℝ}^{+}; M)

are isomorphisms. The rest of the argument is the same as in I.3.14. □

7.13. Let us consider a function ∑_{j} t_{j} : 𝔸 ⟶ 𝔸^{1}, and the corresponding vanishing cycles functor

Φ_{Σ t_{j}} : D^{b}(𝔸) ⟶ D^{b}(𝔸_{(0)})  (138)

where 𝔸_{(0)} = {(t_{j})| ∑_{j} t_{j} = 0}, cf. [KS], 8.6.2.

If K ∈ D(𝔸; S), it is easy to see that the complex Φ_{Σ t_{j}}(K) has the support at the origin. Let us denote by the same letter its stalk at the origin — it is a complex of vector spaces.

7.13.1. Lemma. We have a natural isomorphism

Φ_{Σ t_{j}}(K) ⟶^{∼} Φ_{{0}}^{+}(K)  (139)

Proof is left to the reader. □

7.14. Let us consider the setup of 6.12. Let us denote by the same letter S the stratification of A_{ν} whose strata are subspaces π(S), S being a stratum of the stratification S on ^{π}𝔸. This stratification will be called the principal stratification of A_{ν}.

The function Σ t_{j} is obviously Σ_{π}-equivariant, therefore it induces the function for which we will use the same notation,

Σ t_{j} : A_{ν} ⟶ 𝔸^{1}  (140)

Again, it is easy to see that for K ∈ D^{b}(A_{ν}; S) the complex Φ_{Σt_{j}}(K) has the support at the origin. Let us denote by Φ_{ν}(K) its stalk at the origin.

It is known that the functor of vanishing cycles is t-exact with respect to the middle perversity; whence we get an exact functor

Φ_{ν} : M(A_{ν}; S) ⟶ Vect  (141)

7.14.1. Lemma. Suppose that N is a Σ_{π}- equivariant complex of sheaves over ^{π}𝔸 which belongs to to D^{b}(^{π}𝔸; S) (after forgetting Σ_{π}-action). We have a natural isomorphism

Φ_{ν}((π_{∗}N)^{Σ_{π},−}) ⟶^{∼} (Φ_{Σ t_{j}}(N))^{Σ_{π},−}  (142)

Proof follows from the proper base change for vanishing cycles (see [D3], 2.1.7.1) and the exactness of the functor (·)^{Σ_{π},−}. We leave details to the reader. □

7.14.2. Corollary. For a Σ_{π}-equivariant sheaf N ∈ M(^{π}𝔸_{ν}; S) we have a natural isomorphism of vector spaces

Φ_{ν}(M) ⟶^{∼} (Φ_{{0}}^{+}(N))^{Σ_{π},−}  (143)

where M = (π_{∗}N)^{Σ_{π},−}. □

8. Standard sheaves

Let us keep assumptions and notations of 4.1.

8.1. Let us denote by ^{π}𝔸 the complex affine space with coordinates t_{j}, j ∈ J. We will consider its principal stratification as in 7.1.

Suppose we are given a positive chamber C and a point x = (x_{j})_{j∈J} ∈ C. There extists a unique bijection

σ_{C} : J ⟶^{∼} [N]  (144)

such that for any i, j ∈ J, σ_{C}(i) < σ_{C}(j) iff x_{i} < x_{j}. This bijection does not depend upon the choice of x. This way we can identify the set of all positive chambers with the set of all isomorphisms (144), or, in other words, with the set of all total orderings of J.

Given C and x as above, suppose that we have i, j ∈ J such that x_{i} < x_{j} and there is no k ∈ J such that x_{i} < x_{k} < x_{j}. We will say that i, j are neighbours in C, more precisely that i is a left neighbour of j.

Let x′ = (x′_{j}) be a point with x′_{p} = x_{p} for all p ≠ j, and x′_{j} equal to some number smaller than x_{i} but greater than any x_{k} such that x_{k} < x_{i}. Let ^{ji}C denote the chamber containing x′. Let us introduce a homotopy class of paths ^{C}γ_{ij} connecting x and x′ as shown on Fig. 5(a) below.

On the other hand, suppose that i and 0 are neighbours in C, there is no x_{j} between 0 and x_{i}. Then we introduce the homotopy class of paths from x to itself as shown on Fig. 5 (b).

[figure: i; j; ^{C}γ_{ij}; j′; 0; (a); i; 0; ^{C}γ_{i0}; (b); Fig. 5]

All chambers are contractible. Let us denote by ^{π}𝔸^{∘}_{ℝ}^{+} the union of all positive chambers.

We can apply the discussion I.4.1 and consider the groupoid π_{1}(^{π}𝔸^{∘}, ^{π}𝔸^{∘}_{ℝ}^{+}). It has as the set of objects the set of all positive chambers. The set of morphisms is generated by all morphisms ^{C}γ_{ij} and ^{C}γ_{i0} subject to certain evident braiding relations. We will need only the following particular case.

To define a one-dimensional local system L over ^{π}𝔸^{∘} is the same as to give a set of one-

dimensional vector spaces L_{C}, C ∈ π_{0}(^{π}𝔸^{∘}_{ℝ}^{+}), together with arbitrary invertible linear operators

CT_{ij} : L_{C} ⟶ L_{^{ji}C}  (145)

(”half-monodromies”) defined for chambers where i is a left neighbour of j and

CT_{i0} : L_{C} ⟶ L_{C}  (146)

defined for chambers with neighbouring i and 0.

8.2. Let us fix a weight Λ ∈ X. We define a one-dimensional local system I(^{π}Λ) over ^{π}𝔸^{∘} as follows. Its fibers I(^{π}Λ)_{C} are one-dimensional linear spaces with fixed basis vectors; they will be identified with B.

Monodromies are defined as

CT_{ij} = ζ^{i·j}, ^{C}T_{i0} = ζ^{−2⟨Λ,π(i)⟩},

for i, j ∈ J.

8.3. Let j : ^{π}𝔸^{∘} ⟶ ^{π}𝔸 denote an open embedding. We will study the following objects of M(^{π}𝔸; S):

I(^{π}Λ)_{?} = j_{?}I(^{π}Λ)[N],

where ? =!, ∗. We have a canonical map

m : I(^{π}Λ)_{!} ⟶ I(^{π}Λ)_{∗}  (147)

and by definition I(^{π}Λ)_{!∗} is its image, cf. I.4.5.

COMPUTATIONS FOR I(^{π}Λ)_{!}

8.4. We will use the notations 3.3.2 with I = J and n = 1. For each r ∈ [0, N], let us assign to a map ϱ ∈ P_{r}(J; 1) a point w_{ϱ} = (ϱ(j))_{j∈J} ∈ ^{π}𝔸_{ℝ}. It is easy to see that there exists a unique positive facet F_{ϱ} containing w_{ϱ}, and the rule

ϱ ↦ F_{ϱ}  (148)

establishes an isomorphism between P_{r}(J; 1) and the set of all positive facets of S_{ℝ}. Note that P_{0}(J; 1) consists of one element — the unique map J ⟶ [0]; our stratification has one zero-dimensional facet.

At the same time we have picked a point ^{F_{ϱ}}w := w_{ϱ} on each positive facet F_{ϱ}; this defines cells D_{F}^{+}, S_{F}^{+} (cf. the last remark in 7.2).

8.5. Given ϱ ∈ P_{r}(J; 1) and τ ∈ P_{N}(J; 1), it is easy to see that the chamber C = F_{τ} belongs to Ch(F_{ϱ}) iff τ is a refinement of ϱ in the sense of 3.4. This defines a bijection between the set of all refinements of ϱ and the set of all positive chambers containing F_{ϱ}.

We will denote the last set by Ch^{+}(F_{ϱ}).

8.5.1. Orientations. Let F = F_{ϱ} be a positive facet and C = F_{τ} ∈ Ch^{+}(F). The map τ defines an isomorphism denoted by the same letter

τ : J ⟶^{∼} [N]  (149)

Using τ, the natural order on [N] induces a total order on J. For each a ∈ [r], let m_{a} denote the minimal element of ϱ^{−1}(a), and set

J′ = J_{ϱ≤τ} := J − {m_{1}, …, m_{r}} ⊂ J

Let us consider the map

(x_{j}) ∈ D_{F <C} ↦ {x_{j} − m_{ϱ(j)}|j ∈ J′} ∈ ℝ^{J′}.

It is easy to see that this mapping establishes an isomorphism of the germ of the cell D_{F <C} near the point ^{F} w onto a germ of the cone

{0 ≤ u_{1} ≤ … ≤ u_{N−r}}

in ℝ^{J′} where we have denoted for a moment by (u_{i}) the coordinates in ℝ^{J′} ordered by the order induced from J.

This isomorphism together with the above order defines an orientation on D_{F <C}.

8.6. Basis in Φ_{F}^{+}(I(^{π}Λ)_{!})^{∗}. We follow the pattern of I.4.7. Let F be a positive facet of dimension r. By definition,

Φ_{F}^{+}(I(^{π}Λ)_{!}) = H^{−r}(D_{F}^{+}, S_{F}^{+}; I(^{π}Λ)_{!}) = H^{N−r}(D_{F}^{+}, S_{F}^{+}; j_{!}I(^{π}Λ)) ≅ H^{N−r}(D_{F}^{+}, S_{F}^{+}∪(_{H}H_{ℝ}∩D_{F}^{+}); j_{!}I(^{π}Λ)).

Note that we have

D_{F}^{+} − (S_{F}^{+} ∪ _{H}H_{ℝ}) = ⋃_{C∈Ch^{+}(F)} D^{∘}_{F <C}

(disjoint union), therefore by additivity

H^{N−r}(D_{F}^{+}, S_{F}^{+} ∪ (_{H}H_{ℝ} ∩ D_{F}^{+}); j_{!}I(^{π}Λ)) ≅ ⊕_{C∈Ch^{+}(F)}H^{N−r}(D_{F <C}, S_{F <C}; j_{!}I(^{π}Λ)).

By Poincaré duality,

H^{N−r}(D_{F <C}, S_{F <C}; j_{!}I(^{π}Λ))^{∗} ≅ H^{0}(D^{∘}_{F <C}; I(^{π}Λ)^{−1})

— here we have used the orientations of cells D_{F <C} introduced above. By definition of the local system I, the last space is canonically identified with B.

If F = F_{ϱ}, C = F_{τ}, we will denote by c_{ϱ≤τ} ∈ Φ_{F}^{+}(I(^{π}Λ)_{!})^{∗} the image of

1 ∈ H^{0}(D^{∘}_{F <C}; I(^{π}Λ)^{−1}). Thus the chains c_{ϱ≤τ}, τ ∈ Ord(ϱ), form a basis of the space Φ_{F}^{+}(I(^{π}Λ)_{!})^{∗}.

8.7. Diagrams. It is convenient to use the following diagram notations for chains c_{ϱ≤τ}.

Let us denote elements of J by letters a, b, c, . . .. An r-dimensional chain c_{ϱ≤τ} where ϱ : J ⟶ [0, r], is represented by a picture:

[figure: …; 0; a; 1; b; c; r; d; e; Fig. 6. Chain c_{ϱ≤τ}.]

A picture consists of r + 1 fragments:

[figure: i; a; b; c; Fig. 7. Set ϱ^{−1}(i).]

where i = 0, …, r, the i-th fragment being a blank circle with a number of small vectors going from it. These vectors are in one-to-one correspondence with the set ϱ^{−1}(i); their ends are labeled by elements of this set. Their order (from left to right) is determined by the order on ϱ^{−1}(i) induced by τ. The point 0 may have no vectors (when ϱ^{−1}(0) = ∅); all other points have at least one vector.

8.8. Suppose we have ϱ ∈ P_{r}(J; 1), ϱ′ ∈ P_{r+1}(J; 1). It is easy to see that F_{ϱ} < F_{ϱ′} if and only if there exists i ∈ [0, r] such that ϱ = δ_{i} ∘ ϱ′ where δ_{i} : [0, r + 1] ⟶ [0, r] carries a to a if a ≤ i and to a − 1 if a ≥ i + 1. We will write in this case that ϱ < ϱ′.

Let us compute the dual to the canonical map

+u^{∗} : Φ_{F_{ϱ}}^{+}(I(^{π}Λ)_{!})^{∗} ⟶ Φ_{F_{ϱ′}}^{+}(I(^{π}Λ)_{!})^{∗}.  (150)

Suppose we have τ ∈ Ord(ϱ′); set C = F_{τ}, thus F_{ϱ} < F_{ϱ′} < C. Let us define the sign

sgn(ϱ < ϱ′ ≤ τ) = (−1)^{∑_{j=i+1}^{r+1} card((ϱ′)^{−1}(j)−1)}  (151)

This sign has the following geometrical meaning. The cell D_{F_{ϱ′<C}} lies in the boundary of D_{F_{ϱ<C}}. We have oriented these cells above. Let us define the compatibility of these orientations as follows. Complete an orienting basis of the smaller cell by a vector directed outside the larger cell — if we get the orientation of the larger cell, we say that the orientations are compatible, cf. I.4.6.1.

It is easy to see from the definitions that the sign (151) is equal +1 iff the orientations of D_{F_{ϱ′<C}} and D_{F_{ϱ<C}} are compatible. As a consequence, we get

8.8.1. Lemma. The map (150) has the following matrix:

+u^{∗}(c_{ϱ≤τ}) = ∑ sgn(ϱ < ϱ′ ≤ τ)c_{ϱ′≤τ},

the summation over all ϱ′ such that F_{ϱ} < F_{ϱ′} < F_{τ} and dim F_{ϱ′} = dim F_{ϱ} + 1. □

8.9. Isomorphisms ^{π}ϕ_{!}. We will use notations of 3.4 with I replaced by J, 𝔉 by ^{π}𝔉, with n = 1 and Λ_{0} = ^{π}Λ.

Thus, for any r ∈ [0, N] the set {b_{ϱ≤τ}|ϱ ∈ P_{r}(J; 1), τ ∈ Ord(ϱ)} is a basis of +C^{−r}(^{π}𝔸; I(^{π}Λ)_{!}).

Let us pick η ∈ P_{N}(J). Any τ ∈ P_{N}(J; 1) induces the bijection τ′ : J ⟶^{∼} [N]. We will denote by sgn(τ, η) = ±1 the sign of the permutation τ′η^{−1}.

Let us denote by {b_{ϱ≤τ}|τ ∈ Ord(ϱ)} the basis in Φ_{F_{ϱ}}^{+}(I(^{π}Λ)_{!}) dual to {c_{ϱ≤τ}}.

Let us define isomorphisms

πϕ_{ϱ,!}^{(η)} : Φ_{F_{ϱ}}^{+}(I(^{π}Λ)_{!}) ⟶^{∼} _{ϱ}C_{^{π}𝔉}^{−r}(V (^{π}Λ))  (152)

by the formula

πϕ_{ϱ,!}^{(η)}(b_{ϱ≤τ}) = sgn(τ, η)sgn(ϱ)θ_{ϱ≤τ}  (153)

(see (93)) where

sgn(ϱ) = (−1)^{∑_{i=1}^{r}(r−i+1)·(card(ϱ^{−1}(i))−1)}  (154)

for ϱ ∈ P_{r}(J; 1). Taking the direct sum of ^{π}ϕ_{ϱ,!}^{(η)}, ϱ ∈ P_{r}(J; 1), we get isomorphisms

πϕ_{r,!}^{(η)} : ^{+}C^{−r}(^{π}𝔸; I(^{π}Λ)_{!}) ⟶^{∼} _{χ_{J}} C_{^{π}𝔉}^{−r}(V (^{π}Λ))  (155)

A direct computation using 8.8.1 shows that the maps ^{π}ϕ_{r,!}^{(η)} are compatible with differentials. Therefore, we arrive at

8.10. Theorem. The maps ^{π}ϕ_{r,!}^{(η)} induce an isomorphism of complexes

πϕ_{!}^{(η)} : ^{+}C^{•}(^{π}𝔸; I(^{π}Λ)_{!}) ⟶^{∼} _{χ_{J}} C_{^{π}𝔉}^{•}(V (^{π}Λ)) □  (156)

COMPUTATIONS FOR I(^{π}Λ)_{∗}

8.11. Bases. The Verdier dual to I(^{π}Λ)_{∗} is canonically isomorphic to I(^{π}Λ)_{!}^{−1}. Therefore, by Theorem 7.4 for each positive facet F we have natural isomorphisms

Φ_{F}^{+}(I(^{π}Λ)_{∗})^{∗} ≅ Φ_{F}^{+}(I(^{π}Λ)_{!}^{−1})  (157)

Let {c̃_{ϱ≤τ}|τ ∈ Ord(ϱ)} be the basis in Φ_{F}^{+}(I(^{π}Λ)_{!}^{−1})^{∗} defined in 8.6, with I(^{π}Λ) replaced by I(^{π}Λ)^{−1}. We will denote by {c_{ϱ≤τ,∗}|τ ∈ Ord(ϱ)} the dual basis in Φ_{F}^{+}(I(^{π}Λ)_{∗})^{∗}. Finally, we will denote by {b_{ϱ≤τ,∗}|τ ∈ Ord(ϱ)} the basis in Φ_{F}^{+}(I(^{π}Λ)_{∗}) dual to the previous one.

Our aim in the next subsections will be the description of canonical morphisms m : Φ_{F}^{+}(I(^{π}Λ)_{!}) ⟶ Φ_{F}^{+}(I(^{π}Λ)_{∗}) and of the cochain complex ^{+}C^{•}(^{π}𝔸; I(^{π}Λ)_{∗}) in terms of our bases.

8.12. Example. Let us pick an element, i ∈ I and let π : J := {i} ↪ I. Then we are in a one-dimensional situation, cf. 7.5. The space ^{π}𝔸 has one coordinate t_{i}. By definition, the local system I(^{π}Λ) over ^{π}𝔸^{∘} = ^{π}𝔸 − {0} with the base point w : t_{i} = 1 has the fiber I_{w} = B and monodromy equal to ζ^{−2⟨Λ,i⟩} along a counterclockwise loop.

The stratification S_{ℝ} has a unique 0-dimensional facet F = F_{ϱ} corresponding to the unique element ϱ ∈ P_{0}(J; 1), and a unique 1-dimensional positive facet E = F_{τ} corresponding to the unique element τ ∈ P_{1}(J; 1).

Let us construct the dual chain c_{1}^{∗} := c_{ϱ≤τ,∗}. We adopt the notations of 7.5, in particular of Fig.1. The chain c̃_{1} := c̃_{ϱ≤τ} ∈ H^{1}(^{π}𝔸, {0, w}; j_{!}I(^{π}Λ)^{−1})^{∗} is shown on Fig. 8(a) below. As a first step, we define the dual chain c_{1}^{0} ∈ H^{1}(^{π}𝔸 − {0, w},^{π} 𝔸_{≥r′′}; I(^{π}Λ))^{∗} — it is also shown on Fig. 8(a).

[figure: F; S_{r′′}; E; c̃_{1}; c_{1}^{0}; w; S_{r′}; (a); S_{r′}; ϵc̃_{1}; c_{1}^{∗}; ϵw; (b); Fig. 8.]

Next, we make a clockwise rotation of c_{1}^{0} on π/2, and make a homotopy inside the disk π𝔸_{≤r′} to a chain c_{1}^{∗} ∈ H^{1}(^{π}𝔸_{≤r′} − {0}, ϵ · w; I(^{π}Λ))^{∗} beginning and ending at ϵ · w. This chain is shown on Fig. 8(b) (in a bigger scale). Modulo ”homothety” identification

H^{1}(^{π}𝔸_{≤r′} − {0}, ϵ · w; I(^{π}Λ))^{∗} ≅ H^{1}(^{π}𝔸 − {0}, w; I(^{π}Λ))^{∗} = Φ_{F} (I(^{π}Λ)_{∗})^{∗}

this chain coincides with c_{ϱ≤τ,∗}.

The canonical map

m^{∗} : Φ_{F}^{+}(I(^{π}Λ)_{∗})^{∗} ⟶ Φ_{F}^{+}(I(^{π}Λ)_{!})^{∗}

carries c_{ϱ≤τ,∗} to [⟨Λ, i⟩]_{ζ} · c_{ϱ≤τ}. The boundary map

d^{∗} : Φ_{F}^{+}(I(^{π}Λ)_{∗})^{∗} ⟶ Φ_{E}^{+}(I(^{π}Λ)_{∗})^{∗}

carries c_{ϱ≤τ,∗} to [⟨Λ, i⟩]_{ζ} · c_{τ≤τ,∗}.

8.13. Vanishing cycles at the origin. Let us return to the general situation. First let us consider an important case of the unique 0-dimensional facet — the origin. Let 0 denote the unique element of P_{0}(J; 1). To shorten the notation, let us denote Φ_{F_{0}}^{+} by Φ_{0}^{+}.

The bases in Φ_{0}^{+}(I(^{π}Λ)_{!}), etc. are numbered by all bijections τ : J ⟶^{∼} [N]. Let us pick such a bijection. Let τ^{−1}(i) = {j_{i}}, i = 1, …, N. The chain c_{0≤τ} ∈ Φ_{0}^{+}(I(^{π}Λ)_{!})^{∗} is depicted as follows:

[figure: 0; j_{1}; j_{2}; j_{3}; …; j_{N}; Fig. 9. c_{0≤τ}]

Using considerations completely analogous to the one-dimensional case above, we see that the dual chain c_{0≤τ,∗} ∈ Φ_{0}^{+}(I(^{π}Λ)_{∗})^{∗} is portayed as follows:

[figure: 0; j_{1}; j_{2}; j_{3}; …; j_{N}; Fig. 10. c_{0≤τ,∗}]

The points t_{j_{i}} are travelling independently along the corresponding loops, in the indicated directions. The section of I(^{π}Λ)^{−1} over this cell is determined by the requierement to be equal to 1 when all points are equal to marking points, at the end of their the travel (coming from below).

8.14. Isomorphisms ϕ_{0,∗}^{(η)}. We will use notations of 3.4 and 3.5 with I replaced by J, 𝔉 — by ^{π}𝔉, with n = 1 and Λ_{0} = ^{π}Λ. By definition, C_{^{π}𝔉}^{0}(V (^{π}Λ)) = V (^{π}Λ). The space V (^{π}Λ)_{χ_{J}} admits as a basis (of cardinality N!) the set consisting of all monomials

θ_{0≤τ} = θ_{τ^{−1}(N)}θ_{τ^{−1}(N−1)} · . . . · θ_{τ^{−1}(1)}v_{^{π}Λ},

where τ ranges through the set of all bijections J ⟶^{∼} [N]. By definition, {θ_{0≤τ}^{∗}} is the dual basis of V (^{π}Λ)_{χ_{J}}^{∗}. Let us pick η ∈ P_{N}(J). Let us define an isomorphism

πϕ_{0,∗}^{(η)} : Φ_{0}^{+}(I(^{π}Λ)_{∗}) ⟶^{∼} V (^{π}Λ)_{χ_{J}}^{∗}  (158)

by the formula

πϕ_{0,∗}^{(η)}(b_{0≤τ}) = sgn(τ, η)θ_{0≤τ}^{∗}  (159)

8.15. Theorem. The square

Φ_{0}^{+}(I(^{π}Λ)_{!}) ⟶^{∼}^{^{π}ϕ_{0,!}^{(η)}} V (^{π}Λ)_{χ_{J}}
m ↓  ↓ S_{^{π}Λ}
Φ_{0}^{+}(I(^{π}Λ)_{∗}) ⟶^{∼}^{^{π}ϕ_{0,∗}^{(η)}} V (^{π}Λ)_{χ_{J}}^{∗}

commutes. Proof. This follows directly from the discussion of 8.13 and the definition of the form S_{^{π}Λ}. □

8.16. Let us pass to the setup of 6.12 and 7.14. Let j_{ν} : A^{∘}_{ν} ↪ A_{ν} denote the embedding of the open stratum of the principal stratification. Set

I_{ν}(Λ) = (π_{∗}I(^{π}Λ))^{Σ_{π},−}  (160)

It is a local system over A^{∘}_{ν}. Let us define objects

I_{ν}(Λ)_{?} := j_{ν?}I_{ν}(Λ)[N] ∈ M(A_{ν}; S)  (161)

where ? =!, ∗ or !∗. These objects will be called standard sheaves over A_{ν}. The same reasoning as in 6.13 proves

8.16.1. Lemma. We have natural isomorphisms

I_{ν}(Λ)_{?} ≅ (π_{∗}I(^{π}Λ)_{?})^{Σ_{π},−}  (162)

for ? =!, ∗ or !∗. □

8.17. For a given η ∈ P_{N}(J) the isomorphisms ^{π}ϕ_{0,∗}^{(η)} and ^{π}ϕ_{0,!}^{(η)} are skew Σ_{π}- equivariant. Therefore, after passing to invariants in Theorem 8.15 we get 8.18. Theorem. The maps ^{π}ϕ_{0,∗}^{(η)} and ^{π}ϕ_{0,!}^{(η)} induce isomorphisms included into a commutative square

Φ_{ν}(I_{ν}(Λ)_{!}) ⟶^{∼}^{ϕ_{ν,Λ,!}^{(η)}} V (Λ)_{ν}
m ↓  ↓ S_{Λ}
Φ_{ν}(I_{ν}(Λ)_{∗}) ⟶^{∼}^{ϕ_{ν,Λ,∗}^{(η)}} V (Λ)_{ν}^{∗}

and

ϕ_{ν,Λ,!∗}^{(η)} : Φ_{ν}(I_{ν}(Λ)_{!∗}) ⟶^{∼} L(Λ)_{ν}  (163)

Proof follows from the previous theorem and Lemma 4.7.1(ii). □

8.19. Now suppose we are given an arbitrary r, ϱ ∈ P_{r}(J; 1) and τ ∈ Ord(ϱ). The picture of the dual chain c_{ϱ≤τ,∗} is a combination of Figures 10 and 3. For example, the chain dual to the one on Fig. 6 is portayed as follows:

[figure: …; 0; a; 1; b; c; r; d; e; Fig. 11. Chain c_{ϱ≤τ,∗}.]

8.20. Isomorphisms ^{π}Φ_{∗}. Let us pick η ∈ P_{N}(J). Let us define isomorphisms

πϕ_{ϱ,∗}^{(η)} : Φ_{F_{ϱ}}^{+}(I(^{π}Λ)_{∗}) ⟶^{∼} _{ϱ}C_{^{π}𝔉^{∗}}^{−r}(V (^{π}Λ)^{∗})  (164)

by the formula

πϕ_{ϱ,∗}^{(η)}(b_{ϱ≤τ,∗}) = sgn(τ, η)sgn(ϱ)θ_{ϱ≤τ}^{∗}  (165)

where sgn(ϱ) is defined in (154).

Taking the direct sum of ^{π}ϕ_{ϱ,∗}^{(η)}, ϱ ∈ P_{r}(J; 1), we get isomorphisms

πϕ_{r,∗}^{(η)} : ^{+}C^{−r}(^{π}𝔸; I(^{π}Λ)_{∗}) ⟶^{∼} _{χ_{J}} C_{^{π}𝔉^{∗}}^{−r}(V (^{π}Λ)^{∗})  (166)

8.21. Theorem. The maps ^{π}ϕ_{r,∗}^{(η)} induce an isomorphism of complexes

πϕ_{∗}^{(η)} : ^{+}C^{•}(^{π}𝔸; I(^{π}Λ)_{∗}) ⟶^{∼} _{χ_{J}} C_{^{π}𝔉^{∗}}^{•}(V (^{π}Λ)^{∗})  (167)

which makes the square

+C^{•}(^{π}𝔸; I(^{π}Λ)_{!}) ⟶^{∼}^{^{π}ϕ_{!}^{(η)}}  _{χ_{J}} C_{^{π}𝔉}^{•}(V (^{π}Λ))
m ↓  ↓ S
+C^{•}(^{π}𝔸; I(^{π}Λ)_{∗}) ⟶^{∼}^{^{π}ϕ_{∗}^{(η)}}  _{χ_{J}} C_{^{π}𝔉^{∗}}^{•}(V (^{π}Λ)^{∗})

commutative.

Proof. Compatibility with differentials is verified directly and commutativity of the square are checked directly from the geometric description of our chains (actually, it is sufficient to check one of these claims — the other one follows formally).

Note the geometric meaning of operators t_{i} from (71) — they correspond to the deletion of the i-th loop on Fig. 10. □

8.22. Now let us pass to the situation 8.16. It follows from Theorem 7.12 (after passing to skew Σ_{π}- invariants) that the complexes ^{+}C^{•}(^{π}𝔸; I_{ν?})^{Σ_{π},−} compute the stalk of I_{ν?} at the origin. Let us denote this stalk by I_{ν?,0}.

Therefore, passing to Σ_{π}-invariants in the previous theorem, we get

8.23. Theorem. The isomorphisms ^{π}ϕ_{?}^{(η)} where ? =!, ∗ or !∗, induce isomorphisms in D^{b}(pt) included into a commutative square

I_{ν}(Λ)_{!,0} ⟶^{∼}^{_{Λ}ϕ_{ν,!,0}^{(η)}}  _{ν}C_{𝔉}^{•}(V (Λ))
m ↓  ↓ S
I_{ν}(Λ)_{∗,0} ⟶^{∼}^{_{Λ}ϕ_{ν,∗,0}^{(η)}} _{ν}C_{𝔉^{∗}}^{•}(V (Λ)^{∗})

and

Λϕ_{ν,!∗,0}^{(η)} : I_{ν}(Λ)_{!∗,0} ⟶^{∼} _{ν}C_{𝔣}^{•}(L(Λ)) □  (168)

Chapter 3. Fusion

9. Additivity theorem

9.1. Let us start with the setup of 7.1. For a non-negative integer n let us denote by (n) the set [−n, 0]. Let us introduce the following spaces. 𝔸^{(n)} - a complex affine space with a fixed system of coordinates (t_{i}), i ∈ (n). Let ^{n}J denote the disjoint union (n) ∪ J; ^{n}𝔸 — a complex affine space with coordinates t_{j}, j ∈ ^{n}J.

In general, for an affine space with a distinguished coordinate system of, we will denote by S_{Δ} its diagonal stratification as in 6.1.

Let ^{n}𝔸^{∘} ⊂ ^{n}𝔸, 𝔸^{∘} ^{(n)} ⊂ 𝔸^{(n)} be the open strata of S_{Δ}.

Let ^{n}p : ^{n}𝔸 ⟶ 𝔸^{(n)} be the evident projection; ^{n}𝔹 = ^{n}p^{−1}(𝔸^{∘} ^{(n)}). Given a point z = (z_{i}) ∈ 𝔸^{(n)}, let us denote by z𝔸 the fiber ^{n}p^{−1}(z) and by zS the stratification induced by S_{Δ}. We will consider t_{j}, j ∈ J, as coordinates on z𝔸.

The subscript (.)_{ℝ} will mean as usually ”real points”.

9.2. Let us fix a point z = (z_{−1}, z_{0}) ∈ 𝔸_{ℝ}^{(2)} such that z_{−1} < z_{0}. Let us concentrate on the fiber z𝔸. As an abstract variety it is canonically isomorphic to 𝔸 — a complex affine space with coordinates t_{j}, j ∈ J; so we will suppress z from its notation, keeping it in the notation for the stratification zS where the dependence on z really takes place.

Let us fix a real w > z_{0}. Let us pick two open non-intersecting disks D_{i} ⊂ ℂ with centra at z_{i} and not containing w. Let us pick two real numbers w_{i} > z_{i} such that w_{i} ∈ D_{i}, and paths P_{i} connecting w with w_{i} as shown on Fig. 12 below.

[figure: z_{−1}; D_{−1}; w; U_{−1}; w_{−1}; P_{−1}; z_{0}; D_{0}; w_{0}; P_{0}; U_{0}; Fig. 12.]

Let us denote by Q_{2}(J) the set of all maps ρ : J ⟶ [−1, 0]. Given such a map, let us denote by 𝔸_{ρ} ⊂ 𝔸 an open subvariety consisting of points (t_{j})_{j∈J} such that t_{j} ∈ D_{ρ(j)} for all j. We will denote by the same letter zS the stratification of this space induced by zS.

Set H_{w} := ∪_{j} H_{j}(w); P = P_{−1} ∪ P_{0}; P̃ = {(t_{j}) ∈ 𝔸| there exists j such that t_{j} ∈ P }. Given K ∈ D(𝔸; zS), the restriction map

RΓ(𝔸, H_{w}; K) ⟶ RΓ(𝔸, P̃; K)

is an isomorphism by homotopy. On the other hand, we have restriction maps RΓ(𝔸, P̃; K) ⟶ RΓ(𝔸_{ρ}, P̃_{ρ}; K)

where P̃_{ρ} := P̃ ∩ 𝔸_{ρ}. Therefore we have canonical maps

r_{ρ} : RΓ(𝔸, H_{w}; K) ⟶ RΓ(𝔸_{ρ}, P̃_{ρ}; K)  (169)

9.3. Theorem. For every K ∈ D(𝔸; zS) the canonical map

r = ∑ r_{ρ} : RΓ(𝔸, H_{w}; K) ⟶ ⊕_{ρ∈Q_{2}(J)}RΓ(𝔸_{ρ}, P̃_{ρ}; K)  (170)

is an isomorphism.

9.4. Proof. Let us pick two open subsets U_{−1}, U_{0} ⊂ ℂ as shown on Fig. 12. Set U = U_{−1} ∪ U_{0}, 𝔸_{U} = {(t_{j}) ∈ 𝔸|t_{j} ∈ U for all j}. It is clear that the restriction morphism

RΓ(𝔸, P̃; K) ⟶ RΓ(𝔸_{U}, P̃_{U}; K)

where P̃_{U} := P̃ ∩ 𝔸_{U}, is an isomorphism. For each ρ ∈ Q_{2}(J) set

𝔸_{U,ρ} := {(t_{j}) ∈ 𝔸_{U}|t_{j} ∈ U_{ρ(j)} for all j}; P̃_{U,ρ} := P̃ ∩ 𝔸_{U,ρ}  (171)

We have 𝔸_{U} = ⋃_{ρ} 𝔸_{U,ρ}.

9.4.1. Lemma. For every K ∈ D(𝔸;z S) the sum of restriction maps

q : RΓ(𝔸_{U}, P̃_{U}; K) ⟶ ∑_{ρ∈Q_{2}(J)} RΓ(𝔸_{U,ρ}, P̃_{U,ρ}; K)  (172)

is an isomorphism. Proof. Suppose we have distinct ρ_{1}, …, ρ_{m} such that 𝔸_{U;ρ_{1},... ,ρ_{m}} := 𝔸_{U,ρ_{1}}∩…∩𝔸_{U,ρ_{m}} ≠ ∅; set P̃_{U;ρ_{1},... ,ρ_{m}} := P̃ ∩ 𝔸_{U;ρ_{1},... ,ρ_{m}}. Our lemma follows at once by Mayer-Vietoris argument from the following 9.4.2. Claim. For every m ≥ 2

RΓ(𝔸_{U;ρ_{1},... ,ρ_{m}}, P̃_{U;ρ_{1},... ,ρ_{m}}; K) = 0.

Proof of the claim. It is convenient to use the following notations. If J = A ∪ B is a disjoint union, we will denote by ρ_{A,B} the map J ⟶ [−1, 0] such that ρ^{−1}(−1) = A, ρ^{−1}(0) = B, and by U_{A;B} the subspace 𝔸_{U,ρ_{A,B}}.

Let us prove the claim for the case N = 2. Let J = {i, j}. In this case it is easy to see that the only non-trivial intersections are U^{(1)} = U_{ij;∅} ∩ U_{j;i} and U^{(2)} = U_{ij;∅} ∩ U_{∅;ij}.

To prove our claim for U^{(1)} we will use a shrinking neighbourhood argument based on Lemma I.2.12. Let K′ denote the sheaf on U^{(1)} obtained by extension by zero of K|_{U^{(1)}−P̃}. For each ϵ > 0 let us denote by U_{−1,ϵ} ⊂ ℂ an open domain consisting of points having distance < ϵ from D_{−1} ∪ P. Set

U_{ϵ}^{(1)} = {(t_{i}, t_{j})| t_{i} ∈ U_{−1,ϵ}, t_{j} ∈ U_{0} ∩ U_{−1,ϵ}}.

It is clear that restriction maps RΓ(U^{(1)}; K′) ⟶ RΓ(U_{ϵ}^{(1)}; K′) are isomorphisms. On the other hand, it follows from I.2.12 that

lim_{→}_{ϵ} RΓ(U_{ϵ}^{(1)}; K′) ≅ RΓ(⋂_{ϵ} U_{ϵ}^{(1)}; K′),

and the last complex is zero by the definition of K′ (the point t_{j} is confined to P in ⋂_{ϵ} U_{ϵ}^{(1)}).

The subspace U^{(2)} consists of (t_{i}, t_{j}) such that both t_{i} and t_{j} lie in U_{−1} ∩ U_{0}. This case is even simpler. The picture is homeomorphic to an affine plane with a sheaf smooth along the diagonal stratification; and we are interested in its cohomology modulo the coordinate cross. This is clearly equal to zero, i.e. RΓ(U^{(2)}, P̃ ∩ U^{(2)}; K) = 0.

This proves the claim for N = 2. The case of an arbitrary N is treated in a similar manner, and we leave it to the reader. This completes the proof of the claim and of the lemma. □

9.4.3. Lemma. For every ρ ∈ Q_{2}(J) the restriction map

RΓ(𝔸_{U,ρ}, P̃_{U,ρ}; K) ⟶ RΓ(𝔸_{ρ}, P̃_{ρ}; K)

is an isomorphism.

Proof. Again let us consider the case J = {i, j}. If ρ = ρ_{ij;∅} or ρ_{∅;ij} the statement is obvious. Suppose ρ = ρ_{i;j}. Let us denote by 𝔸′_{U,ρ} ⊂ 𝔸_{U,ρ} the subspace {(t_{i}, t_{j})|t_{i} ∈ D_{−1}, t_{j} ∈ U_{0}}. It is clear that the restriction map

RΓ(𝔸′_{U,ρ}, P̃′_{U,ρ}; K) ⟶ RΓ(𝔸_{ρ}, P̃_{ρ}; K)

where P̃′_{U,ρ} := P̃ ∩ 𝔸′_{U,ρ}, is an isomorphism. Let us consider the restriction

RΓ(𝔸_{U,ρ}, P̃_{U,ρ}; K) ⟶ RΓ(𝔸′_{U,ρ}, P̃′_{U,ρ}; K)  (173)

The cone of this map is isomorphic to RΓ(𝔸_{U,ρ}, P̃_{U,ρ}; M) where the sheaf M has the same singularities as K and in addition is 0 over the closure D̄_{−1}. Now, consider a system of shrinking neighbourhoods of P_{−1} ∪ D̄_{−1} as in the proof af the claim above, we see that RΓ(𝔸_{U,ρ}, P̃_{U,ρ}; M) = 0, i.e. (173) is an isomorphism. This implies the lemma for this case.

The case of arbitrary J is treated exactly in the same way. □

Our theorem is an obvious consequence of two previous lemmas. □

10. Fusion and tensor products

10.1. Fusion functors. The constructions below were inspired by [Dr].

For each integer n ≥ 1 and i ∈ [n], let us define functors

nψ_{i} : D(^{n}𝔸; S_{Δ}) ⟶ D(^{n−1}𝔸; S_{Δ})  (174)

as follows. We have the t-exact nearby cycles functors (see [D3] or [KS], 8.6, but note the shift by [-1]!)

Ψ_{t_{−i}−t_{−i+1}}[−1] : D(^{n}𝔸; S_{Δ}) ⟶ D(𝔸′; S_{Δ})

where 𝔸′ denotes (for a moment) an affine space with coordinates t_{j}, j ∈ ((n)∪J)−{−i}. We can identify the last space with ^{n−1}𝔸 simply by renaming coordinates t_{j} to t_{j+1} for −n ≤ j ≤ −i − 1. By definition, ^{n}ψ_{i} is equal to Ψ_{t_{−i}−t_{−i+1}}[−1] followed by this identification.

10.2. Lemma. (i) For each n ≥ 2, i ∈ [n] have canonical isomorphisms

nα_{i} : ^{n−1}ψ_{i} ∘ ^{n}ψ_{i} ⟶^{∼} ^{n−1}ψ_{i} ∘ ^{n}ψ_{i+1}  (175)

and equalities

n−1ψ_{j} ∘ ^{n}ψ_{i} = ^{n−1}ψ_{i} ∘ ^{n}ψ_{j+1}

for j > i, such that

(ii) (”Stasheff pentagon” identity) the diagram below commutes:

[figure: ^{n−2}ψ_{i} ∘ ^{n−1}ψ_{i} ∘ ^{n}ψ_{i}; ^{n−2}ψ_{i} ∘ ^{n−1}ψ_{i} ∘ ^{n}ψ_{i+1}; ^{n−2}ψ_{i} ∘ ^{n−1}ψ_{i+1} ∘ ^{n}ψ_{i+1}; ^{n−2}ψ_{i} ∘ ^{n−1}ψ_{i+1} ∘ ^{n}ψ_{i+2}; ^{n−2}ψ_{i} ∘ ^{n−1}ψ_{i+1} ∘ ^{n}ψ_{i}; ^{n−2}ψ_{i} ∘ ^{n−1}ψ_{i} ∘ ^{n}ψ_{i+2}; ^{n−2}ψ_{i} ∘ ^{n}α_{i}; ^{n−1}α_{i} ∘ ^{n}ψ_{i+1}; ^{n−2}ψ_{i} ∘ ^{n}α_{i+1}; ^{n−1}α_{i} ∘ ^{n}ψ_{i}; ^{n−1}α_{i} ∘ ^{n}ψ_{i+2}]

□

10.3. Let us define a t-exact functor

ψ : D(^{n}𝔸; S_{Δ}) ⟶ D(𝔸; S)  (176)

as a composition i_{0}^{∗}[−1] ∘ ^{n}ψ_{1} ∘ ^{n−1}ψ_{1} ∘ … ∘ ^{1}ψ_{1}, where

i_{0}^{∗} : D(^{0}𝔸; S_{Δ}) ⟶ D(𝔸; S)

denotes the restriction to the subspace t_{0} = 0. Note that i_{0}^{∗}[−1] is a t-exact equivalence. (Recall that 𝔸 and S denote the same as in 7.1).

STANDARD SHEAVES

The constructions and computations below generalize Section 8.

10.4. Let us make the following assumptions. Let us denote by A the I ×I-matrix (i·j). Let us suppose that det A ≠ 0. There exists a unique ℤ[\frac{1}{det A}]-valued symmetric bilinear form on X (to be denoted by λ, μ ↦ λ·μ) such that the map ℤ[I] ⟶ X, ν ↦ λ_{ν} respects scalar products.

Let us suppose that our field B contains an element ζ′ such that (ζ′)^{det A} = ζ, and fix such ζ′. For a = \frac{c}{det A}, c ∈ ℤ, we set by definition ζ^{a} := (ζ′)^{c}.

10.5. Let us fix ν = ∑ ν_{i}i ∈ ℕ[I] and its unfolding π : J ⟶ I as in 6.12, and an integer n ≥ 1. We will use the preceding notations with this J.

Let us fix n + 1 weights Λ_{0}, Λ_{−1}, …, Λ_{−n} ∈ X. We define a one-dimensional local system I(Λ_{0}, …, Λ_{−n}; ν) over ^{n}𝔸^{∘} exactly in the same manner as in 6.3, with half-monodromies equal to ζ^{π(i)·π(j)} if i, j ∈ J, to ζ^{−⟨Λ_{i},π(j)⟩} if i ∈ (n), j ∈ J and to ζ^{Λ_{i}·Λ_{j}} if i, j ∈ (n).

Let j : ^{n}𝔸^{∘} ⟶ ^{n}𝔸 be the embedding. Let us introduce the sheaves

I(Λ_{0}, …, Λ_{−n}; ν)_{?} := j_{?}I(Λ_{0}, …, Λ_{−n}; ν)[−n − N − 1] ∈ M(^{n}𝔸; S_{Δ}) (177)

where ? =!, ∗ or !∗. Applying the functor ψ, we get the sheaves ψI(Λ_{0}, …, Λ_{−n}; ν)_{?} ∈ M(𝔸; S).

All these objects are naturally Σ_{π}-equivariant. We define the following sheaves on A_{ν}:

ψI_{ν}(Λ_{0}, …, Λ_{−n})_{?} := (π_{∗}ψI(Λ_{0}, …, Λ_{−n}; ν)_{?})^{Σ_{π},−}  (178)

(cf. 8.16).

The following theorem generalizes Theorem 8.18.

10.6. Theorem. Given a bijection η : J ⟶^{∼} [N], we have natural isomorphisms included into a commutative square

Φ_{ν}(^{ψ}I_{ν}(Λ_{0}, …, Λ_{−n})_{!}) ⟶^{∼}^{ϕ_{!}^{(η)}} (V (Λ_{0}) ⊗ … ⊗ V (Λ_{−n}))_{ν}
m ↓  ↓ S_{Λ}
Φ_{ν}(^{ψ}I_{ν}(Λ_{0}, …, Λ_{−n})_{∗}) ⟶^{∼}^{ϕ_{∗}^{(η)}} (V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{−n})^{∗})_{ν}

and

ϕ_{!∗}^{(η)} : Φ_{ν}(^{ψ}I_{ν}(Λ_{0}, …, Λ_{−n})_{!∗}) ⟶^{∼} (L(Λ_{0}) ⊗ … ⊗ L(Λ_{−n}))_{ν}  (179)

A change of η multiplies these isomorphisms by the sign of the corresponding permutation of [N].

Proof. We may suppose that π is injective, i.e. all ν_{i} = 0 or 1. The general case immediately follows from this one after passing to Σ_{π}-skew invariants.

Let us consider the case n = 1. In this case one sees easily from the definitions that we have a canonical isomorphism

Φ_{ν}(^{ψ}I_{ν}(Λ_{0}, Λ_{−1})_{?}) ≅ RΓ(𝔸, H_{w}; I(Λ_{0}, Λ_{−1}; ν)_{?})

in the notations of Additivity Theorem 9.3. On the other hand, the set Q_{2}(J) is in oneto-one correspondence with the set of all decompositions ν = ν_{0} + ν_{−1}, ν_{i} ∈ ℕ[I], and if ρ corresponds to such a decomposition, we have a natural isomorphism

RΓ(𝔸_{ρ}, P̃ ∩ 𝔸_{ρ}; I_{ν}(Λ_{0}, Λ_{−1})_{?}) ≅ Φ_{ν_{0}}(^{ψ}I(Λ_{0}; ν_{0})_{?}) ⊗ Φ_{ν_{−1}}(^{ψ}I(Λ_{−1}; ν_{−1})_{?}) (180)

by the Künneth formula. Therefore, Additivity Theorem implies isomorphisms

Φ_{ν}(^{ψ}I_{ν}(Λ_{0}, Λ_{−1})_{?}) ≅ ⊕_{ν_{0}+ν_{−1}=ν}Φ_{ν_{0}}(^{ψ}I_{ν_{0}}(Λ_{0})_{?}) ⊗ Φ_{ν_{−1}}(^{ψ}I_{ν_{−1}}(Λ_{−1})_{?})

which are the claim of our theorem.

The case n > 2 is obtained similarly by the iterated use of the Additivity Theorem. □

10.7. Next we will consider the stalks (?)_{0} of our sheaves at 0, or what is the same (since they are ℝ^{+∗}-homogeneous), the complexes RΓ(A_{ν}; ?).

The next theorem generalizes Theorem 8.23.

10.8. Theorem. Given a bijection η : J ⟶^{∼} [N], we have natural isomorphisms included into a commutative square

ψI_{ν}(Λ_{0}, …, Λ_{−n})_{!0} ⟶^{∼}^{ϕ_{!,0}^{(η)}}  _{ν}C_{𝔉}^{•}(V (Λ_{0}) ⊗ … ⊗ V (Λ_{−n}))
m ↓  ↓ S_{Λ}
ψI_{ν}(Λ_{0}, …, Λ_{−n})_{∗0} ⟶^{∼}^{ϕ_{∗,0}^{(η)}}  _{ν}C_{𝔉^{∗}}^{•}(V (Λ_{0})^{∗} ⊗ … ⊗ V (Λ_{−n})^{∗})

and

ϕ_{!∗,0}^{(η)} : ^{ψ}I_{ν}(Λ_{0}, …, Λ_{−n})_{!∗0} ⟶^{∼} _{ν}C_{𝔣}^{•}(L(Λ_{0}) ⊗ … ⊗ L(Λ_{−n}))  (181)

A change of η multiplies these isomorphisms by the sign of the corresponding permutation of [N].

Proof. It is not hard to deduce from the previous theorem that we have natural isomorphisms of complexes included into a commutative square

+C(𝔸; ψI(Λ_{0}, …, Λ_{−n}; ν)_{!}) ⟶^{∼}  _{χ_{J}} C_{^{π}𝔉}^{•}(V (^{π}Λ_{0}) ⊗ … ⊗ V (^{π}Λ_{−n}))
m ↓  ↓ S_{Λ}
+C(𝔸; ψI(Λ_{0}, …, Λ_{−n}; ν)_{∗}) ⟶^{∼} _{χ_{J}} C_{^{π}𝔉^{∗}}^{•}(V (^{π}Λ_{0})^{∗} ⊗ … ⊗ V (^{π}Λ_{−n})^{∗})

This implies our claim after passing to Σ_{π}-(skew) invariants. □

Chapter 4. Category C

11. Simply laced case

11.1. From now on untill the end of the paper we will assume, in addition to the assumptions of 1.4, that ζ is a primitive l-th root of unity, where l is a fixed integer l > 3 prime to 2 and 3.

11.2. We will use notations of [L1], Chapters 1, 2, which we briefly recall.

11.2.1. Let (I, ·) be a simply laced Cartan datum of finite type (cf. loc.cit., 1.1.1, 2.1.3), that is, a finite set I and a nondegenerate symmetric bilinear form α, β ↦ α · β on the free abelian group ℤ[I]. This form satisfies conditions

(a) i · i = 2 for any i ∈ I;

(b) i · j ∈ {0, −1} for any i ≠ j in I.

11.2.2. We will consider the simply connected root datum of type (I, ·), that is (see loc. cit., 2.2.2), two free abelian groups Y = ℤ[I] and X = Hom_{ℤ}(Y, ℤ) together with

(a) the canonical bilinear pairing ⟨, ⟩ : Y × X ⟶ ℤ;

(b) an obvious embedding I ↪ Y (i ↦ i) and an embedding I ↪ X (i ↦ i′), such that ⟨i, j′⟩ = i · j for any i, j ∈ I.

We will call X the lattice of weights, and Y the lattice of coroots. An element of X will be typically denoted by λ, μ, ν, …; and an element of Y will be typically denoted by α, β, γ, . . ..

11.3. We consider the finite dimensional algebra U over the field B defined as in the section 1.3 of [AJS]. We also consider the category C of finite dimensional X-graded U-modules defined as in the section 2.3 of [AJS].

11.3.1. The algebra U is given by generators E_{i}, F_{i}, K_{i}^{±1}, i ∈ I, subject to relations

(z) K_{i} · K_{i}^{−1} = 1; K_{i}K_{j} = K_{j}K_{i};

(a) K_{j}E_{i} = ζ^{j·i}E_{i}K_{j};

(b) K_{j}F_{i} = ζ^{−j·i}F_{i}K_{j};

(c) E_{i}F_{j} − F_{j}E_{i} = δ_{ij} \frac{K_{i}−K_{i}^{−1}}{ζ−ζ^{−1}} ;

(d) E_{i}^{l} = F_{j}^{l} = 0;

(e) E_{i}E_{j} − E_{j}E_{i} = 0 if i · j = 0; E_{i}^{2}E_{j} − (ζ + ζ^{−1})E_{i}E_{j}E_{i} + E_{i}E_{j}^{2} = 0 if i · j = −1;

(f) F_{i}F_{j} − F_{j}F_{i} = 0 if i · j = 0; F_{i}^{2}F_{j} − (ζ + ζ^{−1})F_{i}F_{j}F_{i} + F_{i}F_{j}^{2} = 0 if i · j = −1.

The algebra U has a unique B-algebra X-grading U = ⊕U_{μ} for which |E_{i}| = i′, |F_{i}| = −i′, |K_{i}| = 0. We define a comultiplication

Δ : U ⟶ U ⊗ U  (182)

as a unique B-algebra mapping such that Δ(K^{±1}) = K^{±1} ⊗ K^{±1};

Δ(E_{i}) = E_{i} ⊗ K_{i} + 1 ⊗ E_{i};
Δ(F_{i}) = F_{i} ⊗ 1 + K_{i}^{−1} ⊗ F_{i}.

This makes U a Hopf algebra (with obvious unit and counit).

11.3.2. The category C is by definition a category of finite dimensional X-graded Bvector spaces V = ⊕_{μ∈X}V_{μ}, equipped with a left action of U such that the U-action is compatible with the X-grading, and

K_{i}x = ζ^{⟨i,μ⟩}x

for x ∈ V_{μ}, i ∈ I. Since U is a Hopf algebra, C has a canonical structure of a tensor category.

11.4. We define an algebra 𝔲 having generators θ_{i}, ϵ_{i}, K_{i}^{±1}, i ∈ I, subject to relations (z) K_{i} · K_{i}^{−1} = 1; K_{i}K_{j} = K_{j}K_{i};

(a) K_{j}ϵ_{i} = ζ^{j·i}ϵ_{i}K_{j};

(b) K_{j}θ_{i} = ζ^{−j·i}θ_{i}K_{j};

(c) ϵ_{i}θ_{j} − ζ^{i·j}θ_{j}ϵ_{i} = δ_{ij}(1 − K_{i}^{−2})

(d) if f ∈ Ker(S) ⊂ 𝔉 (see (65)) then f = 0;

(e) the same as (d) for the free algebra 𝔈 on the generators ϵ_{i}.

11.4.1. Let us define the comultiplication

Δ : 𝔲 ⟶ 𝔲 ⊗ 𝔲  (183)

by the formulas

Δ(K_{i}^{±1}) = K_{i}^{±1} ⊗ K_{i}^{±1};
Δ(θ_{i}) = θ_{i} ⊗ 1 + K_{i}^{−1} ⊗ θ_{i};
Δ(ϵ_{i}) = ϵ_{i} ⊗ 1 + K_{i}^{−1} ⊗ ϵ_{i}

and the condition that Δ is a morphism of B-algebras. This makes 𝔲 a Hopf algebra (with obvious unit and counit). 𝔲 is an X-graded B-algebra, with an X-grading defined uniquely by the conditions |K_{i}^{±1}| = 0; |θ_{i}| = −i′; |ϵ_{i}| = i′.

11.5. We define C̃ as a category of finite dimensional X-graded vector spaces V = ⊕V_{λ}, equipped with a structure of a left 𝔲-module compatible with X-gradings and such that

K_{i}x = ζ^{⟨i,λ⟩}x

for x ∈ V_{λ}, i ∈ I.

Since 𝔲 is a Hopf algebra, C̃ is a tensor category.

11.6. Recall that for any Λ ∈ X we have defined in 5.2 the X-graded 𝔣-module L(Λ). It is a quotient-module of the Verma module V (Λ), and it inherits its X-grading from the one of V (Λ) (see 2.15). Thus L(Λ) = ⊕L(Λ)_{λ}, and we define the action of generators K_{i} on L(Λ)_{λ} as multiplication by ζ^{⟨i,λ⟩}. Finally, we define the action of generators ϵ_{i} on V (Λ) as in 2.16. These operators on V (Λ) satisfy the relations (a) — (c) above.

We check immediately that this action descends to the quotient L(Λ). Moreover, it follows from Theorem 2.23 that these operators acting on L(Λ) satisfy the relations (a) — (e) above. So we have constructed the action of 𝔲 on L(Λ), therefore we can regard it as an object of C̃.

11.6.1. Lemma. L(Λ) is an irreducible object in C̃.

Proof. Let I(Λ) be the maximal proper homogeneous (with respect to X-grading) submodule of V (Λ) (the sum of all homogeneous submodules not containing v_{Λ}). Then V (Λ)/I(Λ) is irreducible, so it suffices to prove that I(Λ) = ker(S_{Λ}). The inclusion ker(S_{Λ}) ⊂ I(Λ) is obvious. Let us prove the opposite inclusion. Let y ∈ I(Λ). It is enough to check that S_{Λ}(y, x) = 0 for any x ∈ V (Λ) of the form θ_{i_{1}} … θ_{i_{n}}v_{Λ}. By (22) we have S_{Λ}(y, θ_{i_{1}} … θ_{i_{n}}v_{Λ}) = S_{Λ}(ϵ_{i_{n}} … ϵ_{i_{1}}y, v_{Λ}) = 0 since ϵ_{i_{n}} … ϵ_{i_{1}}y ∈ I(Λ). □

11.7. Let us consider elements E_{i}, F_{i} ∈ 𝔲 given by the following formulas:

E_{i} = \frac{ζ^{2}}{ζ − ζ^{−1}} ϵ_{i}K_{i}; F_{i} = θ_{i}  (184)

It is immediate to check that these elements satisfy the relations 11.3.1 (a) — (c).

Moreover, one checks without difficulty that

θ_{i}θ_{j} − θ_{j}θ_{i} ∈ Ker(S) if i · j = 0,

and

θ_{i}^{2}θ_{j} − (ζ + ζ^{−1})θ_{i}θ_{j}θ_{i} + θ_{j}θ_{i}^{2} ∈ Ker(S) if i · j = −1

(cf. [SV2], 1.16). Also, it is immediate that

S(θ_{i}^{a}, θ_{i}^{a}) = ∏_{p=1}^{a} \frac{1 − ζ^{2p}}{1 − ζ^{2}} .

It follows that θ_{i}^{l} ∈ Ker(S) for all i.

It follows easily that the formulas (184) define a surjective morphism of algebras

R : U ⟶ 𝔲  (185)

Moreover, one checks at once that R is a map of Hopf algebras.

Therefore, R induces a tensor functor

Q : C̃ ⟶ C  (186)

which is an embedding of a full subcategory.

11.8. Theorem. Q is an equivalence.

Proof. It is enough to check that C̃ contains enough projectives for C (see e.g. Lemma A.15. of [KL] IV).

First of all, we know from [AJS], section 4.1, that the simple 𝔲-modules L(Λ), Λ ∈ X exhaust the list of simple objects of C. Second, we know from [APK2], Theorem 4.6 and Remark 4.7, that the module L(−ρ) is projective where −ρ ∈ X is characterized by the property ⟨i, −ρ⟩ = −1 for any i ∈ I. Finally we know, say, from [APK2], Remark 4.7, and [APK1], Lemma 9.11, that the set of modules {L(Λ) ⊗ L(−ρ), Λ ∈ X} is an ample system of projectives for C. □

11.9. Denote by 𝔲^{0} (resp., U^{0}) the subalgebra of 𝔲 (resp., of U) generated by K_{i}^{±1}, i ∈ I. Obviously, both algebras are isomorphic to the ring of Laurent polynomials in K_{i}, and the map R induces an identity isomorphism between them.

Denote by 𝔲^{≤0} (resp., by 𝔲^{−}) the subalgebra of 𝔲 generated by θ_{i}, K_{i}^{±1} (resp., by θ_{i}), i ∈ I. The last algebra may be identified with 𝔣. As a vector space, 𝔲^{≤0} is isomorphic to 𝔣 ⊗ 𝔲^{0}.

Denote by U^{−} ⊂ U the subalgebra generated by F_{i}, i ∈ I, and by U^{≤0} ⊂ U the subalgebra generated by F_{i}, K_{i}^{±1}, i ∈ I. As a vector space it is isomorphic to U^{−} ⊗ U^{0}.

11.10. Theorem. (a) R is an isomorphism;

(b) R induces an isomorphism U^{−} ⟶^{∼} 𝔣.

Proof. Evidently it is enough to prove b). We know that R is surjective, and that U^{−} is finite dimensional. So it suffices to prove that dim U^{−} ≤ dim 𝔣. We know by [L1] 36.1.5. that dim U^{−} = dim L(−ρ). On the other hand, the map 𝔣 ⟶ L(−ρ), f ↦ f(v_{−ρ}) is surjective by construction. □

12. Non-simply laced case

In the non-simply laced case all main results of this paper hold true as well. However, the definitions need some minor modifications. In this section we will describe them.

12.1. We will use terminology and notations from [L1], especially from Chapters 1-3. Let us fix a Cartan datum (I, ·) of finite type, not necessarily simply laced, cf. loc. cit., 2.1.3. Let (Y = ℤ[I], X = Hom(Y, ℤ), ⟨, ⟩, I ↪ X, I ↪ Y) be the simply connected root datum associated with (I, ·), loc.cit., 2.2.2.

We set d_{i} := \frac{i·i}{2}, i ∈ I; these numbers are positive integers. We set ζ_{i} := ζ^{d_{i}}.

The embedding I ↪ X sends i ∈ I to i′ such that ⟨d_{j}j, i′⟩ = j · i for all i, j ∈ I.

12.2. The category C is defined in the same way as in the simply laced case, where the definition of the Hopf algebra U should be modified as follows (cf. [AJS], 1.3).

By definition, U has generators E_{i}, F_{i}, K_{i}^{±1}, i ∈ I, subject to relations

(z) K_{i} · K_{i}^{−1} = 1; K_{i}K_{j} = K_{j}K_{i};

(a) K_{j}E_{i} = ζ^{⟨j,i′⟩}E_{i}K_{j};

(b) K_{j}F_{i} = ζ^{−⟨j,i′⟩}F_{i}K_{j};

(c) E_{i}F_{j} − F_{j}E_{i} = δ_{ij} \frac{K̃_{i}−K̃_{i}^{−1}}{ζ_{i}−ζ_{i}^{−1}} ;

(d) E_{i}^{l} = F_{j}^{l} = 0;

(e) ∑_{p=0}^{1−⟨i,j′⟩}(−1)^{p}E_{i}^{(p)}E_{j}E_{i}^{(1−⟨i,j′⟩−p)} = 0 for i ≠ j;

(f) ∑_{p=0}^{1−⟨i,j′⟩}(−1)^{p}F_{i}^{(p)}F_{j}F_{i}^{(1−⟨i,j′⟩−p)} = 0 for i ≠ j,

where we have used the notations: K̃_{i} := K_{i}^{d_{i}}; G_{i}^{(p)} := G_{i}^{p}/[p]_{i}^{!}, G = E or F,

[p]_{i}^{!} := ∏_{a=1}^{p} \frac{ζ_{i}^{a} − ζ_{i}^{−a}}{ζ_{i} − ζ_{i}^{−1}} .

The X-grading on U is defined in the same way as in the simply laced case.

The comultiplication is defined as

Δ(K_{i}) = K_{i} ⊗ K_{i};
Δ(E_{i}) = E_{i} ⊗ K̃_{i} + 1 ⊗ E_{i};
Δ(F_{i}) = F_{i} ⊗ 1 + K̃_{i}^{−1} ⊗ F_{i}.

12.2.1. Remark. This algebra is very close to (and presumably coincides with) the algebra U from [L1], 3.1, specialized to v = ζ. We use the opposite comultiplication, though.

12.3. The definition of the Hopf algebra 𝔲 should be modified as follows. It has generators θ_{i}, ϵ_{i}, K_{i}^{±1}, i ∈ I, subject to relations

(z) K_{i} · K_{i}^{−1} = 1; K_{i}K_{j} = K_{j}K_{i};

(a) K_{j}ϵ_{i} = ζ^{⟨j,i′⟩}ϵ_{i}K_{j}; (b) K_{j}θ_{i} = ζ^{−⟨j,i′⟩}θ_{i}K_{j};

(c) ϵ_{i}θ_{j} − ζ^{i·j}θ_{j}ϵ_{i} = δ_{ij}(1 − K̃_{i}^{−2})

(d) if f ∈ Ker(S) ⊂ 𝔉 (see (65)) then f = 0;

(e) the same as (d) for the free algebra 𝔈 on the generators ϵ_{i}. The comultiplication is defined as

Δ(K_{i}^{±1}) = K_{i}^{±1} ⊗ K_{i}^{±1};

Δ(θ_{i}) = θ_{i} ⊗ 1 + K̃_{i}^{−1} ⊗ θ_{i};
Δ(ϵ_{i}) = ϵ_{i} ⊗ 1 + K̃_{i}^{−1} ⊗ ϵ_{i}

The category C̃ is defined in the same way as in the simply laced case.

12.4. We define an X-grading on the free 𝔉-module V (Λ) with generator v_{Λ} by setting V (Λ)_{Λ} = B · v_{Λ} and assuming that operators θ_{i} decrease the grading by i′. The definition of the form S_{Λ} on V (Λ) should be modified as follows. It is a unique bilinear from such that S_{Λ}(v_{Λ}, v_{Λ}) = 1 and S(θ_{i}x, y) = S(x, ϵ_{i}y) where the operators ϵ_{i} : V (Λ) ⟶ V (Λ) are defined by the requirements ϵ_{i}(v_{Λ}) = 0,

ϵ_{i}(θ_{j}x) = ζ^{i·j}θ_{j}ϵ_{i}(x) + δ_{ij}[⟨i, λ⟩]_{ζ_{i}}x

for x ∈ V (Λ)_{λ}. We define L(Λ) as a quotient V (Λ)/Ker(S). As in the simply laced case, L(Λ) is naturally an object of C, and the same argument proves that it is irreducible.

12.5. We define the morphism

R : U ⟶ 𝔲  (187)

by the formulas

R(E_{i}) = \frac{ζ_{i}^{2}}{ζ_{i} − ζ_{i}^{−1}} ϵ_{i}K̃_{i}; R(F_{i}) = θ_{i}; R(K_{i}) = K_{i}  (188)

Using [L1], 1.4.3, one sees immediately that it is correctly defined morphism of algebras. It follows at once from the definitions that R is a morphism of Hopf algebras. Hence, we get a tensor functor

Q : C̃ ⟶ C  (189)

and the same proof as in 11.8 shows that Q is an equivalence of categories. It is a result of primary importance for us. It implies in particular that all irreducibles in C (as well as their tensor products), come from C̃.

12.6. Suppose we are given Λ_{0}, …, Λ_{−n} ∈ X and ν ∈ ℕ[I]. Let π : J ⟶ I, π : ^{π}𝔸 ⟶ A_{ν} denote the same as in 6.12. We will use the notations for spaces and functors from Section 10.

The definition of the local system I(Λ_{0}, …, Λ_{−n}; ν) from loc. cit. should be modified: it should have half-monodromies ζ_{j}^{−⟨π(j),Λ_{i}⟩} for i ∈ (n), j ∈ J, the other formulas stay without change.

After that, the standard sheaves are defined as in loc. cit. Now we are arriving at the main results of this paper. The proof is the same as the proof of theorems 10.6 and 10.8, taking into account the previous algebraic remarks.

12.7. Theorem. Let L(Λ_{0}), …, L(Λ_{−n}) be irreducibles of C, λ ∈ X, λ = ∑_{m=0}^{n} Λ_{−m} − ∑_{i} ν_{i}i′ for some ν_{i} ∈ ℕ. Set ν = ∑ ν_{i}i ∈ ℕ[I].

Given a bijection η : J ⟶^{∼} [N], we have natural isomorphisms

ϕ_{!∗}^{(η)} : Φ_{ν}(^{ψ}I_{ν}(Λ_{0}, …, Λ_{−n})_{!∗}) ⟶^{∼} (L(Λ_{0}) ⊗ … ⊗ L(Λ_{−n}))_{λ}  (190)

A change of η multiplies these isomorphisms by the sign of the corresponding permutation of [N]. □

12.8. Theorem. In the notations of the previous theorem we have natural isomorphisms

ϕ_{!∗,0}^{(η)} : ^{ψ}I_{ν}(Λ_{0}, …, Λ_{−n})_{!∗0} ⟶^{∼} C_{𝔣}^{•}(L(Λ_{0}) ⊗ … ⊗ L(Λ_{−n}))_{λ}  (191)

where we used the notation C^{•}(…)_{λ} for _{ν}C^{•}(…). A change of η multiplies these isomorphisms by the sign of the corresponding permutation of [N]. □

Part III. TENSOR CATEGORIES

ARISING FROM CONFIGURATION SPACES

1. Introduction

1.1. In Chapter 1 we associate with every Cartan matrix of finite type and a non-zero complex number ζ an abelian artinian category FS. We call its objects finite factorizable sheaves. They are certain infinite collections of perverse sheaves on configuration spaces, subject to a compatibility (”factorization”) and finiteness conditions.

In Chapter 2 the tensor structure on FS is defined using functors of nearby cycles. It makes FS a braided tensor category.

In Chapter 3 we define, using vanishing cycles functors, an exact tensor functor

Φ : FS ⟶ C

to the category C connected with the corresponding quantum group, cf. [AJS], 1.3 and [FS] II, 11.3, 12.2.

In Chapter 4 we prove

1.2. Theorem. Φ is an equivalence of categories.

One has to distinguish two cases.

(i) ζ is not root of unity. In this case it is wellknown that C is semisimple. This case is easier to treat; 1.2 is Theorem 18.4.

(ii) ζ is a root of unity. This is of course the most interesting case; 1.2 is Theorem 17.1.

Φ may be regarded as a way of localizing 𝔲-modules from category C to the origin of the affine line 𝔸^{1}. More generally, in order to construct tensor structure on FS, we define for each finite set K certain categories ^{K}FS along with the functor g_{K} : FS^{K} ⟶ ^{K}FS (see the Sections 9, 10) which may be regarded as a way of localizing K-tuples of 𝔲modules to K-tuples of points of the affine line. In a subsequent paper we plan to show how to localize 𝔲-modules to the points of an arbitrary smooth curve. For example, the case of a projective line is already quite interesting, and is connected with ”semiinfinite” cohomology of quantum groups.

We must warn the reader that the proofs of some technical topological facts are only sketched in this paper. The full details will appear later on.

1.3. The construction of the space A in Section 2 is inspired by the idea of “semiinfinite space of divisors on a curve” one of us learnt from A.Beilinson back in 1990. The construction of the braiding local system I in Section 3 is very close in spirit to P.Deligne’s letter [D1]. In terms of this letter, all the local systems I_{μ}^{α} arise from the semisimple braided tensor category freely generated by the irreducibles i ∈ I with the square of R-matrix: i ⊗ j ⟶ j ⊗ i ⟶ i ⊗ j given by the scalar matrix ζ^{i·j}.

We are very grateful to B.Feigin for the numerous inspiring discussions, and to P.Deligne and L.Positselsky for the useful comments concerning the definition of morphisms in ^{K}FS in 9.6.

1.4. Notations. We will use all the notations of parts I and II. References to loc.cit. will look like Z.1.1 where Z=I or II.

During the whole paper we fix a Cartan datum (I, ·) of finite type and denote by (Y = ℤ[I]; X = Hom(Y, ℤ); I ↪ Y, i ↦ i; I ↪ X, i ↦ i′) the simply connected root datum associated with (I, ·), [L1], 2.2.2. Given α = ∑ a_{i}i ∈ Y, we will denote by α′ the element ∑ a_{i}i′ ∈ X. This defines an embedding

Y ↪ X  (192)

We will use the notation d_{i} := \frac{i·i}{2}. We have ⟨j′, d_{i}i⟩ = i · j. We will denote by A the I × I-matrix (⟨i, j′⟩). We will denote by λ, μ ↦ λ · μ a unique ℤ[\frac{1}{det A}]- valued scalar product on X such that (192) respects scalar products. We have

λ · i′ = ⟨λ, d_{i}i⟩  (193)

for each λ ∈ X, i ∈ I.

We fix a non-zero complex number ζ′ and suppose that our ground field B contains ζ′. We set ζ := (ζ′)^{det A}; for a = \frac{c}{det A}, c ∈ ℤ, we will use the notation ζ^{a} := (ζ′)^{c}.

We will use the following partial orders on X and Y. For α = ∑ a_{i}i, β = ∑ b_{i}i ∈ Y we write α ≤ β if a_{i} ≤ b_{i} for all i. For λ, μ ∈ X we write λ ≥ μ if λ − μ = α′ for some α ∈ Y, α ≥ 0.

1.5. If X_{1}, X_{2} are topological spaces, K_{i} ∈ D(X_{i}), i = 1, 2, we will use the notation K_{1} ⊠ K_{2} for p_{1}^{∗}K_{1} ⊗ p_{2}^{∗}K_{2} (where p_{i} : X_{1} × X_{2} ⟶ X_{i} are projections). If J is a finite set, |J| will denote its cardinality.

For a constructible complex K, SS(K) will denote the singular support of K (microsupport in the terminology of [KS], cf. loc.cit., ch. V).

Chapter 1. Category FS

2. Space A

2.1. We will denote by 𝔸^{1} the complex affine line with a fixed coordinate t. Given real c, c′, 0 ≤ c < c′, we will use the notations D(c) = {t ∈ 𝔸^{1}| |t| < c}; D̄(c) = {t ∈ 𝔸^{1}| |t| ≤ c}; 𝔸_{>c}^{1} := {t ∈ 𝔸^{1}| |t| > c}; D(c, c′) = 𝔸_{>c}^{1} ∩ D(c′).

Recall that we have introduced in II.6.12 configuration spaces A_{α} for α ∈ ℕ[I]. If α = ∑ a_{i}i, the space A_{α} parametrizes configurations of I-colored points t = (t_{j}) on 𝔸^{1}, such that there are precisely a_{i} points of color i.

2.2. Let us introduce some open subspaces of A_{α}. Given a sequence

α⃗ = (α_{1}, …, α_{p}) ∈ ℕ[I]^{p}  (194)

and a sequence of real numbers

d⃗ = (d_{1} …, d_{p−1})  (195)

such that 0 < d_{p−1} < d_{p−2} … < d_{1}, p ≥ 2, we define an open subspace

Aα⃗(d⃗) ⊂ A_{α}  (196)

which parametrizes configurations t such that α_{p} of points t_{j} lie inside the disk D(d_{p−1}), for 2 ≤ i ≤ p − 1, α_{i} of points lie inside the annulus D(d_{i−1}, d_{i}), and α_{1} of points lie inside the ring A_{>d_{1}}^{1}.

For p = 1, we set A^{α}(∅) := A_{α}.

By definition, a configuration space of empty collections of points consists of one point. For example, so is A^{0}(∅).

2.2.1. Cutting. Given i ∈ [p − 1] define subsequences

d⃗_{≤i} = (d_{1}, …, d_{i}); d⃗_{≥i} = (d_{i}, …, d_{p−1})  (197)

and

α⃗_{≤i} = (α_{1}, …, α_{i}, 0); α⃗_{≥i} = (0, α_{i+1}, …, α_{p})  (198)

We have obvious cutting isomorphisms

c_{i} : Aα⃗(d⃗) ⟶^{∼} Aα⃗_{≤i}(d⃗_{≤i}) × Aα⃗_{≥i}(d⃗_{≥i})  (199)

satisfying the following compatibility:

for i < j the square

[figure: Aα⃗_{≤i}(d⃗_{≤i}) × Aα⃗_{≥i;≤j}(d⃗_{≥i;≤j}) × Aα⃗_{≥j}(d⃗_{≥j}); Aα⃗(d⃗); Aα⃗_{≤j}(d⃗_{≤j}) × Aα⃗_{≥j}(d⃗_{≥j}); Aα⃗_{≤i}(d⃗_{≤i}) × Aα⃗_{≥i}(d⃗_{≥i}); c_{j}; c_{i} × id; id × c_{j}; c_{i}]

commutes.

2.2.2. Dropping. For i as above, let ∂_{i}d⃗ denote the subsequence of d⃗ obtained by dropping d_{i}, and set

∂_{i}α⃗ = (α_{1}, …, α_{i−1}, α_{i} + α_{i+1}, α_{i+2}, …, α_{p})  (200)

We have obvious open embeddings

A^{∂_{i}α⃗}(∂_{i}d⃗) ↪ Aα⃗(d⃗)  (201)

2.3. Let us define Aα⃗ ; μ(d⃗) as the space Aα⃗(d⃗) equipped with an additional index μ ∈ X. One should understand μ as a weight assigned to the origin in 𝔸^{1}. We will abbreviate the notation A_{μ}^{α}(∅) to A_{μ}^{α}.

Given a triple (α⃗, d⃗, μ) as above, let us define its i-cutting — two triples (α⃗_{≤i}, d⃗_{≤i}, μ_{≤i}) and (α⃗_{≥i}, d⃗_{≥i}, μ), where

μ⃗_{≤i} = μ⃗ − ( ∑_{j=i+1}^{p} α_{j})′.

We will also consider triples (∂_{i}α⃗, ∂_{i}d⃗, ∂_{i}μ) where ∂_{i}μ = μ if i < p−1, and ∂_{p−1}μ = μ−α′_{p}.

The cutting isomorphisms (199) induce isomorphisms

Aα⃗ ; μ(d⃗) ⟶^{∼} Aα⃗_{≤i} ; μ_{≤i}(d⃗_{≤i}) × Aα⃗_{≥i} ; μ(d⃗_{≥i})  (202)

2.4. For each μ ∈ X, α = ∑ a_{i}i, β = ∑ b_{i}i ∈ ℕ[I],

σ = σ_{μ}^{α,β} : A_{μ}^{α} ↪ A_{μ+β′}^{α+β}  (203)

will denote a closed embedding which adds b_{i} points of color i equal to 0.

For d > 0, A_{μ+β′}^{(α,β)}(d) is an open neighbourhood of σ(A_{μ}^{α}) in A_{μ+β′}^{β+α}.

2.5. By definition, A is a collection of all spaces Aα⃗ ; μ(d⃗) as above, together with the cutting isomorphisms (202) and the closed embeddings (203).

2.6. Given a coset c ∈ X/Y (where we regard Y as embedded into X by means of a map i ↦ i′), we define A_{c} as a subset of A consisting of A_{μ}^{α} such that μ ∈ c. Note that the closed embeddings σ, as well as cutting isomorphisms act inside A_{c}. This subset will be called a connected component of A. The set of connected components will be denoted π_{0}(A). Thus, we have canonically π_{0}(A) ≅ X/Y.

2.7. We will be interested in two stratifications of spaces A_{μ}^{α}. We will denote by A^{•}_{μ}^{α} ⊂ A_{μ}^{α} the complement

A_{μ}^{α} − ⋃_{β<α} σ(A_{μ−β′+α′}^{β}).

We define a toric stratification of A_{μ}^{α} as

A_{μ}^{α} = ∐ σ(A^{•}_{μ−β′+α′}^{β}).

Another stratification of A_{μ}^{α} is the principal stratification defined in II.7.14. Its open stra-

tum will be denoted by A^{∘}_{μ}^{α} ⊂ A_{μ}^{α}. Unless specified otherwise, we will denote the prinicipal stratification on spaces A_{μ}^{α}, as well as the induced stratifications on its subspaces, by S.

The sign ∘ (resp., •) over a subspace of A_{μ}^{α} will denote the intersection of this subspace

with A^{∘}_{μ}^{α} (resp., with A^{•}_{μ}^{α}).

3. Braiding local system I

3.1. Local systems I_{μ}^{α}. Let us recall some definitions from II. Let α = ∑_{i} a_{i}i ∈ ℕ[I] be given. Following II.6.12, let us choose an unfolding of α, i.e. a set J together with a map π : J ⟶ I such that |π^{−1}(i)| = a_{i} for all i. We define the group Σ_{π} := {σ ∈ Aut(J)| σ ∘ π = π}.

We define ^{π}𝔸 as an affine space with coordinates t_{j}, j ∈ J; it is equipped with the principal stratification defined by hyperplanes t_{j} = 0 and t_{i} = t_{j}, cf. II.7.1. The group Σ_{π} acts on ^{π}𝔸 by permutations of coordinates, respecting the stratification. By definition, A_{α} = ^{π}𝔸/Σ_{π}. We will denote by the same letter π the canonical projection ^{π}𝔸 ⟶ A_{α}.

If ^{π}𝔸^{∘} ⊂ ^{π}𝔸 denotes the open stratum of the principal stratification, π(^{π}𝔸^{∘}) = A^{∘}_{π}, and

the restriction of π to ^{π}𝔸^{∘} is unramified covering.

Suppose a weight μ ∈ X is given. Let us define a one dimensional local system ^{π}I_{μ} over

π𝔸^{∘} by the procedure II.8.1. Its fiber over each positive chamber C ∈ π_{0}(^{π}𝔸^{∘}_{ℝ}) is identified with B; and monodromies along standard paths shown on II, Fig. 5 (a), (b) are given by the formulas

CT_{ij} = ζ^{−π(i)·π(j)}, ^{C}T_{i0} = ζ^{2μ·π(i)′}  (204)

respectively (cf. (193)). (Note that, by technical reasons, this definition differs by the sign from that of II.8.2 and II.12.6).

We have a canonical Σ_{π}-equivariant structure on ^{π}I_{μ}, i.e. a compatible system of isomorphisms

i_{σ} : ^{π}I_{μ} ⟶^{∼} σ^{∗} ^{π}I_{μ}, σ ∈ Σ_{π},  (205)

defined uniquely by the condition that

(i_{σ})_{C} = id_{B} : (^{π}I_{μ})_{C} = B ⟶^{∼} (σ^{∗} ^{π}I_{μ})_{σC} = B

for all (or for some) chamber C. As a consequence, the group Σ_{π} acts on the local system π_{∗} ^{π}I_{μ}.

Let sgn : Σ_{π} ⟶ {±1} denote the sign character. We define a one-dimensional local

system I_{μ}^{α} over A^{∘}_{μ}^{α} = A^{∘}_{α} as follows:

I_{μ}^{α} := (π_{∗} ^{π}I_{μ})^{sgn}  (206)

where the superscript (•)^{sgn} denotes the subsheaf of sections x such that σx = sgn(σ)x for all σ ∈ Σ_{π}. Cf. II.8.16.

Alternatively, we can define this local system as follows. By the descent, there exists a unique local system I′_{μ} over A_{α} such that π^{∗}I′_{μ} is equal to ^{π}I_{μ} with the equivariant structure described above. In fact,

I′_{μ} = (π_{∗} ^{π}I_{μ})^{Σ_{π}}

where the superscript (•)^{Σ_{π}} denotes invariants. We have

I_{μ}^{α} = I′_{μ} ⊗ S}\  (207)

where S}\ denotes the one-dimensional local system over A_{α} associated with the sign representation π_{1}(A_{α}) ⟶ Σ_{π} ⟶^{sgn} {±1}.

This definition does not depend (up to a canonical isomorphism) upon the choice of an unfolding.

3.2. For each triple (α⃗, d⃗, μ) as in the previous section, let us denote by Iα⃗ ; μ(d⃗) the re-

striction of I_{μ}^{α} to the subspace A^{∘}α⃗ ; μ(d⃗) ⊂ A^{∘}^{α}(μ) where α ∈ ℕ[I] is the sum of components of α⃗.

Let us define factorization isomorphisms

ϕ_{i} = ϕα⃗ ; μ;i(d⃗) : Iα⃗ ; μ(d⃗) ⟶^{∼} Iα⃗_{≤i} ; μ_{≤i}(d⃗_{≤i}) ⊠ Iα⃗_{≥i} ; μ(d⃗_{≥i})  (208)

(we are using identifications (202)). By definition, we have canonical identifications of the stalks of all three local systems over a point with real coordinates, with B. We define (208) as a unique isomorphism acting as identity when restricted to such a stalk. We will omit irrelevant indices from the notation for ϕ if there is no risk of confusion.

3.3. Associativity. These isomorphisms have the following associativity property. For all i < j, diagrams

[figure: I(d⃗_{≤i}) ⊠ I(d⃗_{≥i;≤j}) ⊠ I(d⃗_{≥j}); I(d⃗); I(d⃗_{≤j}) ⊠ I(d⃗_{≥j}); I(d⃗_{≤i}) ⊠ I(d⃗_{≥i}); ϕ_{j}; ϕ_{i} ⊠ id; id ⊠ ϕ_{j}; ϕ_{i}]

are commutative. In fact, it is enough to check the commutativity restricted to some fiber (•)_{C}, where it is obvious.

3.4. The collection of local systems I = {I_{μ}^{α}} together with factorization isomorphisms (208) will be called the braiding local system (over A^{∘}). The couple (A, I) will be called the semi-infinite configuration space associated with the Cartan datum (I, ·) and parameter ζ.

3.5. Let j : A^{∘}α⃗ ; μ(d⃗) ↪ A^{•}α⃗ ; μ(d⃗) denote an embedding; let us define a preverse sheaf

I^{•}α⃗ ; μ(d⃗) := j_{!∗}Iα⃗ ; μ(d⃗)[dim Aα⃗ ; μ] ∈ M(A^{•}α⃗ ; μ(d⃗); S)

By functoriality, factorization isomorphisms (208) induce analogous isomorphisms (denoted by the same letter)

ϕ_{i} = ϕα⃗ ; μ;i(d⃗) : I^{•}α⃗ ; μ(d⃗) ⟶^{∼} I^{•}α⃗_{≤i} ; μ_{≤i}(d⃗_{≤i}) ⊠ I^{•}α⃗_{≥i} ; μ(d⃗_{≥i})  (209)

satisfying the associativity property completely analogous to 3.3.

4. Factorizable sheaves

4.1. The aim of this section is to define certain B-linear category F̃S. Its objects will be called factorizable sheaves (over (A, I)). By definition, F̃S is a direct product of B-categories F̃S_{c}, where c runs through π_{0}(A) (see 2.6). Objects of F̃S_{c} will be called factorizable sheaves supported at A_{c}. In what follows we pick c, and denote by X_{c} ⊂ X the corresponding coset modulo Y.

4.2. Definition. A factorizable sheaf X over (A, I) supported at A_{c} is the following collection of data:

(a) a weight λ ∈ X_{c}; it will be denoted by λ(X);

(b) for each α ∈ ℕ[I], a sheaf X^{α} ∈ M(A_{λ}^{α}; S); we will denote by X α⃗(d⃗) perverse sheaves over Aα⃗ ; λ(d⃗) obtained by taking the restrictions with respect to the embeddings Aα⃗ ; λ(d⃗) ↪ A_{λ}^{α};

(c) for each α, β ∈ ℕ[I], d > 0, a factorization isomorphism

ψ^{α,β}(d) : X^{(α,β)}(d) ⟶^{∼} I^{•}_{λ−β′}^{(α,0)}(d) ⊠ X^{(0,β)}(d)  (210)

such that (associativity) for each α, β, γ ∈ ℕ[I], 0 < d_{2} < d_{1}, the square below must commute:

[figure: X^{(α,β,γ)}(d_{1}, d_{2}); I^{•}_{λ−γ′}^{(α,β,0)}(d_{1}, d_{2}) ⊠ X^{(0,γ)}(d_{2}); I^{•}_{λ−β′−γ′}^{(α,0)}(d_{1}) ⊠ X^{(0,β,γ)}(d_{1}, d_{2}); I^{•}_{λ−β′−γ′}^{(α,0)}(d_{1}) ⊠ I^{•}_{λ−γ′}^{(0,β,0)}(d_{1}, d_{2}) ⊠ X^{(0,γ)}(d_{2}); ψ; ϕ ⊠ id; id ⊠ ψ; ψ]

4.2.1. Remark that with these definitions, the braiding local system I resembles a “coalgebra”, and a factorizable sheaf — a “comodule” over it.

4.3. Remark. Note an immediate corollary of the factorization axiom. We have isomorphisms

X^{(α,0)}(d) ≅ X^{0} ⊗ I^{•}_{λ}^{(α,0)}(d)  (211)

(where X^{0} is simply a vector space). Our next aim is to define morphisms between factorizable sheaves.

4.4. Let X be as above. For each μ ≥ λ, μ = λ + β′, and α ∈ ℕ[I], let us define a sheaf X_{μ}^{α} ∈ M(A_{μ}^{α}; S) as σ_{∗}X^{α−β}. For example, X_{λ}^{α} = X^{α}. By taking restriction, the sheaves X α⃗ ; μ(d⃗) ∈ M(Aα⃗ ; μ(d⃗); S) are defined.

Suppose X , Y are two factorizable sheaves supported at A_{c}, λ = λ(X), ν = λ(Y). Let μ ∈ X, μ ≥ λ, μ ≥ ν, α, β ∈ ℕ[I]. By definition we have canonical isomorphisms

θ = θ_{μ}^{β,α} : Hom_{A_{μ}^{α}}(X_{μ}^{α}, Y_{μ}^{α}) ⟶^{∼} Hom_{A_{μ+β′}^{α+β}}(X_{μ+β′}^{α+β}, Y_{μ+β′}^{α+β})  (212)

The maps (210) induce analogous isomorphisms

ψ_{μ}^{α,β}(d) : X_{μ}^{(α,β)}(d) ⟶^{∼} I^{•}_{μ−β′}^{(α,0)}(d) ⊠ X_{μ}^{(0,β)}(d)  (213)

which satisfy the same associativity property as in 4.2.

For α ≥ β let us define maps

τ_{μ}^{αβ} : Hom_{A_{μ}^{α}}(X_{μ}^{α}, Y_{μ}^{α}) ⟶ Hom_{A_{μ}^{β}}(X_{μ}^{β}, Y_{μ}^{β})  (214)

as compositions

Hom_{A_{μ}^{α}}(X_{μ}^{α}, Y_{μ}^{α}) ⟶^{res} Hom_{A_{μ}^{(α−β,β)}(d)}(X_{μ}^{(α−β,β)}(d), Y_{μ}^{(α−β,β)}(d)) ⟶^{∼}^{ψ} (215)

⟶^{∼}^{ψ} Hom_{A_{μ−β′}^{(α−β,0)}(d)}(I^{•}_{μ−β′}^{(α−β,0)}(d), I^{•}_{μ−β′}^{(α−β,0)}(d)) ⊗ Hom_{A_{μ}^{(0,β)}(d)}(X_{μ}^{(0,β)}(d), Y_{μ}^{(0,β)}(d)) =

= B ⊗_{B} Hom_{A_{μ}^{(0,β)}(d)}(X_{μ}^{(0,β)}(d), Y_{μ}^{(0,β)}(d)) = Hom_{A_{μ}^{(0,β)}(d)}(X_{μ}^{(0,β)}(d), Y_{μ}^{(0,β)}(d)) ⟶^{∼}

⟶^{∼} Hom_{A_{μ}^{β}}(X_{μ}^{β}, Y_{μ}^{β})

where we have chosen some d > 0, the first map is the restriction, the second one is induced by the factorization isomorphism, the last one is inverse to the restriction. This definition does not depend on the choice of d.

The associativity axiom implies that these maps satisfy an obvious transitivity property. They are also compatible in the obvious way with the isomorphisms θ.

We define the space Hom_{_{FS}_{c}}(X , Y) as the following inductive-projective limit

Hom_{_{FS}_{c}}(X , Y) := lim_{→}_{μ}lim_{←}_{β}Hom(X_{μ}^{β}, Y_{μ}^{β})  (216)

where the inverse limit is over β ∈ ℕ[I], the transition maps being τ_{μ}^{αβ}, μ being fixed, and the direct limit over μ ∈ X such that μ ≥ λ, μ ≥ ν, the transition maps being induced by (212).

With these spaces of homomorphisms, factorizable sheaves supported at A_{c} form a Blinear category to be denoted by F̃S_{c} (the composition of morphisms is obvious).

As we have already mentioned, the category F̃S is by definition a product ∏_{c∈π_{0}(A)} F̃S_{C}. Thus, an object X of F̃S is a direct sum ⊕_{c∈π_{0}(A)}X_{c}, where X_{c} ∈ F̃S_{c}. If X ∈ F̃S_{c}, Y ∈ F̃S_{c′}, then

Hom_{FS}(X , Y) = Hom_{_{FS}_{c}}(X , Y)

if c = c′, and 0 otherwise.

4.5. Let Vect_{f} denote the category of finite dimensional B-vector spaces. Recall that in II.7.14 the functors of ”vanishing cycles at the origin”

Φ_{α} : M(A_{μ}^{α}; S) ⟶ Vect_{f}

have been defined.

Given X ∈ F̃S_{c}, let us define for each λ ∈ X_{c} a vector space

Φ_{λ}(X) := Φ_{α}(X_{λ+α′}^{α})  (217)

where α ∈ ℕ[I] is such that λ + α′ ≥ λ(X). If λ ∈ X − X_{c}, we set Φ_{λ}(X) = 0.

Due to the definition of the sheaves X_{μ}^{α}, 4.4, this vector space does not depend on a choice of α, up to a unique isomorphism.

This way we get an exact functor

Φ : F̃S_{c} ⟶ Vect_{f}^{X}  (218)

to the category of X-graded vector spaces with finite dimensional components which induces an exact functor

Φ : F̃S ⟶ Vect_{f}^{X}  (219)

4.6. Lemma. If Φ(X) = 0 then X = 0.

Proof. We may suppose that X ∈ F̃S_{c} for some c. Let λ = λ(X). Let us prove that for every α = ∑ a_{i}i ∈ ℕ[I], X^{α} = 0. Let us do it by induction on |α| := ∑ a_{i}. We have X^{0} = Φ_{λ}(X) = 0 by assumption.

Given an arbitrary α, it is easy to see from the factorizability and induction hypothesis that X^{α} is supported at the origin of A_{α}. Since Φ_{λ−α}(X) = 0, we conclude that X^{α} = 0. □

5. Finite sheaves

5.1. Definition. A factorizable sheaf X is called finite if Φ(X) is finite dimensional.

This is equivalent to saying that there exists only finite number of α ∈ ℕ[I] such that Φ_{α}(X^{α}) ≠ 0 (or SS(X^{α}) contains the conormal bundle to the origin 0 ∈ A_{λ}^{α}, where λ := λ(X)).

5.2. Definition. The category of finite factorizable sheaves (FFS for short) is a full subcategory FS ⊂ F̃S whose objects are finite factorizable sheaves.

We set FS_{c} := FS ∩ F̃S_{c} for c ∈ π_{0}(A).

This category is our main character. It is clear that FS is a strictly full subcategory of F̃S closed with respect to taking subobjects and quotients.

The next stabilization lemma is important.

5.3. Lemma. Let X , Y be two FFS’s supported at the same connected component of A. For a fixed μ ≥ λ(X), λ(Y) there exists α ∈ ℕ[I] such that for any β ≥ α the transition map

τ_{μ}^{βα} : Hom_{A_{μ}^{β}}(X_{μ}^{β}, Y_{μ}^{β}) ⟶ Hom_{A_{μ}^{α}}(X_{μ}^{α}, Y_{μ}^{α})

is an isomorphism.

Proof. Let us introduce a finite set

N_{μ}(Y) := {α ∈ ℕ[I]| Φ_{α}(Y_{μ}^{α}) ≠ 0}.

Let us pick β ∈ ℕ[I]. Consider a non-zero map f : X_{μ}^{β} ⟶ Y_{μ}^{β}. For each α ≤ β we have a map f^{α} := τ_{μ}^{βα}(f) : X_{μ}^{α} ⟶ Y_{μ}^{α}. Let us consider subsheaves Z^{α} := Im(f^{α}) ⊂ Y_{μ}^{α}. These subsheaves satisfy an obvious factorization property.

Let us consider the toric stratification of A_{μ}^{β}. For each α ≤ β set A^{α} := σ(A_{μ+α′−β′}^{α}) ⊂ A_{μ}^{β};

A^{•}^{α} := σ(A^{•}_{μ+α′−β′}^{α}). Thus, the subspaces A^{•}^{α} are strata of the toric stratification. We have α_{1} ≤ α_{2} iff A^{α_{1}} ⊂ A^{α_{2}}.

Let γ denote a maximal element in the set {α| Z^{β}|_{A^{•}}^{α} ≠ 0}. Then it is easy to see that Z^{β−γ} is a non-zero scyscraper on A^{β−γ} supported at the origin. Therefore, Φ_{β−γ}(Z^{β−γ}) ≠ 0, whence Φ_{β−γ}(Y^{β−γ}) ≠ 0, i.e. β − γ ∈ N_{μ}(Y).

Suppose that for some α ≤ β, τ_{μ}^{βα}(f) = 0. Then

Z^{β}|_{A^{(β−α,α)}(d)} = 0

for every d > 0. It follows that if Z^{β}|_{A^{•}}_{δ} ≠ 0 then δ ≱ β − α.

Let us apply this remark to δ equal to γ above. Suppose that γ = ∑ c_{i}i, β = ∑ b_{i}i, α = ∑ a_{i}i. There exists i such that c_{i} < b_{i} − a_{i}. Recall that β − γ ∈ N_{μ}(Y). Consequently, we have

5.3.1. Corollary. Suppose that α ≥ δ for all δ ∈ N_{μ}(Y). Then all the maps

τ_{μ}^{βα} : Hom(X_{μ}^{β}, Y_{μ}^{β}) ⟶ Hom(X_{μ}^{α}, Y_{μ}^{α}),

β ≥ α, are injective. □

Since all the spaces Hom(X_{μ}^{α}, Y_{μ}^{α}) are finite dimensional due to the constructibility of our sheaves, there exists an α such that all τ_{μ}^{βα} are isomorphisms. Lemma is proven. □

5.4. For λ ∈ X_{c} let us denote by FS_{c;≤λ} ⊂ FS_{c} the full subcategory whose objects are FFS’s X such that λ(X_{c}) ≤ λ. Obviously FS_{c} is a filtered union of these subcategories.

We have obvious functors

p_{λ}^{β} : FS_{c;≤λ} ⟶ M(A_{λ}^{β}; S), X ↦ X_{λ}^{β}  (220)

The previous lemma claims that for every X , Y ∈ FS_{c;≤λ} there exists α ∈ ℕ[I] such that for every β ≥ α the map

p_{λ}^{β} : Hom_{FS}(X , Y) ⟶ Hom_{A_{λ}^{β}}(X_{λ}^{β}, Y_{λ}^{β})

is an isomorphism. (Obviously, a similar claim holds true for any finite number of FFS’s.)

5.5. Lemma. FS is an abelian artinian category.

Proof. FS is abelian by Stabilization lemma. Each object has finite length by Lemma 4.6. □

6. Standard sheaves

6.1. For Λ ∈ X, let us define factorizable sheaves M(Λ), DM(Λ)_{ζ^{−1}} and L(Λ) as follows. (The notation DM(Λ)_{ζ^{−1}} will be explained in 13.3 below).

Set

λ(M(Λ)) = λ(DM(Λ)_{ζ^{−1}}) = λ(L(Λ)) = Λ.

For α ∈ ℕ[I] let j denote the embedding A^{•}_{α} ↪ A_{α}. We define

M(Λ)^{α} = j_{!}I^{•}_{Λ}^{α}; DM(Λ)_{ζ^{−1}}^{α} = j_{∗}I^{•}_{Λ}^{α}; L(Λ)^{α} = j_{!∗}I^{•}_{Λ}^{α}.

The factorization isomorphisms are defined by functoriality from these isomorphisms for

I^{•}.

Thus, the collections {M(Λ)^{α}}_{α}, etc. form factorizable sheaves to be denoted by M(Λ), DM(Λ)_{ζ^{−1}} and L(Λ) respectively. Obviously, we have a canonical morphism

m : M(Λ) ⟶ DM(Λ)_{ζ^{−1}}  (221)

and L(Λ) is equal to its image.

6.2. Theorem. (i) The factorizable sheaves L(Λ) are finite.

(ii) They are irreducible objects of FS, non-isomorphic for different Λ, and they exhaust all irreducibles in FS, up to isomorphism.

Proof. (i) follows from II.8.18.

(ii) Since the sheaves L(Λ)^{α} are irreducible as objects of M(A_{α}; S), the irreducibility of L(Λ) follows easily. It is clear that they are non-isomorphic (consider the highest component).

Suppose X is an irreducible FFS, λ = λ(X). Let α ∈ ℕ[X] be a minimal among β such that Φ_{λ−β}(X) ≠ 0; set Λ = λ − α. By factorizability and the universal property of !-extension, there exists a morphism if FS’s f : M(Λ) ⟶ X such that Φ_{Λ}(f) ≠ 0 (hence is a monomorphism). It follows from irreducibility of L(Λ) that the composition Ker(f) ⟶ M(Λ) ⟶ L(Λ) is equal to zero, hence f factors through a non-zero morphism L(Λ) ⟶ X which must be an isomorphism. □

6.3. Let us look more attentively at the sheaf L(0).

Let Ã^{α} ⊂ A^{α} denote the open stratum of the diagonal stratification, i.e. the complement to the diagonals. Thus, A^{∘}^{α} ⊂ Ã^{α}. Let Ĩ^{α} denote the local system over Ã^{α} defined in the same way as local systems I_{μ}^{α}, but using only ”diagonal” monodromies, cf. II.6.3.

One sees immediately that L(0)^{α} is equal to the middle extenstion of Ĩ^{α}.

Chapter 2. Tensor structure

7. Marked disk operad

7.1. Let K be a finite set. If T is any set, we will denote by T^{K} the set of all mappings K ⟶ T; elements of T^{K} will be denoted typically by x⃗ = (x_{k})_{k∈K}.

We will use the following partial orders on X^{K}, ℕ[I]^{K}. For λ⃗ = (λ_{k}), μ⃗ = (μ_{k}) ∈ X^{K}, we write λ⃗ ≥ μ⃗ iff λ_{k} ≥ μ_{k} for all k. An order on ℕ[I]^{K} is defined in the same manner.

For α⃗ = (α_{k}) ∈ ℕ[I]^{K} we will use the notation α for the sum of its components ∑_{k∈K} α_{k}; the same agreement will apply to X^{K}.

𝔸^{K} will denote the complex affine space with fixed coordinates u_{k}, k ∈ K; 𝔸^{∘}^{K} ⊂ 𝔸^{K} will denote the open stratum of the diagonal stratification.

7.2. Trees. We will call a tree a couple

τ = (σ, d⃗)  (222)

where σ, to be called the shape of τ,

σ = (K_{p} ⟶^{ρ_{p−1}} K_{p−1} ⟶^{ρ_{p−2}} … ⟶^{ρ_{1}} K_{1} ⟶^{ρ_{0}} K_{0})

is a sequence of epimorhisms of finite sets, such that card(K_{0}) = 1, d⃗ = (d_{0}, d_{1}, …, d_{p}), to be called the thickness of τ — a tuple of real numbers such that d_{0} = 1 > d_{1} > … > d_{p} ≥ 0.

We will use a notation ρ_{ab} for composition K_{a} ⟶ K_{a−1} ⟶ … ⟶ K_{b}, a > b.

A number p ≥ 0 will be called the height of τ and denoted ht(τ). Elements k ∈ K_{i} will be called branches of height i; d_{i} will be called the thickness of k. A unique branch of height 0 will be called bole and denoted by ∗(τ).

The set K_{p} will be called the base of τ and denoted K_{τ}; we will also say that τ is K_{p}-based; we will denote d_{p} by d_{τ}. We will use notation K(τ) for the set ∐_{i=0}^{p} K_{i} and K′(τ) for ∐_{i=0}^{p−1} K_{i}.

A tree of height one will be called elementary. A tree τ whose branches of height ht(τ) have thickness 0, will be called grown up; otherwise it will be called young. We will assign to every tree τ a grown up tree τ̃ by changing the thickness of the thinnest branches to zero.

Thus, an elementary tree is essentially a finite set and a real 0 ≤ d < 1; a grown up elementary tree is essentally a finite set.

7.2.1. Cutting. Suppose we have a tree τ as above, and an integer i, 0 < i < p. We define the operation of cutting τ at level i. It produces from τ new trees τ_{≤i} and τ_{≥k}, k ∈ K_{i}. Namely,

τ_{≤i} = (σ_{≤i}, d⃗_{≤i}),

where σ_{≤i} = (K_{i} ⟶ K_{i−1} ⟶ … ⟶ K_{0}) and d⃗_{≤i} = (d_{0}, d_{1}, …, d_{i}).

Second, for k ∈ K_{i}

τ_{≥k} = (σ_{≥k}, d⃗_{>i})

where

σ_{≥k} = (ρ_{pi}^{−1}(k) ⟶ ρ_{p−1,i}^{−1}(k) ⟶ … ⟶ ρ_{i}^{−1}(k) ⟶ {k}),
d_{>i} = (1, d_{i+1}d_{i}^{−1}, …, d_{p}d_{i}^{−1}).

7.2.2. For 0 < i ≤ p we will denote by ∂_{i}τ a tree (∂_{i}σ, ∂_{i}d⃗) where

∂_{i}σ = (K_{p} ⟶ … ⟶ K̂_{i} ⟶ … ⟶ K_{0}),

and ∂_{i}d⃗ is obtained from d⃗ by omitting d_{i}.

7.3. Operad of disks. For r ∈ ℝ_{≥0} ∪ {∞}, z ∈ ℂ, we define an open disk D(z; r) := {u ∈ 𝔸^{1}| |u − z| < r}, and a closed disk D̄(z; r) := {u ∈ 𝔸^{1}| |u − z| ≤ r}.

For a tree (222) we define a space

O(τ) = O(σ; d⃗)

parametrizing all collections D⃗ = (D̄_{k})_{k∈K_{τ}} of closed disks, such that D_{∗(τ)} = D(0; 1), for k ∈ K_{i} the disk D̄_{k} has radius d_{i}, for fixed i ∈ [p] the disks D̄_{k}, k ∈ K_{i}, do not intersect, and for each i ∈ [0, p − 1] and each k ∈ K_{i+1} we have D̄_{k} ⊂ D_{ρ_{i}(k)}.

Sometimes we will call such a collection a configuration of disks shaped by a tree τ.

7.3.1. Given such a configuration, we will use the notation

D^{∘}_{k}(τ) = D_{k} − ⋃_{l∈ρ_{i}^{−1}(k)}(D̄_{l})  (223)

if k ∈ K_{i} and i < p, and we set D^{∘}_{k}(τ) = D_{k} if i = p.

If τ = (K ⟶ {∗}; d) is an elementary tree, we will use the notation O(K; d) for O(τ); if d = 0, we will abrreviate the notation to O(K).

We have obvious embeddings

O(K; d) ↪ O(K)  (224)

and

O(K) ↪ 𝔸^{∘}^{K}  (225)

this one is a homotopy equivalence.

We have open embeddings

O(τ) ↪ O(τ̃)  (226)

obtained by changing the radius of smallest discs to zero.

7.3.2. Substitution. For each tree τ and 0 < i < ht(τ) we have the following substitution isomorphisms

O(τ) ≅ O(τ_{≤i}) × ∏_{k∈K_{i}} O(τ_{≥k})  (227)

In fact, a configuration of disks shaped by a tree τ is the same as a configuration shaped by τ_{≤i}, and for each k ∈ K_{i} a configuration shaped by τ_{>k} inside D_{k} (playing the role of D_{0}; here we have to make a dilation by d_{i}^{−1}).

These isomorphisms satisfy obvious quadratic relations connected with pairs 0 < i < j < ht(τ). We leave their formulation to the reader.

7.4. Enhanced trees. We will call an enhanced tree a couple (τ, α⃗) where τ is a tree and α⃗ ∈ ℕ[I]^{K′(τ)}. Vector α⃗ will be called enhancement of τ.

Let us define cutting for enhanced trees. Given τ and i as in 7.2.1, let us note that K′(τ_{≤i}) and K′(τ_{≥k}) are subsets of K′(τ). We define

α⃗_{≤i} ∈ ℕ[I]^{K′(τ_{≤i})}, α⃗_{≥k} ∈ ℕ[I]^{K′(τ_{≥k})}

as the corresponding subsequences of α⃗.

Let us define operations ∂_{i} for enhanced trees. Namely, in the setup of 7.2.2, we define ∂_{i}α⃗ = (α′_{k}) ∈ ℕ[I]^{K′(∂_{i}τ)} as follows. If i = p then K′(∂_{p}τ) ⊂ K′(τ), and we define ∂_{p}α⃗ as a corresponding subsequence. If i < p, we set α′_{k} = α_{k} if k ∈ K_{j}, j > i or j < i − 1. If j = i − 1, we set

α′_{k} = α_{k} + ∑_{l∈ρ_{i−1}^{−1}(k)} α_{l}.

7.5. Enhanced disk operad. Given an enhanced tree (τ, α⃗), let us define a configuration space Aα⃗(τ) as follows. Its points are couples (D⃗, t), where D⃗ ∈ O(τ) and t = (t_{j}) is an α-colored configuration in 𝔸^{1} (see II.6.12) such that

for each k ∈ K′(τ) exactly α_{k} points lie inside D^{∘}_{k}(τ) if k ∉ K_{τ} (resp., inside D_{k}(τ) if k ∈ K_{τ}) (see 7.3.1).

In particular, all points lie inside D_{∗(τ)} = D(0; 1) and outside ⋃_{k∈K_{τ}} D̄_{k} if τ is young. This space is an open subspace of the product O(τ) × A_{α}.

We will also use a notation

A^{α}(L; d) := A^{α}(L ⟶ {∗}; d)

for elementary trees and A^{α}(L) for A^{α}(L; 0).

The isomorphisms (227) induce isomorphisms

Aα⃗(τ) ≅ Aα⃗_{≤i}(τ_{≤i}) × ∏_{k∈K_{i}} Aα⃗_{≥k}(τ_{≥k})  (228)

We have embeddings

d_{i} : Aα⃗(τ) ⟶ A^{∂_{i}α⃗}(∂_{i}τ), 0 < i ≤ p,  (229)

— dropping all disks D_{k}, k ∈ K_{i}.

We have obvious open embeddings

Aα⃗(τ) ↪ A^{α}(K_{τ}; d_{τ}) ↪ A^{α}(K_{τ})  (230)

7.6. Marked trees. We will call a marked tree a triple (τ, α⃗, μ⃗) where (τ, α⃗) is an enhanced tree, and μ⃗ ∈ X^{K_{τ}}. We will call ht(τ) the height of this marked tree.

Let us define operations ∂_{i}, 0 < i ≤ p = ht(τ) for marked trees. Namely, for i < p we set ∂_{i}μ⃗ = μ⃗. For i = p we define ∂_{p}μ⃗ as (μ′_{k})_{k∈K_{p−1}}, where

μ′_{k} = ∑_{l∈ρ_{p−1}^{−1}(k)} μ_{l} − α′_{l}.

Let us define cutting for marked trees. Namely, for 1 ≤ i < p we define μ⃗_{≤i} as ∂_{i+1} … ∂_{p−1}∂_{p}μ⃗.

Next, for k ∈ K_{i} we have K_{τ_{≥k}} ⊂ K_{τ}, and we define μ⃗_{≥k} as a corresponding subsequence of μ⃗.

7.7. Marked disk operad. Now we can introduce our main objects. For each marked tree (τ, α⃗, μ⃗) we define Aα⃗ ; μ⃗(τ) as a topological space Aα⃗(τ) defined above, together with a marking μ⃗ of the tree τ considered as an additional index assigned to this space.

We will regard Aα⃗ ; μ⃗(τ) as a space whose points are configurations (D⃗, t) ∈ Aα⃗(τ), together with a marking of smallest disks D_{k}, k ∈ K_{τ}, by weights μ_{k}.

As above, we will use abbreviations Aα ; μ⃗(L; d) for Aα ; μ⃗(L ⟶ {∗}; d) (where μ⃗ ∈ X^{L}) and

Aα ; μ⃗(L) for Aα ; μ⃗(L; 0).

We have natural open embeddings

d_{i} : Aα⃗ ; μ⃗ ⟶ A_{∂_{i}μ⃗}^{∂_{i}α⃗}(∂_{i}τ), 0 < i ≤ p,  (231)

and

Aα⃗(τ) ↪ A^{α}(K_{τ}; d_{τ}) ↪ A^{α}(K_{τ})  (232)

induced by the corresponding maps without marking.

The substitution isomorphisms (228) induce isomorphisms

Aα⃗ ; μ⃗(τ) ≅ Aα⃗_{≤i} ; μ⃗_{≤i}(τ_{≤i}) × ∏_{k∈K_{i}} Aα⃗_{≥k} ; μ⃗_{≥k}(τ_{≥k})  (233)

7.8. We define closed embeddings

σ = σ_{μ⃗;β⃗}^{α} : Aα ; μ⃗(K) ⟶ A_{μ⃗+β⃗′}^{α+β}(K)  (234)

where β⃗ = (β_{k})_{k∈K}, β_{k} = ∑_{i} b_{k}^{i} · i and β = ∑_{k} β_{k}. By definition, σ leaves points u_{k} intact (changing their markings) and adds b_{k}^{i} copies of points of color i equal to u_{k}.

7.9. Stratifications. We set

A^{•}α ; μ⃗(K) := Aα ; μ⃗(K) − ⋃_{γ⃗>0} σ(A_{μ⃗−γ⃗}^{α−γ}(K))

We define a toric stratification of Aα ; μ⃗(K) as

Aα ; μ⃗(K) = ∐_{β⃗<α⃗} σ(A^{•}_{μ⃗−α′+β⃗′}^{β}(K))

A principal stratification on Aα ; μ⃗(K) is defined as follows. The space Aα ; μ⃗(K) is a quotient

of 𝔸^{∘}^{K} × 𝔸^{J} where π : J ⟶ I is an unfolding of α (cf. II.6.12). We define the principal

stratification as the image of the diagonal stratification on 𝔸^{∘}^{K} × 𝔸^{J} under the canonical

projection 𝔸^{∘}^{K} × 𝔸^{J} ⟶ Aα ; μ⃗(K). We will denote by A^{∘}α ; μ⃗(K) the open stratum of the principal stratification.

8. Cohesive local systems ^{K}I

8.1. Let us fix a non-empty finite set K. Suppose we are given μ⃗ ∈ X^{K} and α ∈ ℕ[I]. Let us pick an unfolding of α, π : J ⟶ I. Let

πα ; μ⃗ : (D(0; 1)^{K} × D(0; 1)^{J})^{∘} ⟶ A^{∘}α ; μ⃗(K)  (235)

denote the canonical projection (here (D(0; 1)^{K} × D(0; 1)^{J})^{∘} denotes the open stratum of the diagonal stratification).

Let us define a one dimensional local system ^{π}Iμ⃗ by the same procedure as in 3.1. Its fiber over each positive chamber C ∈ π_{0}((D(0; 1)^{K} × D(0; 1)^{J})_{ℝ}^{∘}) is identified with B. Monodromies along the standard paths are given by the formulas

CT_{ij} = ζ^{−π(i)·π(j)}, ^{C}T_{ik} = ζ^{2μ_{k}·π(i)′}, ^{C}T_{km} = ζ^{−μ_{k}·μ_{m}},  (236)

i, j ∈ J, i ≠ j; k, m ∈ K, k ≠ m. Here ^{C}T_{ij} and ^{C}T_{km} are half-circles, and ^{C}T_{ik} are full circles. This definition essentially coincides with II.12.6, except for an overall sign.

We define a one-dimensional local system Iα ; μ⃗(K) over A^{∘}α ; μ⃗(K) as

Iα ; μ⃗ := (π_{∗} ^{π}Iμ⃗)^{sgn}  (237)

where the superscript (•)^{sgn} has the same meaning as in 3.1. For each non-empty subset L ⊂ K we can take a part of weights μ⃗_{L} = (μ_{k})_{k∈L} and get a local system Iα ; μ⃗(L) over A^{∘}α ; μ⃗(L). For each marked tree (τ, α⃗, μ⃗) with K_{τ} ⊂ K, we define the local system Iα⃗ ; μ⃗(τ) as the restriction of Iα ; μ⃗(K_{τ}) with respect to embedding (232). 8.2. Factorization. . The same construction as in 3.2 defines factorization isomorphisms

ϕ_{i} = ϕα⃗ ; i;μ⃗(τ) : Iα⃗ ; μ⃗(τ) ≅ Iα⃗_{≤i} ; μ⃗_{≤i}(τ_{≤i}) ⊠ × _{k∈K_{i}}Iα⃗_{≥k} ; μ⃗_{≥k}(τ_{≥k})  (238)

They satisfy the property of 8.3. Associativity. For all 0 < i < j < p squares

[figure: I(τ_{≤i}) ⊠ × _{k∈K_{i}} I(τ_{≥k;≤j}) ⊠ × _{l∈K_{j}} I(τ_{≥l}); I(τ); I(τ_{≤j}) ⊠ × _{l∈K_{j}} I(τ_{≥l}); I(τ_{≤i}) ⊠ × _{k∈K_{i}} I(τ_{≥k}); ϕ_{j}; ϕ_{i} ⊠ id; id ⊠ ϕ_{j}; ϕ_{i}]

commute. (To unburden the notation we have omitted irrelevant indices — they are restored uniquely.) 8.4. The collection of local systems ^{K}I = {Iα ; μ⃗(L), L ⊂ K}, together with the factorization isomorphisms defined above, will be called the cohesive local system over ^{K} A^{∘}. 8.5. Let us define perverse sheaves

I^{•}α⃗ ; μ⃗(τ) := j_{!∗}Iα⃗ ; μ⃗(τ)[dim Aα⃗ ; μ⃗(τ)] ∈ M(A^{•}α⃗ ; μ⃗(τ); S)

where j : A^{∘}α⃗ ; μ⃗(τ) ↪ A^{•}α⃗ ; μ⃗(τ) denotes the embedding. By functoriality, the factorization isomorphisms (238) induce isomorphisms

ϕ_{i} = ϕα⃗ ; i;μ⃗(τ) : I^{•}α⃗ ; μ⃗(τ) ≅ I^{•}α⃗_{≤i} ; μ⃗_{≤i}(τ_{≤i}) ⊠ × _{k∈K_{i}} I^{•}α⃗_{≥k} ; μ⃗_{≥k}(τ_{≥k})  (239)

These isomorphisms satisfy an associativity property completely analogous to 8.3; one should only replace I by I^{•} in the diagrams.

9. Factorizable sheaves over ^{K}A

We keep the assumptions of the previous section.

9.1. The first goal of this section is to define a B-linear category ^{K} F̃S whose objects will be called factorizable sheaves (over (^{K}A, ^{K}I)). Similarly to F̃S, this category is by definition a product of B-categories

K F̃S = ∏_{c⃗∈π_{0}(A)^{K}} ^{K} F̃Sc⃗.  (240)

Objects of ^{K} F̃Sc⃗ will be called factorizable sheaves supported at c⃗.

9.2. Definition. A factorizable sheaf X over (^{K}A, ^{K}I) supported at c⃗ = (c_{k}) ∈ π_{0}(A)^{K} is the following collection of data:

(a) a K-tuple of weights λ⃗ = (λ_{k}) ∈ X^{K} such that λ_{k} ∈ X_{c_{k}}, to be denoted by λ⃗(X);

(b) for each α ∈ ℕ[I] a sheaf X^{α}(K) ∈ M(Aα ; λ⃗(K); S).

Taking restrictions, as in 8.1, we get for each K-based enhanced tree (τ, α⃗) sheaves

X α⃗(τ) ∈ M(Aα⃗ ; λ⃗(τ); S).

(c) For each enhanced tree (τ, α⃗) of height 2, τ = (K ⟶^{id} K ⟶ {∗}; (1, d, 0)), α⃗ = (α, β⃗) where α ∈ ℕ[I]; β⃗ ∈ ℕ[I]^{K}, a factorization isomorphism

ψ(τ) : X^{(α,β⃗)}(τ) ≅ I^{•}_{λ⃗(τ)_{≤1}}^{α}(τ_{≤1}) ⊠ X^{(0,β⃗)}(τ)  (241)

These isomorphisms should satisfy

Associativity axiom.

For all enhanced trees (τ, α⃗) of height 3, τ = (K ⟶^{id} K ⟶^{id} K ⟶ {∗}; (1, d_{1}, d_{2}, 0)), α⃗ = (α, β⃗, γ⃗) where α ∈ ℕ[I]; β⃗, γ⃗ ∈ ℕ[I]^{K}, the square

[figure: X^{(α,β⃗,γ⃗)}(τ); I^{•}_{λ⃗(τ)_{≤2}}^{(α,β⃗)}(τ_{≤2}) ⊠ X^{(0,0,γ⃗)}(τ); I^{•}_{λ⃗(τ)_{≤1}}^{α}(τ_{≤1}) ⊠ X^{(0,β⃗,γ⃗)}(τ); I^{•}_{λ⃗(τ)_{≤1}}^{α}(τ_{≤1}) ⊠ × _{k∈K} I^{•}_{λ⃗(τ)_{≥k;≤2}}^{β_{k}}(τ_{≥k;≤2}) ⊠ X^{(0,0,γ⃗)}(τ); ψ_{2}; ϕ_{1} ⊠ id; id ⊠ ψ_{2}; ψ_{1}]

commutes.

9.3. Let X be as above. For each μ⃗ ∈ X^{K}, μ⃗ ≥ λ⃗, so that μ⃗ = λ⃗+β⃗′ for some β⃗ ∈ ℕ[I]^{K}, and α ∈ ℕ[I], let us define a sheaf X α ; μ⃗(K) ∈ M(Aα ; μ⃗(K); S) as σ_{∗}X^{α−β}(K). For example,

X α ; λ⃗(K) = X^{α}(K).

Taking restrictions, the sheaves X α⃗ ; μ⃗(τ) ∈ M(Aα⃗ ; μ⃗(τ); S) for all K-based trees τ are defined.

9.4. Suppose X , Y are two factorizable sheaves supported at c⃗, λ⃗ = λ⃗(X), ν⃗ = λ⃗(Y). Let μ⃗ ∈ X^{K}, μ⃗ ≥ λ⃗, μ⃗ ≥ ν⃗. By definition, we have canonical isomorphisms

θ = θ_{μ⃗;β⃗}^{α} : Hom_{Aα ; μ⃗(K)}(X α ; μ⃗(K), Yα ; μ⃗(K)) ⟶^{∼} Hom_{A_{μ⃗+β⃗′}^{α+β}(K)}(X_{μ⃗+β⃗′}^{α+β}(K), Y_{μ⃗+β⃗′}^{α+β}(K)) (242)

for each α ∈ ℕ[I], β⃗ ∈ ℕ[I]^{K}.

9.4.1. Suppose we are given β⃗ = (β_{k}) ∈ ℕ[I]^{K}. Let β = ∑_{k} β_{k} as usually. Choose a real d, 0 < d < 1.

Consider a marked tree (τ_{d}, (0, β⃗), μ⃗) where

τ_{d} = (K ⟶^{id} K ⟶ {∗}; (0, d, 1)).

We have the restriction homomorphism

ξ_{μ⃗;β⃗;d} : Hom_{Aβ ; μ⃗(K)}(X β ; μ⃗(K), Yβ ; μ⃗(K)) ⟶ Hom_{A(0,β⃗) ; μ⃗(τ_{d})}(X (0,β⃗) ; μ⃗(τ_{d}), Y(0,β⃗) ; μ⃗(τ_{d})) (243)

Suppose we are given ′β⃗ = (′β_{k}) ∈ ℕ[I]^{K} such that ′β⃗ ≤ β⃗. Let ′β = ∑_{k} ′β_{k} as usually. Choose a real ε, 0 < ε < d.

The restriction and the factorization isomorphisms ψ induce the map

η_{μ⃗; ′β⃗;ε}^{β⃗;d} : Hom_{A(0,β⃗) ; μ⃗(τ_{d})}(X (0,β⃗) ; μ⃗(τ_{d}), Y(0,β⃗) ; μ⃗(τ_{d})) ⟶ Hom_{A(0, ′β⃗) ; μ⃗(τ_{ε})}(X (0, ′β⃗) ; μ⃗(τ_{ε}), Y(0, ′β⃗) ; μ⃗(τ_{ε})) (244)

The associativity axiom implies that these maps satisfy an obvious transitivity property.

We define the space Hom_{_{K} _{FS}}(X , Y) as the following inductive-projective limit

Hom_{_{K} _{FS}}(X , Y) := lim_{→} lim_{←} Hom_{Aα ; μ⃗(K)}(X α ; μ⃗(K), Yα ; μ⃗(K))  (245)

where the inverse limit is understood as follows. Its elements are collections of maps

{f_{K}^{α} : X α ; μ⃗(K) ⟶ Yα ; μ⃗(K)}

given for all α ∈ ℕ[I], β⃗ ∈ ℕ[I]^{K}, such that for every α, ′β⃗ ≤ β⃗, 0 < ε < d < 1 as above, we have

η_{μ⃗; ′β⃗;ε}^{β⃗;d}ξ_{μ⃗;β⃗;d}(f_{K}^{β}) = ξ_{μ⃗; ′β⃗;ε}(f_{K}^{′β})

μ⃗ being fixed. The direct limit is taken over μ⃗ ∈ X^{K} such that μ⃗ ≥ λ⃗, μ⃗ ≥ ν⃗, the transition maps being induced by (242).

With these spaces of homomorphisms, factorizable sheaves supported at c⃗ form a Blinear category to be denoted by ^{K} F̃Sc⃗. As we have already mentioned, the category of factorizable sheaves ^{K} F̃S is by definition the product (240).

FINITE SHEAVES

9.5. Definition. A sheaf X ∈ ^{K} F̃Sc⃗ is called finite if there exists only finitely many β⃗ ∈ ℕ[I]^{K} such that the singular support of X α ; λ⃗(K) contains the conormal bundle to σ_{λ⃗−β⃗′;β⃗}^{α−β}(A_{λ⃗−β⃗′}^{α−β}) (see (234)) for α ≥ β = ∑_{k} β_{k}.

A sheaf X = ⊕c⃗Xc⃗ ∈ ^{K} F̃S, Xc⃗ ∈ ^{K} F̃Sc⃗ is called finite if all Xc⃗ are finite.

9.6. Suppose we are given finite sheaves X , Y ∈ ^{K}FSc⃗; and μ⃗ ≥ λ⃗(X), λ⃗(Y). As in the proof of the Lemma 5.3, one can see that there exists ′β⃗ ∈ ℕ[I]^{K} such that for any β⃗ ≥ ′β⃗ the map

η_{μ⃗;β⃗;ε}^{′β⃗;d} : Hom_{A(0,β⃗) ; μ⃗(τ_{d})}(X (0,β⃗) ; μ⃗(τ_{d}), Y(0,β⃗) ; μ⃗(τ_{d})) ⟶ Hom_{A(0, ′β⃗) ; μ⃗(τ_{ε})}(X (0, ′β⃗) ; μ⃗(τ_{ε}), Y(0, ′β⃗) ; μ⃗(τ_{ε})) (246)

is an isomorphism. We will identify all the spaces Hom_{A(0,β⃗) ; μ⃗(τ_{d})}(X (0,β⃗) ; μ⃗(τ_{d}), Y(0,β⃗) ; μ⃗(τ_{d}))

with the help of the above isomorphisms, and we will denote this stabilized space by \overline{Hom}^{K}_{FS}(X , Y). Evidently, it does not depend on a choice of ′β⃗.

Quite similarly to the loc.cit one can see that for any β⃗ ≥ ′β⃗ the map

ξ_{μ⃗;β⃗;d} : Hom_{Aβ ; μ⃗(K)}(X β ; μ⃗(K), Yβ ; μ⃗(K)) ⟶ Hom_{A(0,β⃗) ; μ⃗(τ_{d})}(X (0,β⃗) ; μ⃗(τ_{d}), Y(0,β⃗) ; μ⃗(τ_{d})) (247)

is an injection.

Thus we may view Hom_{Aβ ; μ⃗(K)}(X β ; μ⃗(K), Yβ ; μ⃗(K)) as the subspace of \overline{Hom}^{K}_{FS}(X , Y).

We define Hom^{K}_{FS}(X , Y) ⊂ \overline{Hom}^{K}_{FS}(X , Y) as the projective limit of the system of subspaces Hom_{Aβ ; μ⃗(K)}(X β ; μ⃗(K), Yβ ; μ⃗(K)), β⃗ ≥ ′β⃗.

With such definition of morphisms finite factorizable sheaves supported at c⃗ form an abelian category to be denoted by ^{K}FSc⃗. We set by definition

KFS = ∏_{c⃗∈π_{0}(A)^{K}} ^{K}FSc⃗  (248)

10. Gluing

10.1. Let

A_{μ;1}^{α} ⊂ A_{μ}^{α}

denote an open configuration subspace parametrizing configurations lying entirely inside the unit disk D(0; 1). Due to monodromicity, the restriction functors

M(A_{μ}^{α}; S) ⟶ M(A_{μ;1}^{α}; S)

are equivalences.

Let {∗} denote a one-element set. We have closed embeddings

i : A_{μ;1}^{α} ↪ A_{μ}^{α}({∗}),

which identify the first space with the subspace of the second one consisting of configurations with the small disk centered at 0. The inverse image functors

i^{∗}[−1] : M(A_{μ}^{α}({∗}); S) ⟶ M(A_{μ}^{α}; S)  (249)

are equivalences, again due to monodromicity. Thus, we get equivalences

M(A_{μ}^{α}; S) ⟶^{∼} M(A_{μ}^{α}({∗}; S)

which induce canonical equivalences

F̃S ⟶^{∼} F̃S^{{∗}}  (250)

and

FS ⟶^{∼} FS^{{∗}}  (251)

Using these equivalences, we will sometimes identify these categories.

10.2. Tensor product of categories. Let B_{1}, B_{2} be B-linear abelian categories. Their tensor product category B_{1}⊗B_{2} is defined in §5 of [D2]. It comes together with a canonical right biexact functor B_{1}×B_{2} ⟶ B_{1}⊗B_{2}, and it is the initial object among such categories.

10.2.1. Basic Example. Let M_{i}, i = 1, 2, be complex algebraic varieties equipped with algebraic Whitney stratifications S_{i}. Let B_{i} = M(M_{i}; S_{i}). Then

B_{1} ⊗ B_{2} = M(M_{1} × M_{2}; S_{1} × S_{2}).

The canonical functor B_{1} × B_{2} ⟶ B_{1} ⊗ B_{2} sends (X_{1}, X_{2}) to X_{1} ⊠ X_{2}.

10.2.2. Recall the notations of 9.4.1. Let us consider the following category FS^{⊗K}. Its objects are the collections of perverse sheaves X (0,β⃗) ; μ⃗(τ_{d}) on the spaces A(0,β⃗) ; μ⃗(τ_{d}) for sufficiently small d, satisfying the usual factorization and finiteness conditions. The morphisms are defined via the inductive-projective system with connecting maps η_{μ⃗; ′β⃗;ε}^{β⃗;d}. Using the above Basic Example, one can see easily that the category FS^{⊗K} is canonically equivalent to FS ⊗ … ⊗ FS (K times) which justifies its name.

By definition, the category ^{K}FS comes together with the functor p_{K} : ^{K}FS ⟶ FS^{⊗K} injective on morphisms. In effect,

Hom^{K}_{FS}(X , Y) ↪ \overline{Hom}^{K}_{FS}(X , Y) = Hom_{FS}^{⊗K}(p_{K}(X), p_{K}(Y)).

Let us construct a functor in the opposite direction.

10.3. Gluing of factorizable sheaves. For each 0 < d < 1 let us consider a tree

τ_{d} = (K ⟶^{id} K ⟶ {∗}; (1, d, 0)).

Suppose we are given α ∈ ℕ[I]. Let V(α) denote the set of all enhancements α⃗ = (α_{∗}; (α_{k})_{k∈K}) of τ such that α_{∗} + ∑_{k∈K} α_{k} = α. Obviously, the open subspaces Aα⃗(τ_{d}) ⊂ A^{α}(K), for varying d and α⃗ ∈ V(α), form an open covering of A^{α}(K).

Suppose we are given a collection of factorizable sheaves X_{k} ∈ FS_{c_{k}}, k ∈ K. Set λ⃗ = (λ(X_{k})) ∈ X^{K}. For each d, α⃗ as above consider a sheaf

X α⃗(τ_{d}) := I^{•}α_{∗} ; λ⃗_{≤1}(τ_{d;≤1}) ⊠ × _{k∈K} X_{k}^{α_{k}}

over Aα⃗ ; λ⃗(τ_{d}).

Non-trivial pairwise intersections of the above open subspaces look as follows. For 0 < d_{2} < d_{1} < 1, consider a tree of height 3

ς = ς_{d_{1},d_{2}} = (K ⟶^{id} K ⟶^{id} K ⟶ {∗}; (1, d_{1}, d_{2}, 0)).

We have ∂_{1}ς = τ_{d_{2}}, ∂_{2}ς = τ_{d_{1}}. Let β⃗ = (β_{∗}, (β_{1;k})_{k∈K}, (β_{2;k})_{k∈K}) be an enhancement of ς. Set α⃗_{1} = ∂_{2}β⃗, α⃗_{2} = ∂_{1}β⃗. Note that β⃗ is defined uniquely by α⃗_{1}, α⃗_{2}. We have

Aβ⃗ ; λ⃗(ς) = Aα⃗_{1} ; λ⃗(τ_{d_{1}}) ∩ Aα⃗_{2} ; λ⃗(τ_{d_{2}}).

Due to the factorization property for sheaves I^{•} and X_{k} we have isomorphisms

X α⃗_{1}(τ_{d_{1}})|_{Aβ⃗ ; λ⃗(ς)} ≅ I^{•}_{∂_{3}λ⃗}^{∂_{3}β⃗}(∂_{3}ς) ⊠ × _{k∈K}(X_{k})_{λ_{k}}^{β_{2;k}},

and

X α⃗_{2}(τ_{d_{2}})|_{Aβ⃗ ; λ⃗(ς)} ≅ I^{•}_{∂_{3}λ⃗}^{∂_{3}β⃗}(∂_{3}ς) ⊠ × _{k∈K}(X_{k})_{λ_{k}}^{β_{2;k}}

Taking composition, we get isomorphisms

ϕ_{d_{1},d_{2}}^{α⃗_{1},α⃗_{2}} : X α⃗_{1}(τ_{d_{1}})|_{Aα⃗_{1} ; λ⃗(τ_{d_{1}})∩Aα⃗_{2} ; λ⃗(τ_{d_{2}})} ⟶^{∼} X α⃗_{2}(τ_{d_{2}})|_{Aα⃗_{1} ; λ⃗(τ_{d_{1}})∩Aα⃗_{2} ; λ⃗(τ_{d_{2}})}  (252)

From the associativity of the factorization for the sheaves I^{•} and X_{k} it follows that the isomorphisms (252) satisfy the cocycle condition; hence they define a sheaf X^{α}(K) over

Aα ; λ⃗(K).

Thus, we have defined a collection of sheaves {X^{α}(K)}. Using the corresponding data for the sheaves X_{k}, one defines easily factorization isomorphisms 9.2 (d) and check that they satisfy the associativity property. One also sees immediately that the collection of sheaves {X^{α}(K)} is finite. We leave this verification to the reader.

This way we get maps

∏_{k} Ob(FS_{c_{k}}) ⟶ Ob(^{K}FSc⃗), c⃗ = (c_{k})

which extend by additivity to the map

g_{K} : Ob(FS^{K}) ⟶ Ob(^{K}FS)  (253)

To construct the functor

g_{K} : FS^{K} ⟶ ^{K}FS  (254)

it remains to define g_{K} on morphisms.

Given two collections of finite factorizable sheaves X_{k}, Y_{k} ∈ FS_{c_{k}}, k ∈ K, let us choose λ⃗ = (λ_{k})_{k∈K} such that λ_{k} ≥ λ(X_{k}), λ(Y_{k}) for all k ∈ K. Suppose we have a collection of morphisms f_{k} : X_{k} ⟶ Y_{k}, k ∈ K; that is the maps f_{k}^{α_{k}} : (X_{k})_{λ_{k}}^{α_{k}} ⟶ (Y_{k})_{λ_{k}}^{α_{k}} given for any α_{k} ∈ ℕ[I] compatible with factorizations.

Given α ∈ ℕ[I] and an enhancement α⃗ ∈ V(α) as above we define the morphism f α⃗(τ_{d}) : X α⃗(τ_{d}) ⟶ Yα⃗(τ_{d}) over Aα⃗ ; λ⃗(τ_{d}) as follows: it is the tensor product of the identity on

I^{•}α_{∗} ; λ⃗_{≤1}(τ_{d;≤1}) with the morphisms f_{k}^{α_{k}} : (X_{k})_{λ_{k}}^{α_{k}} ⟶ (Y_{k})_{λ_{k}}^{α_{k}}.

One sees easily as above that the morphisms f α⃗(τ_{d}) glue together to give a morphism f^{α}(K) : X^{α}(K) ⟶ Y^{α}(K); as α varies they provide a morphism f(K) : g_{K}((X_{k})) ⟶ g_{K}((Y_{k})). Thus we have defined the desired functor g_{K}. Obviously it is K-exact, so by universal property it defines the same named functor

g_{K} : FS^{⊗K} ⟶ ^{K}FS  (255)

By the construction, the composition p_{K} ∘ g_{K} : FS^{⊗K} ⟶ FS^{⊗K} is isomorphic to the identity functor. Recalling that p_{K} is injective on morphisms we see that g_{K} and p_{K} are quasiinverse. Thus we get

10.4. Theorem. The functors p_{K} and g_{K} establish a canonical equivalence

KFS ⟶^{∼} FS^{⊗K} □

11. Fusion

BRAIDED TENSOR CATEGORIES

In this part we review the definition of a braided tensor category following Deligne, [D1].

11.1. Let C be a category, Y a locally connected locally simply connected topological space. By a local system over Y with values in C we will mean a locally constant sheaf over Y with values in C. They form a category to be denoted by Locsys(Y; C).

11.1.1. We will use the following basic example. If X is a complex algebraic variety with a Whitney stratification S then the category M(X × Y; S × S_{Y;tr}) is equivalent to Locsys(Y; M(X; S)). Here S_{Y;tr} denotes the trivial stratification of Y, i.e. the first category consists of sheaves smooth along Y.

11.2. Let π : K ⟶ L be an epimorphism of non-empty finite sets. We will use the notations of 7.3. For real ϵ, δ such that 1 > ϵ > δ > 0, consider a tree

τ_{π;ϵ,δ} = (K ⟶^{π} L ⟶ {∗}; (1, ϵ, δ))

We have an isomorphism which is a particular case of (227):

O(τ_{π;ϵ,δ}) ≅ O(L; ϵ) × ∏_{l∈L} O(K_{l}; δϵ^{−1})

where K_{l} := π^{−1}(l).

11.2.1. Lemma There exists essentially unique functor

r_{π} : Locsys(O(K); C) ⟶ Locsys(O(L) × ∏_{l} O(K_{l}); C)

such that for each ϵ, δ as above the square

[figure: Locsys(O(K); C); r_{π}; Locsys(O(L) × ∏_{l} O(K_{l}); C); Locsys(τ_{π;ϵ,δ}; C); ∼; Locsys(O(L; ϵ) × ∏_{l} O(K_{l}; δϵ^{−1}); C)]

commutes. Proof follows from the remark that O(L) is a union of its open subspaces

O(L) = ⋃_{ϵ>0} O(L; ϵ). □

11.3. Let C be a category. A braided tensor structure on C is the following collection of data.

(i) For each non-empty finite set K a functor

⊗_{K} : C^{K} ⟶ Locsys(O(K); C), {X_{k}} ↦ ⊗_{K} X_{k}

from the K-th power of C to the category of local systems (locally constant sheaves) over the space O(K) with values in C (we are using the notations of 7.3). We suppose that ⊗_{{∗}} X is the constant local system with the fiber X. (ii) For each π : K ⟶ L as above a natural isomorphism

ϕ_{π} : (⊗_{K} X_{k})|_{O(L)×∏ O(K_{l})} ⟶^{∼} ⊗_{L} (⊗_{K_{l}} X_{k}).

To simplify the notation, we will write this isomorphism in the form

ϕ_{π} : ⊗_{K} X_{k} ⟶^{∼} ⊗_{L} (⊗_{K_{l}} X_{k}),

implying that in the left hand side we must take restriction. These isomorphisms must satisfy the following Associativity axiom. For each pair of epimorphisms K ⟶^{π} L ⟶^{ρ} M the square

[figure: ⊗_{K} X_{k}; ⊗_{M}(⊗_{K_{m}} X_{k}); ⊗_{L}(⊗_{K_{l}} X_{k}); ⊗_{M}(⊗_{L_{m}}(⊗_{K_{l}} X_{k})); ϕ_{ρπ}; ϕ_{π}; ⊗_{M} ϕ_{π|_{Km}}; ϕ_{ρ}]

where K_{m} := (ρπ)^{−1}(m), L_{m} := ρ^{−1}(m), commutes.

11.4. The connection with the conventional definition is as follows. Given two objects X_{1}, X_{2} ∈ Ob C, define an object X_{1}⊗^{•}X_{2} as the fiber of ⊗_{{1,2}}X_{k} at the point (1/3, 1/2).

We have natural isomorphisms

A_{X_{1},X_{2},X_{3}} : X_{1}⊗^{•}(X_{2}⊗^{•}X_{3}) ⟶^{∼} (X_{1}⊗^{•}X_{2})⊗^{•}X_{3}

coming from isomorphisms ϕ associated with two possible order preserving epimorphic maps {1, 2, 3} ⟶ {1, 2}, and

R_{X_{1},X_{2}} : X_{1}⊗^{•}X_{2} ⟶^{∼} X_{2}⊗^{•}X_{1}

coming from the standard half-circle monodromy. Associativity axiom for ϕ is equivalent to the the usual compatibilities for these maps.

11.5. Now suppose that the data 11.3 is given for all (possibly empty) tuples and all (not necessarily epimorphic) maps. The space O(∅) is by definition one point, and a local system ⊗_{∅} over it is simply an object of C; let us denote it 1 and call a unit of our tensor structure. In this case we will say that C is a braided tensor category with unit.

In the conventional language, we have natural isomorphisms

1⊗^{•}X ⟶^{∼} X

(they correspond to {2} ↪ {1, 2}) satisfying the usual compatibilities with A and R.

FUSION FUNCTORS

11.6. Let

A_{α;1} ⊂ A_{α}

denote the open subspace parametrizing configurations lying inside the unit disk D(0; 1).

Let K be a non-empty finite set. Obviously, A^{α}(K) = A_{α;1} × O(K), and we have the projection

A^{α}(K) ⟶ O(K).

Note also that we have an evident open embedding O(K) ↪ D(0; 1)^{K}.

Our aim in this part is to define certain fusion functors

Ψ_{K} : D(A^{α}(K)) ⟶ D^{mon}(A^{α}({∗}) × O(K))

where (•)^{mon} denotes the full subcategory of complexes smooth along O(K). The construction follows the classical definition of nearby cycles functor, [D3].

11.7. Poincaré groupoid. We start with a topological notation. Let X be a connected locally simply connected topological space. Let us denote by { \atop X × X} the space whose points are triples (x, y, p), where x, y ∈ X; p is a homotopy class of paths in X connecting x with y. Let

c_{X} : { \atop X × X} ⟶ X × X  (256)

be the evident projection. Note that for a fixed x ∈ X, the restriction of c_{X} to c_{X}^{−1}(X×{x}) is a universal covering of X with a group π_{1}(X; x).

11.8. Consider the diagram with cartesian squares

[figure: A^{α}(\{∗\}) × O(K); { \atop A_{α}(K)} × O(K); A^{α}(K) × O(K); { \atop A^{α}(K) × O(K)}; D(0, 1) × O(K); D(0; 1)^{K} × O(K); O(K) × O(K); { \atop O(K) × O(K)}; Δ̃; Δ; j̃; j; c̃; c]

where we have denoted { \atop A^{α}(K)} := A_{α;1} × D(0; 1)^{K}. Here Δ is induced by the diagonal embedding D(0; 1) ↪ D(0; 1)^{K}, j — by the open embedding O(K) ↪ D(0; 1)^{K}, c is the map (256). The upper horizontal arrows are defined by pull-back.

We define Ψ_{K} as a composition

Ψ_{K} = Δ̃^{∗}j̃_{∗}c̃_{∗}c̃^{∗}p^{∗}[1]

where p : A^{α}(K) × O(K) ⟶ A^{α}(K) is the projection.

This functor is t-exact and induces an exact functor

Ψ_{K} : M(A^{α}(K); S) ⟶ M(A^{α}({∗}) × O(K); S × S_{tr})  (257)

where S_{tr} denotes the trivial stratification of O(K).

11.9. Set

A^{α}(K)_{d} := A_{α;1} × O(K; d)

The category M(A^{α}(K); S) is equivalent to the ”inverse limit” ”lim_{←}”M(A^{α}(K)_{d}; S).

Let π : K ⟶ L be an epimorphism. Consider a configuration space

A^{α}(τ_{π;d}) := A_{α;1} × O(τ_{π;d})

where τ_{π;d} := τ_{π;d,0}. An easy generalization of the definition of Ψ_{K} yields a functor

Ψ_{π;d} : M(A^{α}(τ_{π;d})) ⟶ M(A^{α}(L)_{d} × ∏_{l∈L} O(K_{l}))

(In what follows we will omit for brevity stratifications from the notations of abelian categories M(•), implying that we use the principal stratification on all configuration spaces A^{α}(•) and the trivial stratification on spaces O(•), i.e. our sheaves are smooth along these spaces.) Passing to the limit over d > 0 we conclude that there exists essentially unique functor

Ψ_{K⟶L} : M(A^{α}(K)) ⟶ M(A^{α}(L) × ∏_{l} O(K_{l}))  (258)

such that all squares

[figure: M(A^{α}(K)); Ψ_{K⟶L}; M(A^{α}(L) × ∏_{l} O(K_{l})); M(A^{α}(τ_{π;d})); Ψ_{π;d}; M(A^{α}(L)_{d} × ∏_{l} O(K_{l}))]

commute (the vertical arrows being restrictions). If L = {∗}, we return to Ψ_{K}.

11.10. Lemma. All squares

[figure: M(A^{α}(K)); Ψ_{K}; M(A^{α}({∗}) × O(K)); M(A^{α}(L) × ∏_{l} O(K_{l})); Ψ_{L}; M(A^{α}({∗}) × O(L) × ∏_{l} O(K_{l})); Ψ_{K⟶L}; r_{π}]

2-commute. More precisely, there exist natural isomorphisms

ϕ_{K⟶L} : r_{π} ∘ Ψ_{K} ⟶^{∼} Ψ_{L} ∘ Ψ_{K⟶L.}

These isomorphisms satisfy a natural cocycle condition (associated with pairs of epimorphisms K ⟶ L ⟶ M).

11.11. Applying the functors Ψ_{K} componentwise, we get functors

Ψ_{K} : ^{K}FS ⟶ Locsys(O(K); FS);

taking composition with the gluing functor g_{K}, (254), we get functors

⊗_{K} : FS^{K} ⟶ Locsys(O(K); FS)  (259)

It follows from the previous lemma that these functors define a braided tensor structure on FS.

11.12. Let us define a unit object in FS as 1_{FS} = L(0) (cf. 6.3). One can show that it is a unit for the braided tensor structure defined above.

Chapter 3. Functor Φ

12. Functor Φ

12.1. Recall the category C defined in II.11.3.2 and II.12.2.

Our main goal in this section will be the construction of a tensor functor Φ : FS ⟶ C.

12.2. Recall that we have already defined in 4.5 a functor

Φ : FS ⟶ Vect_{f}^{X}.

Now we will construct natural transformations

ϵ_{i} : Φ_{λ}(X) ⟶ Φ_{λ+i′}(X)

and

θ_{i} : Φ_{λ+i′}(X) ⟶ Φ_{λ}(X).

We may, and will, assume that X ∈ FS_{c} for some c. If λ ∉ X_{c} then there is nothing to do.

Suppose that λ ∈ X_{c}; pick α ∈ ℕ[I] such that λ + α′ ≥ λ(X). By definition.

Φ_{λ}(X) = Φ_{α}(X_{λ+α′}^{α})

where Φ_{α} is defined in II.7.14 (the definition will be recalled below).

12.3. Pick an unfolding π : J ⟶ I of α, II.6.12; we will use the same notation for the canonical projection

π : ^{π}𝔸 ⟶ A_{λ+α′}^{α} = A_{α}.

Let N be the dimension of A_{α}.

12.4. Recall some notations from II.8.4. For each r ∈ [0, N] we have denoted by P_{r}(J; 1) the set of all maps

ϱ : J ⟶ [0, r]

such that ϱ(J) contains [r]. Let us assign to such ϱ the real point w_{ϱ} = (ϱ(j))_{j∈J} ∈ ^{π}𝔸.

There exists a unique positive facet of S_{ℝ}, F_{ϱ} containing w_{ϱ}. This establishes a bijection between P_{r}(J; 1) and the set F⊣⌋_{r} of r-dimensional positive facets. At the same time we have fixed on each F_{ϱ} a point w_{ϱ}. This defines cells D_{ϱ}^{+} := D_{F_{ϱ}}^{+}, S_{ϱ}^{+} := S_{F_{ϱ}}^{+}, cf. II.7.2.

Note that this ”marking” of positive facets is Σ_{π}-invariant. In particular, the group Σ_{π} permutes the above mentioned cells.

We will denote by {0} the unique zero-dimensional facet.

12.5. Given a complex K from the bounded derived category D^{b}(A_{α}), its inverse image π^{∗}K is correctly defined as an element of the equivariant derived category D^{b}(^{π}𝔸, Σ_{π}) obtained by localizing the category of Σ_{π}-equivariant complexes on ^{π}𝔸. The direct image π_{∗} acts between equivariant derived categories

π_{∗} : D^{b}(^{π}𝔸, Σ_{π}) ⟶ D^{b}(A_{α}, Σ_{π})

(the action of Σ_{π} on A_{π} being trivial).

We have the functor of Σ_{π}-invariants

(•)^{Σ_{π}} : D^{b}(A_{α}, Σ_{π}) ⟶ D^{b}(A_{α})  (260)

12.5.1. Lemma. For every K ∈ D^{b}(A_{α}) the canonical morphism

K ⟶ (π_{∗}π^{∗}K)^{Σ_{π}}

is an isomorphism.

Proof. The claim may be checked fiberwise. Taking of a fiber commutes with taking Σ_{π}-invariants since our group Σ_{π} is finite and we are living over the field of characteristic zero, hence (•)^{Σ_{π}} is exact. After that, the claim follows from definitions. □

12.5.2. Corollary. For every K ∈ D^{b}(A_{α}) we have canonical isomorphism

RΓ(A_{α}; K) ⟶^{∼} RΓ(^{π}𝔸; π^{∗}K)^{Σ_{π}} □  (261)

12.6. Following II.7.13, consider the sum of coordinates function

∑ t_{j} : ^{π}𝔸 ⟶ 𝔸^{1};

and for L ∈ D(^{π}𝔸; S) let Φ_{∑ t_{j}}(L) denote the fiber at the origin of the corresponding vanishing cycles functor. If H ⊂ ^{π}𝔸 denotes the inverse image (∑ t_{j})^{−1}({1}) then we have canonical isomorphisms

Φ_{∑ t_{j}}(L) ≅ RΓ(^{π}𝔸, H; L) ≅ Φ_{{0}}^{+}(L)  (262)

The first one follows from the definition of vanishing cycles and the second one from homotopy argument.

Note that the if L = π^{∗}K for some K ∈ D(A_{α}; S) then the group Σ_{π} operates canonically on all terms of (262), and the isomorphisms are Σ_{π}-equivariant.

Let us use the same notation

∑ t_{j} : A_{α} ⟶ 𝔸^{1}

for the descended function, and for K ∈ D(A_{α}; S) let Φ_{∑ t_{j}}(K) denote the fiber at the origin of the corresponding vanishing cycles functor. If K belongs to M(A_{α}; S) then Φ_{∑ t_{j}}(K) reduces to a single vector space and this is what we call Φ_{α}(K).

If H̄ denotes π(H) = (∑ t_{j})^{−1}({1}) ⊂ A_{α} then we have canonical isomorphism

Φ_{∑ t_{j}}(K) ≅ RΓ(A_{α}, H̄; K)

12.7. Corollary. (i) For every K ∈ D(A_{α}; S) we have a canonical isomorphism

i_{π} : Φ_{∑ t_{j}}(K) ⟶^{∼} Φ_{{0}}^{+}(π^{∗}K)^{Σ_{π}}.  (263)

(ii) This isomorphism does not depend on the choice of an unfolding π : J ⟶ I.

Let us explain what (ii) means. Suppose π′ : J′ ⟶ I be another unfolding of α. There exists (a non unique) isomorphism

ρ : J ⟶^{∼} J′

such that π′ ∘ ρ = π. It induces isomorphisms

ρ^{∗} : Σ_{π′} ⟶^{∼} Σ_{π}

(conjugation by ρ), and

ρ^{∗} : Φ_{{0}}^{+}((π′)^{∗}K) ⟶^{∼} Φ_{{0}}^{+}(π^{∗}K)

such that

ρ^{∗}(σx) = ρ^{∗}(σ)ρ^{∗}(x),

σ ∈ Σ_{π′}, x ∈ Φ_{{0}}^{+}((π′)^{∗}K). Passing to invariants, we get an isomorphism

ρ^{∗} : Φ_{{0}}^{+}((π′)^{∗}K)^{Σ_{π′}} ⟶^{∼} Φ_{{0}}^{+}(π^{∗}K)^{Σ_{π}}.

Now (ii) means that i_{π} ∘ρ^{∗} = i_{π′}. As a consequence, the last isomorphism does not depend on the choice of ρ.

Proof. Part (i) follows from the preceding discussion and 12.5.2.

As for (ii), it suffices to prove that any automorphism ρ : J ⟶^{∼} J respecting π induces the identity automorphism of the space of invariants Φ_{{0}}^{+}(π^{∗}K)^{Σ_{π}}. But the action of ρ on the space Φ_{{0}}^{+}(π^{∗}K) comes from the action of Σ_{π} on this space, and our claim is obvious. □

In computations the right hand side of (263) will be used as a de facto definition of Φ_{α}.

12.8. Suppose that α = ∑ a_{i}i; pick an i such that a_{i} > 0.

Let us introduce the following notation. For a partition J = J_{1} ∐ J_{2} and a positive d let 𝔸^{J_{1},J_{2}}(d) denote an open suspace of ^{π}𝔸 consisting of all points t = (t_{j}) such that |t_{j}| > d (resp., |t_{j}| < d) if j ∈ J_{1} (resp., j ∈ J_{2}).

We have obviously

π^{−1}(A^{i,α−i}(d)) = ∐_{j∈π^{−1}(i)} 𝔸^{{j},J−{j}}(d)  (264)

12.9. For j ∈ π^{−1}(i) let π_{j} : J − {j} ⟶ I denote the restriction of π; it is an unfolding of α − i. The group Σ_{π_{j}} may be identified with the subgroup of Σ_{π} consisting of automorphisms stabilizing j. For j′, j′′ ∈ J let (j′j′′) denotes their transposition. We have

Σ_{π_{j′′}} = (j′j′′)Σ_{π_{j′}}(j′j′′)^{−1}  (265)

For a fixed j ∈ π^{−1}(i) we have a partition into cosets

Σ_{π} = ∐_{j′∈π^{−1}(i)} Σ_{π_{j}}(jj′)  (266)

12.10. For j ∈ J let F_{j} denote a one-dimensional facet corresponding to the map ϱ_{j} : J ⟶ [0, 1] sending all elements to 0 except for j being sent to 1 (cf. 12.4).

For K ∈ D(A_{α}; S) we have canonical and variation maps

v_{j} : Φ_{{0}}^{+}(π^{∗}K) ⟵^{⟶} Φ_{F_{j}}^{+}(π^{∗}K) : u_{j}

defined in II, (89), (90). Taking their sum, we get maps

v_{i} : Φ_{{0}}^{+}(π^{∗}K) ⟵^{⟶} ⊕_{j∈π^{−1}(i)} Φ_{F_{j}}^{+}(π^{∗}K) : u_{i}  (267)

Note that the group Σ_{π} operates naturally on both spaces and both maps v_{i} and u_{i} respect this action.

Let us consider more attentively the action of Σ_{π} on ⊕_{j∈π^{−1}(i)} Φ_{F_{j}}^{+}(π^{∗}K). A subgroup Σ_{π_{j}} respects the subspace Φ_{F_{j}}^{+}(π^{∗}K). A transposition (j′j′′) maps Φ_{F_{j′}}^{+}(π^{∗}K) isomorphically onto Φ_{F_{j′′}}^{+}(π^{∗}K).

Let us consider the space of invariants

(⊕_{j∈π^{−1}(i)} Φ_{F_{j}}^{+}(π^{∗}K))^{Σ_{π}}

For every k ∈ π^{−1}(i) the obvious projection induces isomorphism

(⊕_{j∈π^{−1}(i)} Φ_{F_{j}}^{+}(π^{∗}K))^{Σ_{π}} ⟶^{∼} (Φ_{F_{k}}^{+}(π^{∗}K))^{Σ_{π_{k}}};

therefore for two different k, k′ ∈ π^{−1}(i) we get an isomorphism

i_{kk′} : (Φ_{F_{k}}^{+}(π^{∗}K))^{Σ_{π_{k}}} ⟶^{∼} (Φ_{F_{k′}}^{+}(π^{∗}K))^{Σ_{π_{k′}}}  (268)

Obviously, it is induced by transposition (kk′).

12.11. Let us return to the situation 12.2 and apply the preceding discussion to K = X_{λ+α′}^{α}. We have by definition

Φ_{λ}(X) = Φ_{∑ t_{j}}(X_{λ+α′}^{α}) ≅ Φ_{{0}}^{+}(π^{∗}X_{λ+α′}^{α})^{Σ_{π}}  (269)

On the other hand, let us pick some k ∈ π^{−1}(i) and a real d, 0 < d < 1. The subspace

F_{k}^{⊥}(d) ⊂ 𝔸^{{j},J−{j}}(d)  (270)

consisting of points (t_{j}) with t_{k} = 1, is a transversal slice to the face F_{k}. Consequently, the factorization isomorphism for X_{λ+α′}^{α} lifted to 𝔸^{{j},J−{j}}(d) induces isomorphism

Φ_{F_{k}}^{+}(π^{∗}X_{λ+α′}^{α}) ≅ Φ_{{0}}^{+}(π_{k}^{∗}X_{λ+α′}^{α−i}) ⊗ (I_{λ+i′}^{i})_{{1}} = Φ_{{0}}^{+}(π_{k}^{∗}X_{λ+α′}^{α−i})

Therefore we get isomorphisms

Φ_{λ+i′}(X) = Φ_{{0}}^{+}(π_{k}^{∗}X_{λ+α′}^{α−i})^{Σ_{π_{k}}} ≅ Φ_{F_{k}}^{+}(π^{∗}X_{λ+α′}^{α})^{Σ_{π_{k}}} ≅  (271)
≅ (⊕_{j∈π^{−1}(i)} Φ_{F_{j}}^{+}(π^{∗}X_{λ+α′}^{α}))^{Σ_{π}}

It follows from the previous discussion that this isomorphism does not depend on the intermediate choice of k ∈ π^{−1}(i).

Now we are able to define the operators θ_{i}, ϵ_{i}:

ϵ_{i} : Φ_{λ}(X) ⟵^{⟶} Φ_{λ+i′}(X) : θ_{i}

By definition, they are induced by the maps u_{i}, v_{i} (cf. (267)) respectively (for K = X_{λ+α′}^{α}) after passing to Σ_{π}-invariants and taking into account isomorphisms (269) and (271).

12.12. Theorem. The operators ϵ_{i} and θ_{i} satisfy the relations II.12.3, i.e. the previous construction defines functor

Φ̃ : FS ⟶ C̃

where the category C̃ is defined as in loc. cit.

Proof will occupy the rest of the section.

12.13. Let 𝔲^{+} (resp., 𝔲^{−}) denote the subalgebra of 𝔲 generated by all ϵ_{i} (resp., θ_{i}). For β ∈ ℕ[I] let 𝔲_{β}^{±} ⊂ 𝔲^{±} denote the corresponding homogeneous component.

The proof will go as follows. First, relations II.12.3 (z), (a), (b) are obvious. We will do the rest in three steps.

Step 1. Check of (d). This is equivalent to showing that the action of operators θ_{i} correctly defines maps

𝔲_{β}^{−} ⊗ Φ_{λ+β′}(X) ⟶ Φ_{λ}(X)

for all β ∈ ℕ[I].

Step 2. Check of (e). This is equivalent to showing that the action of operators ϵ_{i} correctly defines maps

𝔲_{β}^{+} ⊗ Φ_{λ}(X) ⟶ Φ_{λ+β′}(X)

for all β ∈ ℕ[I].

Step 3. Check of (c).

12.14. Let us pick an arbitrary β = ∑ b_{i}i ∈ ℕ[I] and α ∈ ℕ[I] such that λ + α′ ≥ λ(X) and α ≥ β. We pick the data from 12.3. In what follows we will generalize the considerations of 12.8 — 12.11.

Let U(β) denote the set of all subsets J′ ⊂ J such that |J′ ∩ π^{−1}(i)| = b_{i} for all i. Thus, for such J′, π_{J′} := π|_{J′} : J′ ⟶ I is an unfolding of β and π_{J−J′} — an unfolding of α − β. We have a disjoint sum decomposition

π^{−1}(A^{β,α−β}(d)) = ∐_{J′∈U(β)} 𝔸^{J′,J−J′}(d)  (272)

(cf. (264)).

12.15. For J′ ∈ U(β) let F_{J′} denote a one-dimensional facet corresponding to the map ϱ_{J′} : J ⟶ [0, 1] sending j ∉ J′ to 0 j ∈ J′ — to 1 (cf. 12.4).

For K ∈ D(A_{α}; S) we have canonical and variation maps

v_{J′} : Φ_{{0}}^{+}(π^{∗}K) ⟵^{⟶} Φ_{F_{J′}}^{+}(π^{∗}K) : u_{J′}

Taking their sum, we get maps

v_{β} : Φ_{{0}}^{+}(π^{∗}K) ⟵^{⟶} ⊕_{J′∈U(β)} Φ_{F_{J′}}^{+}(π^{∗}K) : u_{β}  (273)

The group Σ_{π} operates naturally on both spaces and both maps v_{β} and u_{β} respect this action.

A subgroup Σ_{J′} respects the subspace Φ_{F_{J′}}^{+}(π^{∗}K). The projection induces an isomorphism

(⊕_{J′∈U(β)} Φ_{F_{J′}}^{+}(π^{∗}K))^{Σ_{π}} ⟶^{∼} Φ_{F_{J′}}^{+}(π^{∗}K)^{Σ_{J′}}.

We have the crucial

12.16. Lemma. Factorization isomorphism for X induces canonical isomorphism

𝔲_{β}^{−} ⊗ Φ_{λ+β′}(X) ≅ (⊕_{J′∈U(β)} Φ_{F_{J′}}^{+}(π^{∗}X_{λ+α′}^{α}))^{Σ_{π}}  (274)

Proof. The argument is the same as in 12.11, using II, Thm. 6.16. □

12.17. As a consequence, passing to Σ_{π}-invariants in (273) (for K = X_{λ+α′}^{α}) we get the maps

ϵ_{β} : Φ_{λ}(X) ⟵^{⟶} 𝔲_{β}^{−} ⊗ Φ_{λ+β′}(X) : θ_{β}

12.18. Lemma. The maps θ_{β} provide Φ(X) with a structure of a left module over the negative subalgebra 𝔲^{−}.

Proof. We must prove the associativity. It follows from the associativity of factorization isomorphisms. □

This lemma proves relations II.12.3 (d) for operators θ_{i}, completing Step 1 of the proof of our theorem.

12.19. Now let us consider operators

ϵ_{β} : Φ_{λ}(X) ⟶ 𝔲_{β}^{−} ⊗ Φ_{λ+β′}(X).

By adjointness, they induce operators

𝔲_{β}^{−∗} ⊗ Φ_{λ}(X) ⟶ Φ_{λ+β′}(X)

The bilinear form S, II.2.10, induces isomorhisms

S : 𝔲_{β}^{−∗} ⟶^{∼} 𝔲_{β}^{−};

let us take their composition with the isomorphism of algebras

𝔲^{−} ⟶^{∼} 𝔲^{+}

sending θ_{i} to ϵ_{i}. We get isomorphisms

S′ : 𝔲_{β}^{−∗} ⟶^{∼} 𝔲_{β}^{+}

Using S′, we get from ϵ_{β} operators

𝔲_{β}^{+} ⊗ Φ_{λ}(X) ⟶ Φ_{λ+β′}(X)

12.19.1. Lemma. The above operators provide Φ(X) with a structure of a left 𝔲^{+}- module.

For β = i they coincide with operators ϵ_{i} defined above.

This lemma completes Step 2, proving relations II.12.3 (e) for generators ϵ_{i}.

12.20. Now we will perform the last Step 3 of the proof, i.e. prove the relations II.12.3 (c) between operators ϵ_{i}, θ_{j}. Consider a square

[figure: Φ_{λ+i′}(X); Φ_{λ}(X); Φ_{λ−j′+i′}(X); Φ_{λ−j′}(X); ϵ_{i}; θ_{j}; ϵ_{i}; θ_{j}]

We have to prove that

ϵ_{i}θ_{j} − ζ^{i·j}θ_{j}ϵ_{i} = δ_{ij}(1 − ζ^{−2λ·i′})  (275)

(cf. (193)).

As before, we may and will suppose that X ∈ FS_{c} and λ ∈ X_{c} for some c. Choose α ∈ ℕ[I] such that α ≥ i, α ≥ j and λ − j′ + α′ ≥ λ(X). The above square may be identified with the square

[figure: Φ_{α−i−j}(X_{λ−j′+α′}^{α−i−j}); Φ_{α−j}(X_{λ−j′+α′}^{α−j}); Φ_{α−i}(X_{λ−j′+α′}^{α−i}); Φ_{α}(X_{λ−j′+α′}^{α}); ϵ_{i}; θ_{j}; ϵ_{i}; θ_{j}]

12.21. Choose an unfolding π : J ⟶ I of α; let Σ = Σ_{π} be its automorphism group, and π : 𝔸 = ^{π}𝔸 ⟶ A_{α} denote the corresponding covering. We will reduce our proof to certain statements about (vanishing cycles of) sheaves on 𝔸.

Let us introduce a vector space

V = H^{0}Φ_{{0}}^{+}(π^{∗}X_{λ−j′+α′}^{α});

the group Σ operates on it, and we have

Φ_{α}(X_{λ−j′+α′}^{α}) ≅ V^{Σ}  (276)

For each k ∈ J we have a positive one-dimensional facet F_{k} ⊂ 𝔸_{ℝ} defined as in 12.10. Denote

V_{k} = H^{0}Φ_{F_{k}}^{+}(π^{∗}X_{λ−j′+α′}^{α});

we have canonically

Φ_{α−p}(X_{λ−j′+α′}^{α−p}) ≅ (⊕_{k∈π^{−1}(p)}V_{k})^{Σ}  (277)

for each p ∈ I, p ≤ α, cf. 12.11.

12.21.1. We have to extend considerations of 12.11 to two-dimensional facets. For each pair of different k, l ∈ J such that π(k) = i, π(l) = j, let F_{kl} denote a two-dimensional positive facet corresponding to the map ϱ : J ⟶ [0, 2] sending k to 1, l — to 2 and all other elements — to zero (cf. 12.4). Set

V_{kl} = H^{0}Φ_{F_{kl}}^{+}(π^{∗}X_{λ−j′+α′}^{α}).

Again, due to equivariance of our sheaf, the group Σ operates on ⊕ V_{kl} in such a way that

σ(V_{kl}) = V_{σ(k)σ(l)}.

Let

π_{kl} : J − {k, l} ⟶ I

be the restriction of π. It is an unfolding of α − i − j; let Σ_{kl} denote its automorphism group. Pick d_{1}, d_{2} such that 0 < d_{2} < 1 < d_{1} < 2. The subspace

F_{kl}^{⊥} ⊂ 𝔸^{{l},{k},J−{k,l}}(d_{1}, d_{2})  (278)

consisting of all points (t_{j}) such that t_{k} = 1 and t_{l} = 2 is a transversal slice to f_{kl}. Consequently, factorization axiom implies canonical isomorphism

Φ_{F_{kl}}^{+}(π^{∗}X_{λ−j′+α′}^{α}) ≅ Φ_{{0}}^{+}(π_{kl}^{∗}X_{λ−j′+α′}^{α−i−j}) ⊗ (I_{λ+i′}^{i})_{{1}} ⊗ (I_{λ}^{j})_{{2}} = Φ_{{0}}^{+}(π_{kl}^{∗}X_{λ−j′+α′}^{α−i−j})

Symmetry. Interchanging k and l, we get isomorphisms

t : V_{kl} ⟶^{∼} V_{lk}  (279)

Passing to Σ-invariants, we get isomorphisms

Φ_{α−i−j}(X_{λ−j′+α′}^{α−i−j}) = Φ_{{0}}^{+}(π_{kl}^{∗}X_{λ−j′+α′}^{α−i−j})^{Σ_{kl}} ≅  (280)
≅ Φ_{F_{kl}}^{+}(π^{∗}X_{λ−j′+α′}^{α})^{Σ_{kl}} ≅ (⊕_{k∈π^{−1}(i),l∈π^{−1}(j),k≠l} V_{kl})^{Σ}

(cf. (271)).

12.22. The canonical and variation maps induce linear operators

v_{k} : V ⟵^{⟶} V_{k} : u^{k}

and

v_{lk}^{k} : V_{k} ⟵^{⟶} V_{lk} : u_{k}^{lk}

which are Σ-equivariant in the obvious sense. Taking their sum, we get operators

V ⟵^{⟶} ⊕_{k∈π^{−1}(p)}V_{k} ⟵^{⟶} ⊕_{l∈π^{−1}(q),k∈π^{−1}(p)}V_{lk}

which induce, after passing to Σ-invariants, operators ϵ_{p}, ϵ_{q} and θ_{p}, θ_{q}.

Our square takes a form

[figure: (⊕_{k∈π^{−1}(j),l∈π^{−1}(i),k≠l}V_{lk})^{Σ}; (⊕_{k∈π^{−1}(j),l∈π^{−1}(i),k≠l}V_{kl})^{Σ}; (⊕_{k∈π^{−1}(j)}V_{k})^{Σ}; (⊕_{l∈π^{−1}(i)}V_{l})^{Σ}; V^{Σ}; ⟶^{∼}^{t}; ∑ v_{lk}^{k}; ∑ u^{k}; ∑ v_{l}; ∑ u_{l}^{kl}]

Now we will formulate two relations between u and v which imply the necessary relations between ϵ and θ.

12.23. Lemma. Suppose that i = j and π(k) = i. Consider operators

u^{k} : V_{k} ⟵^{⟶} V : v_{k}.

The composition v_{k}u^{k} is equal to the multiplication by 1 − ζ^{−2λ·i′}.

Proof. Consider the transversal slice F_{k}^{⊥}(d) to the face F_{k} as in 12.11. It follows from the definition of the canonical and variation maps, II.7.10, that composition v_{k}u^{k} is equal to 1 − T^{−1} where T is the monodromy acquired by of Φ_{{0}}^{+}(π_{k}^{∗}X_{λ−i′+α′}^{α−i}) when the point t_{k} moves around the disc of radius d where all other points are living. By factorization, T = ζ^{2λ·i′}. □

12.24. Lemma. For k ∈ π^{−1}(j), l ∈ π^{−1}(i), k ≠ l, consider the pentagon

[figure: V_{lk}; V_{kl}; V_{k}; V_{l}; V; ⟶^{∼}^{t}; v_{lk}^{k}; u^{k}; v_{l}; u_{l}^{kl}]

We have

v_{l}u^{k} = ζ^{i·j}u_{k}^{kl} ∘ t ∘ v_{lk}^{k}.

This lemma is a consequence of the following more general statement.

12.25. Let K ∈ D(A; S) be arbitrary. We have naturally

Φ_{F_{kl}}^{+}(K) ≅ Φ_{{0}}^{+}(K|_{F_{kl}^{⊥}(d_{1},d_{2})}[−2])

Let

t : Φ_{F_{kl}}^{+}(K) ⟶^{∼} Φ_{F_{lk}}^{+}(K)

be the monodromy isomorphism induced by the travel of the point t_{l} in the upper half plane to the position to the left of t_{k} (outside the disk of radius d_{1}).

12.25.1. Lemma. The composition

v_{F_{l}}^{{0}} ∘ u_{{0}}^{F_{k}} : Φ_{F_{k}}^{+}(K) ⟶ Φ_{F_{l}}^{+}(K)

is equal to u_{F_{l}}^{F_{kl}} ∘ t ∘ v_{F_{lk}}^{F_{k}}.

12.26. It remains to note that due to 12.22 the desired relation (275) is a formal consequence of lemmas 12.23 and 12.24. This completes the proof of Theorem 12.12. □

12.27. Taking composition of Φ̃ with an inverse to the equivalence Q, II (143), we get a functor

Φ : FS ⟶ C  (281)

13. Main properties of Φ

TENSOR PRODUCTS

13.1. Theorem. Φ is a tensor functor, i.e. we have natural isomorphisms

Φ(X ⊗^{•}Y) ⟶^{∼} Φ(X) ⊗ Φ(Y)

satisfying all necessary compatibilities.

Proof follows from the Additivity theorem, II.9.3. □

DUALITY

13.2. Let C̃_{ζ^{−1}} denote the category C̃ with the value of parameter ζ changed to ζ^{−1}. The notations FS_{ζ^{−1}}, etc. will have the similar meaning.

Let us define a functor

D : C̃^{opp} ⟶ C̃_{ζ^{−1}}  (282)

as follows. By definition, for V = ⊕V_{λ} ∈ Ob C̃^{opp} = Ob C̃,

(DV)_{λ} = V_{λ}^{∗},

and operators

θ_{i,DV} : (DV)_{λ} ⟵^{⟶} (DV)_{λ−i′} : ϵ_{i,DV}

are defined as

ϵ_{i,DV} = θ_{i,V}^{∗}; θ_{i,DV} = −ζ^{2λ·i′}ϵ_{i,V}^{∗} .

On morphisms D is defined in the obvious way. One checks directly that D is well defined, respects tensor structures, and is an equivalence.

13.3. Let us define a functor

D : FS^{opp} ⟶ FS_{ζ^{−1}}  (283)

as follows. For X ∈ Ob FS^{opp} = Ob FS we set λ(DX) = λ(X); (DX)^{α} = D(X^{α}) where D in the right hand side is Verdier duality. Factorization isomorphisms for DX are induced in the obvious way from factorization isomorphisms for X. The value of D on morphisms is defined in the obvious way.

D is a tensor equivalence.

13.4. Theorem. Functor Φ̃ commutes with D.

Proof. Our claim is a consequence of the following topological remarks.

13.5. Consider a standard affine space 𝔸 = 𝔸^{J} with a principal stratification S as in II.7. Let K ∈ D(𝔸; S), let F_{j} be the one-dimensional facet corresponding to an element j ∈ J as in 12.10. Consider a transversal slice F_{j}^{⊥}(d) as in 12.11. We have canonically

Φ_{F_{j}}^{+}(K) ≅ Φ_{{0}}(K|_{F_{j}^{⊥}(d)}[−1]);

when the point t_{j} moves counterclockwise around the disk of radius d, Φ_{j}^{+}(K) acquires monodromy

T_{j} : Φ_{j}^{+}(K) ⟶^{∼} Φ_{j}^{+}(K).

13.5.1. Lemma. Consider canonical and variation maps

v_{DK} : Φ_{{0}}^{+}(DK) ⟵^{⟶} Φ_{F_{j}}^{+}(DK) : u_{DK}

Let us identify Φ_{{0}}^{+}(DK), Φ_{F_{j}}^{+}(DK) with Φ_{{0}}^{+}(K)^{∗} and Φ_{F_{j}}^{+}(K)^{∗} respectively. Then the maps become

v_{DK} = u_{K}^{∗}; u_{DK} = −v_{K}^{∗} ∘ T_{j}^{∗}

The theorem follows from this lemma. □

STANDARD OBJECTS 13.6. Theorem. We have naturally

Φ(L(Λ)) ≅ L(Λ)

for all Λ ∈ X.

Combining this with Theorem 6.2, we get

13.7. Theorem. Φ induces bijection between sets of isomorphism classes of irreducibles. □

13.8. Verma modules. Let 𝔲^{≥0} ⊂ 𝔲 denote the subalgebra generated by ϵ_{i}, K_{i}, K^{−1}, i ∈ I. For Λ ∈ X, let χ_{Λ} denote a one-dimensional representation of 𝔲^{≥0} generated by a vector v_{Λ}, with the action

ϵ_{i}v_{Λ} = 0, K_{i}v_{Λ} = ζ^{⟨Λ,i⟩}v_{Λ}.

Let M(Λ) denote the induced 𝔲-module

M(Λ) = 𝔲 ⊗_{𝔲^{≥0}} χ_{Λ}.

Equipped with an obvious X-grading, M(Λ) becomes an object of C̃. We will also use the same notation for the corresponding object of C.

13.9. Theorem. The factorizable sheaves M(Λ) are finite. We have naturally

Φ(M(Λ)) ≅ M(Λ).

Proof is given in the next two subsections.

13.10. Let us consider the space 𝔸 as in 13.5. Suppose that K ∈ D(𝔸; S) has the form K = j_{!}j^{∗}K where

j : 𝔸 − ⋃_{j∈J}{t_{j} = 0} ↪ 𝔸.

Let F_{Δ} be the positive facet whose closure is the main diagonal.

13.10.1. Lemma. The canonical map

u : Φ_{F_{Δ}}^{+}(K) ⟶ Φ_{{0}}^{+}(K)

is an isomorphism.

Proof. Pick j_{0} ∈ J, and consider a subspace Y = {t_{j_{0}} = 0} ∪ {t_{j_{0}} = 1} ⊂ 𝔸. Set K′ = k_{!}k^{∗}K where

k : 𝔸 − ⋃_{j∈J}{t_{j} = 0} − ⋃_{j∈J}{t_{j} = 1} ↪ 𝔸.

We have

Φ_{{0}}^{+}(K) = RΓ(𝔸; K′)

On the other hand by homotopy we have

Φ_{F_{Δ}}^{+}(K) ≅ RΓ({t_{j_{0}} = c}; K′)[−1]

where c is any real between 0 and 1. Let us compute RΓ(𝔸; K′) using the Leray spectral sequence of a projection

p : 𝔸 ⟶ 𝔸^{1}, (t_{j}) ↦ t_{j_{0}}.

The complex p_{∗}K′ is equal to zero at the points {0} and {1}, and is constant with the fiber RΓ({t_{j_{0}} = c}; K′) over c. It follows that

RΓ(𝔸; K′) ≅ RΓ(𝔸^{1}; p_{∗}K′) ≅ RΓ({t_{j_{0}} = c}; K′)[−1],

and the inverse to this isomorphism may be identified with u. □

13.11. Suppose we have α ∈ ℕ[I], let π : J ⟶ I; π : 𝔸 ⟶ A_{α} be the corresponding unfolding. Let us apply the previous lemma to K = π^{∗}M(Λ)^{α}. Note that after passing to Σ_{π}-invariants, the map u becomes

𝔲_{α}^{−} ⟶ Φ̃_{α}(M(Λ))

by Theorem II.6.16. This identifies homogeneous components of Φ̃_{α}(M(Λ)) with the components of the Verma module. After that, the action of ϵ_{i} and θ_{i} is identified with the action of 𝔲 on it. The theorem is proven. □

Chapter 4. Equivalence

14. Truncation functors

14.1. Recall the notations of 5.4. We fix a coset X_{c} ⊂ X, and we denote the subcategory FS_{≤λ;c} ⊂ FS by FS_{≤λ} for simplicity untill further notice.

Given λ ∈ X, we will denote by C_{≤λ} ⊂ C the full subcategory of all 𝔲-modules V such that V_{μ} ≠ 0 implies μ ≤ λ. We denote by q_{λ} the embedding functor C_{λ} ↪ C. Obviously, Φ(FS_{≤λ}) ⊂ C_{λ}.

In this section we will construct functors

σ_{λ}^{!}, σ_{λ}^{∗} : FS ⟶ FS_{≤λ}

and

q_{λ}^{!}, q_{λ}^{∗} : C ⟶ C_{≤λ},

such that σ_{λ}^{!} (resp., σ_{λ}^{∗}) is right (resp., left) adjoint to σ_{λ} and q_{λ}^{!} (resp., q_{λ}^{∗}) is right (resp., left) adjoint to q_{λ}.

14.2. First we describe σ_{λ}^{∗}, σ_{λ}^{!}. Given a factorizable sheaf X = {X^{α}} with λ(X) = μ ≥ λ we define FS’s Y := σ_{λ}^{∗}X and Z := σ_{λ}^{!}X as follows.

We set λ(Y) = λ(Z) = λ. For α ∈ ℕ[I] we set

Y^{α} = L^{0}σ^{∗}X^{α+μ−λ}

if α + μ − λ ∈ ℕ[I] and 0 otherwise, and

Z^{α} = R^{0}σ^{!}X^{α+μ−λ}

if α + μ − λ ∈ ℕ[I] and 0 otherwise. Here σ denotes the canonical closed embedding (cf. 2.4)

σ : A_{λ}^{α} ↪ A_{μ}^{α+μ−λ},

and L^{0}σ^{∗} (resp., R^{0}σ^{!}) denotes the zeroth perverse cohomology of σ^{∗} (resp., of σ^{!}).

The factorization isomorphisms for Y and Z are induced from those for X; associativity is obvious.

14.3. Lemma. Let M ∈ FS_{≤λ}, X ∈ FS. Then

Hom_{FS}(X , M) = Hom_{FS_{≤λ}}(σ_{λ}^{∗}X , M)

and

Hom_{FS}(M, X) = Hom_{FS_{≤λ}}(M, σ_{λ}^{!}X)

Proof. Let μ = λ(X). We have

Hom_{A_{μ}^{α}}(X_{μ}^{α}, M_{μ}^{α}) = Hom_{A_{μ}^{α−μ+λ}}(σ_{λ}^{∗}X_{λ}^{α−μ+λ}, M_{λ}^{α−μ+λ})

by the usual adjointness. Passing to projective limit in α, we get the desired result for σ^{∗}. The proof for σ^{!} is similar. □

14.4. Given λ ≤ μ ∈ X_{c}, we denote by σ_{λ≤μ} the embedding of the full subcategory

σ_{λ≤μ} : FS_{≤λ} ↪ FS_{≤μ}.

Obviously, the functor

σ_{λ≤μ}^{∗} := σ_{λ}^{∗} ∘ σ_{μ} : FS_{≤μ} ⟶ FS_{≤λ}

is left adjoint to σ_{λ≤μ}. Similarly, σ_{λ≤μ}^{!} := σ_{λ}^{!} ∘ σ_{μ} is the right adjoint to σ_{λ≤μ}.

For λ ≤ μ ≤ ν we have obvious transitivities

σ_{λ≤μ}^{∗}σ_{μ≤ν}^{∗} = σ_{λ≤ν}^{∗}; σ_{λ≤μ}^{!}σ_{μ≤ν}^{!} = σ_{λ≤ν}^{!}.

14.5. For each α ∈ ℕ[I] and i ∈ I such that α ≥ i let j_{ν−i′≤ν}^{α} denote the open embedding

j_{ν−i′≤ν}^{α} : A_{ν}^{α} − σ(A_{ν−i′}^{α−i}) ↪ A_{ν}^{α}.

Note that the complement of this open subspace is a divisor, so the corresponding extension by zero and by ∗ functors are t-exact, cf. [BBD], 4.1.10 (i). Let us define functors

j_{ν−i′≤ν!}, j_{ν−i′≤ν∗} : FS_{≤ν} ⟶ FS_{≤ν}

as follows. For a factorizable sheaf X = {X_{ν}^{α}} ∈ FS_{≤ν} we set

(j_{ν−i′≤ν!}X)_{ν}^{α} = j_{ν−i′≤ν!}^{α}j_{ν−i′≤ν}^{α∗}X_{ν}^{α}

and

(j_{ν−i′≤ν∗}X)_{ν}^{α} = j_{ν−i′≤ν∗}^{α}j_{ν−i′≤ν}^{α∗}X_{ν}^{α},

the factorization isomorphisms being induced from those for X.

14.6. Lemma. We have natural in X ∈ FS_{≤ν} exact sequences

j_{ν−i′,ν!}X ⟶ X ⟶^{a} σ_{ν−i′,ν}^{∗}X ⟶ 0

and

0 ⟶ σ_{ν−i′,ν}^{!}X ⟶^{a′} X ⟶ j_{ν−i′,ν∗}X

where the maps a and a′ are the adjunction morphisms.

Proof. Evidently follows from the same claim at each finite level, which is [BBD], 4.1.10 (ii). □

14.7. Recall (see 13.3) that we have the Verdier duality functor

D : FS^{opp} ⟶ FS_{ζ^{−1}}.

By definition, D(FS_{≤λ}^{opp}) ⊂ FS_{ζ^{−1};≤λ} for all λ.

We have functorial isomorphisms

D ∘ σ_{λ}^{∗} ≅ σ_{λ}^{!} ∘ D; D ∘ σ_{ν−i′,ν}^{∗} ≅ σ_{ν−i′,ν}^{!} ∘ D

and

D ∘ j_{ν−i′,ν∗} ≅ j_{ν−i′,ν!} ∘ D

After applying D, one of the exact sequences in 14.6 becomes another one.

14.8. Let us turn to the category C. Below we will identify C with C̃ using the equivalence Q, cf. II.12.5. In other words, we will regard objects of C as 𝔲-modules.

For λ ∈ X_{c} functors q_{λ}^{!} and q_{λ}^{∗} are defined as follows. For V ∈ C, q_{λ}^{!}V (resp., q_{λ}^{∗}V) is the maximal subobject (resp., quotient) of V belonging to the subcategory C_{λ}.

For λ ≤ μ ∈ X_{c} let q_{λ≤μ} denotes an embedding of a full subcategory

q_{λ≤μ} : C_{≤λ} ↪ C_{≤μ}

Define q_{λ≤μ}^{!} := q_{λ}^{!} ∘ q_{μ}; q_{λ≤μ}^{∗} := q_{λ}^{!} ∘ q_{μ}. Obviously, the first functor is right adjoint, and the second one is left adjoint to q_{λ≤μ}. They have an obvious transitivity property.

14.9. Recall that in 13.2 the weight preserving duality equivalence

D : C^{opp} ⟶ C_{ζ^{−1}}

is defined. By definition, D(C_{≤λ}^{opp}) ⊂ C_{ζ^{−1};≤λ} for all λ.

We have functorial isomorphisms

D ∘ q_{λ}^{∗} ≅ q_{λ}^{!} ∘ D; D ∘ q_{ν−i′,ν}^{∗} ≅ q_{ν−i′,ν}^{!} ∘ D.

14.10. Given i ∈ I, let us introduce a ”Levi” subalgebra 𝔩_{i} ⊂ 𝔲 generated by θ_{j}, ϵ_{j}, j ≠ i, and K_{i}^{±}. Let 𝔭_{i} ⊂ 𝔲 denote the ”parabolic” subalgebra generated by 𝔩_{i} and ϵ_{i}.

The subalgebra 𝔩_{i} projects isomorphically to 𝔭_{i}/(ϵ_{i}) where (ϵ_{i}) is a two-sided ideal generated by ϵ_{i}. Given an 𝔩_{i}-module V, we can consider it as a 𝔭_{i}-module by restriction of scalars for the projection 𝔭_{i} ⟶ 𝔭_{i}/(ϵ_{i}) ≅ 𝔩_{i}, and form the induced 𝔲-module Ind_{𝔭_{i}}^{𝔲}V — ”generalized Verma”.

14.11. Given an 𝔲-module V ∈ C_{≤ν}, let us consider a subspace

iV = ⊕_{α∈ℕ[I−{i}]}V_{ν−α′} ⊂ V.

It is an X-graded 𝔭_{i}-submodule of V. Consequently, we have a canonical element

π ∈ Hom_{C}(Ind_{π_{i}}^{𝔲} _{i}V, V) = Hom_{𝔭_{i}}(_{i}V, V)

corresponding to the embedding _{i}V ↪ V.

We will also consider the dual functor

V ↦ D^{−1}Ind_{𝔭_{i}}^{𝔲} _{i}(DV).

By duality, we have a natural morphism in C, V ⟶ D^{−1}Ind_{𝔭_{i}}^{𝔲} _{i}(DV).

14.12. Lemma. We have natural in V ∈ C_{≤ν} exact sequences

Ind_{𝔭_{i}}^{𝔲} _{i}V ⟶^{π} V ⟶ q_{ν−i′≤ν}^{∗}V ⟶ 0

and

0 ⟶ q_{ν−i′≤ν}^{!}V ⟶ V ⟶ D^{−1}Ind_{𝔭_{i}}^{𝔲} _{i}(DV).

where the arrows V ⟶ q_{ν−i′≤ν}^{∗}V and q_{ν−i′≤ν}^{!}V ⟶ V are adjunction morphisms.

Proof. Let us show the exactness of the first sequence. By definition, q_{ν−i′≤ν}^{∗}V is the maximal quotient of V lying in the subcategory C_{ν−i′≤ν} ⊂ C. Obviously, Coker π ∈ C_{ν−i′,ν}. It remains to show that for any morphism h : V ⟶ W with W ∈ C_{ν−i′}, the composition h ∘ π : Ind_{𝔭_{i}}^{𝔲} _{i}V ⟶ W is zero. But Hom_{C}(Ind_{𝔭_{i}}^{𝔲} _{i}V, W) = Hom_{𝔭_{i}}(_{i}V, W) = 0 by weight reasons.

The second exact sequence is the dual to the first one. □

14.13. Lemma. We have natural in X ∈ FS_{ν} isomorphisms

Φj_{ν−i′≤ν!}X ⟶^{∼} Ind_{𝔭_{i}}^{𝔲} _{i}(ΦX)

and

Φj_{ν−i′≤ν∗}X ⟶^{∼} D^{−1}Ind_{𝔭_{i}}^{𝔲} _{i}(DΦX)

such that the diagram

[figure: Φj_{ν−i′,ν!}X; ΦX; Φj_{ν−i′,ν∗}X; Ind_{𝔭_{i}}^{𝔲} _{i}(ΦX); ΦX; D^{−1}Ind_{𝔭_{i}}^{𝔲} _{i}(DΦX)]

commutes.

14.14. Lemma. Let λ ∈ X_{c}. We have natural in X ∈ FS isomorphisms

Φσ_{λ}^{∗}X ≅ q_{λ}^{∗}ΦX

and

Φσ_{λ}^{!}X ≅ q_{λ}^{!}ΦX

Proof follows at once from lemmas 14.13, 14.6 and 14.12. □

15. Rigidity

15.1. Lemma. Let X ∈ FS_{≤0}. Then the natural map

a : Hom_{FS}(L(0), X) ⟶ Hom_{C}(L(0), Φ(X))

is an isomorphism.

Proof. We know already that a is injective, so we have to prove its surjectivity. Let K(0) (resp., K(0)) denote the kernel of the projection M(0) ⟶ L(0) (resp., M(0) ⟶ L(0)). Consider a diagram with exact rows:

[figure: 0; Hom(L(0), X); Hom(M(0), X); Hom(K(0), X); 0; Hom(L(0), Φ(X)); Hom(M(0), Φ(X)); Hom(K(0), Φ(X)); a; b]

All vertical rows are injective. On the other hand, Hom(M(0), Φ(X)) = Φ(X)_{0}. The last space is isomorphic to a generic stalk of X_{0}^{α} for each α ∈ ℕ[I], which in turn is isomorphic to Hom_{FS}(M(0), X) by the universal property of the shriek extension. Consequently, b is isomorphism by the equality of dimensions. By diagram chase, we conclude that a is isomorphism. □

15.2. Lemma. For every X ∈ FS the natural maps

Hom_{FS}(L(0), X) ⟶ Hom_{C}(L(0), Φ(X))

and

Hom_{FS}(X , L(0)) ⟶ Hom_{C}(Φ(X), L(0))

are isomorphisms.

Proof. We have

Hom_{FS}(L(0), X) = Hom_{FS}(L(0), σ_{0}^{!}X)

(by lemma 14.3)

= Hom_{C}(L(0), Φ(σ_{0}^{!}X))

(by the previous lemma)

= Hom_{C}(L(0), q_{0}^{!}Φ(X))

(by lemma 14.14)

= Hom_{C}(L(0), Φ(X)).

This proves the first isomorphism. The second one follows by duality. □

15.3. Recall (see [KL]IV, Def. A.5) that an object X of a tensor category is called rigid if there exists another object X^{∗} together with morphisms

i_{X} : 1 ⟶ X ⊗ X^{∗}

and

e_{X} : X^{∗} ⊗ X ⟶ 1

such that the compositions

X = 1 ⊗ X ⟶^{i_{X}⊗id} X ⊗ X^{∗} ⊗ X ⟶^{id⊗e_{X}} X

and

X^{∗} = X^{∗} ⊗ 1 ⟶^{id⊗i_{X}} X^{∗} ⊗ X ⊗ X^{∗} ⟶^{e_{X}⊗id} X^{∗}

are equal to id_{X} and id_{X^{∗}} respectively. A tensor category is called rigid if all its objects are rigid.

15.4. Theorem. All irreducible objects L(λ), λ ∈ X, are rigid in FS.

Proof. We know (cf. [AJS], 7.3) that C is rigid. Moreover, there exists an involution λ ↦ λ̄ on X such that L(λ)^{∗} = L(λ̄). Let us define L(λ)^{∗} := L(λ̄); i_{L(λ)} corresponds to i_{L(Λ)} under identification

Hom_{FS}(L(0), L(λ) ⊗ L(λ̄)) = Hom_{C}(L(0), L(λ) ⊗ L(λ̄))

and e_{L(λ)} corresponds to e_{L(λ)} under identification

Hom_{FS}(L(λ̄) ⊗ L(λ), L(0)) = Hom_{C}(L(λ̄) ⊗ L(λ), L(0)),

cf. 15.2. □

16. Steinberg sheaf

In this section we assume that l is a positive number prime to 2, 3 and that ζ′ is a primitive (l · det A)-th root of 1 (recall that ζ = (ζ′)^{det A}).

We fix a weight λ_{0} ∈ X such that ⟨i, λ_{0}⟩ = −1(mod l) for any i ∈ I. Our goal in this section is the proof of the following

16.1. Theorem. The FFS L(λ_{0}) is a projective object of the category FS.

Proof. We have to check that Ext^{1}(L(λ_{0}), X) = 0 for any FFS X. By induction on the length of X it is enough to prove that Ext^{1}(L(λ_{0}), L) = 0 for any simple FFS L.

16.2. To prove vanishing of Ext^{1} in FS we will use the following principle. Suppose Ext^{1}(X , Y) ≠ 0, and let

0 ⟶ Y ⟶ Z ⟶ X ⟶ 0

be the corresponding nonsplit extension. Let us choose a weight λ which is bigger than of λ(X), λ(Y), λ(Z). Then for any α ∈ ℕ[I] the sequence

0 ⟶ Y_{λ}^{α} ⟶ Z_{λ}^{α} ⟶ X_{λ}^{α} ⟶ 0

is also exact, and for α ≫ 0 it is also nonsplit (see lemma 5.3). That is, for α ≫ 0 we have Ext^{1}(X_{λ}^{α}, Y_{λ}^{α}) ≠ 0 in the category of all perverse sheaves on the space A_{λ}^{α}. This latter Ext can be calculated purely topologically. So its vanishing gives a criterion of Ext^{1}-vanishing in the category FS.

16.3. In calculating Ext^{1}(L(λ_{0}), L(μ)) we will distinguish between the following three cases:

a) μ ≱ λ_{0};

b) μ = λ_{0};

c) μ > λ_{0}.

16.4. Let us treat the case a).

16.4.1. Lemma. For any α ∈ ℕ[I] the sheaf L(λ_{0})_{λ_{0}}^{α} is the shriek-extension from the open stratum of toric stratification of A_{λ_{0}}^{α}.

Proof. Due to the factorization property it is enough to check that the stalk of L(λ_{0})_{λ_{0}}^{α} at the point {0} ∈ A_{λ_{0}}^{α} vanishes for any α ∈ ℕ[I], α ≠ 0. By the Theorem II.8.23, we have (L(λ_{0})_{λ_{0}}^{α})_{0} = _{α}C_{𝔣}^{•}(L(λ_{0})) ≃ 0 since L(λ_{0}) is a free 𝔣-module by [L1] 36.1.5 and Theorem II.11.10(b) and, consequently, C_{𝔣}^{•}(L(λ_{0})) ≃ H_{𝔣}^{0}(L(λ_{0})) = B and has weight zero. □

Returning to the case a), let us choose ν ∈ X, ν ≥ λ_{0}, ν ≥ μ. For any α, the sheaf L′ := L(μ)_{ν}^{α} is supported on the subspace

A′ := σ(A_{μ}^{α+μ−ν}) ⊂ A_{ν}^{α}

and L′′ := L(λ_{0})_{ν}^{α} — on the subspace

A′′ := σ(A_{λ_{0}}^{α+λ_{0}−ν}) ⊂ A_{ν}^{α}.

Let i denote a closed embedding

i : A′′ ↪ A_{ν}^{α}

and j an open embedding

j : A^{∘}′′ := A′′ − A′′ ∩ A′ ↪ A′′.

We have by adjointness

RHom_{A_{ν}^{α}}(L′′, L′) = RHom_{A′′}(L′′, i^{!}L′) =

(by the previous lemma)

= RHom_{A^{∘}}′′(j^{∗}L′′, j^{∗}i^{!}L′) = 0

since obviously j^{∗}i^{!}L′ = 0. This proves the vanishing in the case a).

16.5. In case (b), suppose

0 ⟶ L(λ_{0}) ⟶ X ⟶ L(λ_{0}) ⟶ 0

is a nonsplit extension in FS. Then for α ≫ 0 the restriction of X_{λ_{0}}^{α} to the open toric stratum of A_{λ_{0}}^{α} is a nonsplit extension

0 ⟶ I^{•}_{λ_{0}}^{α} ⟶ X^{•} _{λ_{0}}^{α} ⟶ I^{•}_{λ_{0}}^{α} ⟶ 0

(in the category of all perverse sheaves on A^{•}_{λ_{0}}^{α}) (we can restrict to the open toric stratum because of lemma 16.4.1). This contradicts to the factorization property of FFS X. This contradiction completes the consideration of case (b).

16.6. In case (c), suppose Ext^{1}(L(λ_{0}), L(μ)) ≠ 0 whence Ext^{1}(L(λ_{0})_{μ}^{α}, L(μ)_{μ}^{α}) ≠ 0 for some α ∈ ℕ[I] by the principle 16.2. Here the latter Ext is taken in the category of all perverse sheaves on A_{μ}^{α}. We have μ − λ_{0} = β′ for some β ∈ ℕ[I], β ≠ 0.

Let us consider the closed embedding

σ : A′ := A_{λ_{0}}^{α−β} ↪ A_{μ}^{α};

let us denote by j an embedding of the open toric stratum

j : A^{•}′ := A^{•}_{λ_{0}}^{α−β} ↪ A′.

As in the previous case, we have

RHom_{A_{μ}^{α}}(L(λ_{0})_{μ}^{α}, L(μ)_{μ}^{α}) = RHom_{A′}(L(λ_{0})_{μ}^{α}, σ^{!}L(μ)_{μ}^{α}) = RHom_{A^{•}}′(j^{∗}L(λ_{0})_{μ}^{α}, j^{∗}σ^{!}L(μ)_{μ}^{α}).

We claim that j^{∗}σ^{!}L(μ)_{μ}^{α} = 0. Since the sheaf L(μ)_{μ}^{α} is Verdier auto-dual up to replacing ζ by ζ^{−1}, it suffices to check that j^{∗}σ^{∗}L(μ)_{μ}^{α} = 0.

To prove this vanishing, by factorization property of L(μ), it is enough to check that the stalk of the sheaf L(μ)_{μ}^{β} at the point {0} ∈ A_{μ}^{β} vanishes.

By the Theorem II.8.23, we have (L(μ)_{μ}^{β})_{0} = _{β}C_{𝔣}^{•}(L(μ)). By the Theorem II.11.10 and Shapiro Lemma, we have _{β}C_{𝔣}^{•}(L(μ)) ≃ C_{U}^{•}(M(λ_{0}) ⊗ L(μ)).

By the Theorem 36.1.5. of [L1], the canonical projection M(λ_{0}) ⟶ L(λ_{0}) is an isomorphism. By the autoduality of L(λ_{0}) we have C_{U}^{•}(L(λ_{0})⊗L(μ)) ≃ RHom_{U}^{•}(L(λ_{0}), L(μ)) ≃ 0 since L(λ_{0}) is a projective U-module, and μ ≠ λ_{0}.

This completes the case c) and the proof of the theorem. □

17. Equivalence

We keep the assumptions of the previous section.

17.1. Theorem. Functor Φ : FS ⟶ C is an equivalence.

17.2. Lemma. For any λ ∈ X the FFS L(λ_{0})⊗^{•}L(λ) is projective.

As λ ranges through X, these sheaves form an ample system of projectives in FS.

Proof. We have

Hom(L(λ_{0})⊗^{•}L(λ),?) = Hom(L(λ_{0}), L(λ)^{∗} ⊗^{•}?)

by the rigidity, and the last functor is exact since ⊗^{•} is a biexact functor in FS, and L(λ_{0}) is projective. Therefore, L(λ_{0})⊗^{•}L(λ) is projective.

To prove that these sheaves form an ample system of projectives, it is enough to show that for each μ ∈ X there exists λ such that Hom(L(λ_{0})⊗^{•}L(λ), L(μ)) ≠ 0. We have

Hom(L(λ_{0})⊗^{•}L(λ), L(μ)) = Hom(L(λ), L(λ_{0})^{∗}⊗^{•}L(μ)).

Since the sheaves L(λ) exhaust irreducibles in FS, there exists λ such that L(λ) embeds into L(λ_{0})^{∗}⊗^{•}L(μ), hence the last group is non-zero. □

17.3. Proof of 17.1. As λ ranges through X, the modules Φ(L(λ_{0})⊗^{•}L(λ)) = L(λ_{0}) ⊗ L(λ) form an ample system of projectives in C. By the Lemma A.15 of [KL]IV we only have to show that

Φ : Hom_{FS}(L(λ_{0})⊗^{•}L(λ), L(λ_{0})⊗^{•}L(μ)) ⟶ Hom_{C}(L(λ_{0}) ⊗ L(λ), L(λ_{0}) ⊗ L(μ))

is an isomorphism for any λ, μ ∈ X. We already know that it is an injection. Therefore, it remains to compare the dimensions of the spaces in question. We have

dim Hom_{FS}(L(λ_{0})⊗^{•}L(λ), L(λ_{0})⊗^{•}L(μ)) = dim Hom_{FS}(L(λ_{0}), L(λ_{0})⊗^{•}L(μ)⊗^{•}L(λ)^{∗})

by rigidity,

= [L(λ_{0})⊗^{•}L(μ)⊗^{•}L(λ)^{∗} : L(λ_{0})]

because L(λ_{0}) is its own indecomposable projective cover in FS,

= [L(λ_{0}) ⊗ L(μ) ⊗ L(λ)^{∗} : L(λ_{0})]

since Φ induces an isomorphism of K-rings of the categories FS and C,

= dim Hom_{C}(L(λ_{0}) ⊗ L(λ), L(λ_{0}) ⊗ L(μ))

by the same argument applied to C. The theorem is proven. □

18. The case of generic ζ

In this section we suppose that ζ is not a root of unity.

18.1. Recall the notations of II.11,12. We have the algebra U defined in II.12.2, the algebra 𝔲 defined in II.12.3, and the homomorphism R : U ⟶ 𝔲 defined in II.12.5.

18.1.1. Lemma. The map R : U ⟶ 𝔲 is an isomorphism.

Proof follows from [R], no. 3, Corollaire. □

18.2. We call Λ ∈ X dominant if ⟨i, Λ⟩ ≥ 0 for any i ∈ I. An irreducible U-module L(Λ) is finite dimensional if only if Λ is dominant. Therefore we will need a larger category O containing all irreducibles L(Λ).

Define O as a category consisting of all X-graded U-modules V = ⊕_{μ∈X}V_{μ} such that

a) V_{μ} is finite dimensional for any μ ∈ X;

b) there exists λ = λ(V) such that V_{μ} = 0 if μ ≱ λ(V).

18.2.1. Lemma The category O is equivalent to the usual category O_{𝔤} over the classical finite dimensional Lie algebra 𝔤.

Proof. See [F1]. □

18.3. Let W denote the Weyl group of our root datum. For w ∈ W, λ ∈ X let w · λ denote the usual action of W on X centered at −ρ.

Finally, for Λ ∈ X let M(Λ) ∈ O denote the U^{−}-free Verma module with highest weight Λ.

18.3.1. Corollary. Let μ, ν ∈ X be such that W ·μ ≠ W ·ν. Then Ext^{•}(M(ν), L(μ)) = 0.

Proof. M(ν) and L(μ) have different central characters. □

18.4. Theorem. Functor Φ : FS ⟶ C is an equivalence.

Proof. We know that Φ(L(Λ)) ≃ L(Λ) for any Λ ∈ X. So Φ(X) is finite dimensional iff all the irreducible subquotients of X are of the form L(λ), λ dominant. By virtue of Lemma 18.2.1 above the category C is semisimple: it is equivalent to the category of finite dimensional 𝔤-modules. It consists of finite direct sums of modules L(λ), λ dominant. So to prove the Theorem it suffices to check semisimplicity of FS. Thus the Theorem follows from

18.5. Lemma. Let μ, ν ∈ X be the dominant weights. Then Ext^{1}(L(μ), L(ν)) = 0.

Proof. We will distinguish between the following two cases:

(a) μ = ν;

(b) μ ≠ ν.

In calculating Ext^{1} we will use the principle 16.2. The argument in case (a) is absolutely similar to the one in section 16.5, and we leave it to the reader.

In case (b) either μ − ν ∉ Y ⊂ X — and then the sheaves L(ν) and L(μ) are supported on the different connected components of A, whence Ext^{1} obviously vanishes, — or there exists λ ∈ X such that λ ≥ μ, ν. Let us fix such λ. Suppose Ext^{1}(L(μ), L(ν)) ≠ 0. Then according to the principle 16.2 there exists α ∈ ℕ[I] such that Ext^{1}(L(μ)_{λ}^{α}, L(ν)_{λ}^{α}) ≠ 0. The latter Ext is taken in the category of all perverse sheaves on A_{λ}^{α}.

We have Ext^{1}(L(μ)_{λ}^{α}, L(ν)_{λ}^{α}) = R^{1}Γ(A_{λ}^{α}, D(L(μ)_{λ}^{α} ⊗ DL(ν)_{λ}^{α})) where D stands for Verdier duality, and ⊗ denotes the usual tensor product of constructible comlexes.

We will prove that

L(μ)_{λ}^{α} ⊗ DL(ν)_{λ}^{α} = 0  (284)

and hence we will arrive at the contradiction. Equality (284) is an immediate corollary of the lemma we presently formulate.

For β ≤ α let us consider the canonical embedding

σ : A^{•}^{α−β} ↪ A^{α}

and denote its image by A^{•}′ (we omit the lower case indices).

18.5.1. Lemma. (i) If σ^{∗}L(μ)_{λ}^{α} ≠ 0 then λ − β ∈ W · μ.

(ii) If σ^{!}L(μ)_{λ}^{α} ≠ 0 then λ − β ∈ W · μ.

To deduce Lemma 18.5 from this lemma we notice first that the sheaf L(μ)_{λ}^{α} is Verdier autodual up to replacing ζ by ζ^{−1}. Second, since the W -orbits of ν and μ are disjoint, we

see that over any toric stratum A^{•}′ ⊂ A^{α} at least one of the factors of (284) vanishes.

It remains to prove Lemma 18.5.1. We will prove (i), while (ii) is just dual. Let us denote β + μ − λ by γ. If γ ∉ ℕ[I] then (i) is evident. Otherwise, by the factorizability condition it is enough to check that the stalk of L(μ)_{μ}^{γ} at the origin in A_{μ}^{γ} vanishes if μ − γ ∉ W · μ. Let us denote μ − γ by χ.

By the Theorem II.8.23, we have (L(μ)_{μ}^{γ})_{0} = _{γ}C_{U^{−}}^{•}(L(μ)) ≃ C_{U}^{•}(M(χ) ⊗ L(μ)) which is dual to Ext_{U}^{•}(M(χ), L(μ)). But the latter Ext vanishes by the Corollary 18.3.1 since W · χ ≠ W · μ. □

This completes the proof of Lemma 18.5 together with Theorem 18.4.

Part IV. LOCALIZATION ON ℙ^{1}

1. Introduction

1.1. Given a collection of m finite factorizable sheaves {X_{k}}, we construct here some perverse sheaves over configuration spaces of points on a projective line ℙ^{1} with m additional marked points.

We announce here (with sketch proof) the computation of the cohomology spaces of these sheaves. They turn out to coincide with certain ”semiinfinite” Tor spaces of the corresponding 𝔲-modules introduced by S.Arkhipov. For a precise formulation see Theorem 8.11.

This result is strikingly similar to the following hoped-for picture of affine Lie algebra representation theory, explained to us by A.Beilinson. Let M_{1}, M_{2} be two modules over an affine Lie algebra 𝔤̂ on the critical level. One hopes that there is a localization functor which associates to these modules perverse sheaves Δ(M_{1}), Δ(M_{2}) over the semiinfinite flag space Ĝ/B̂^{0}. Suppose that Δ(M_{1}) and Δ(M_{2}) are equivariant with respect to the opposite Borel subgroups of Ĝ. Then the intersection S of their supports is finite dimensional, and one hopes that

R^{•}Γ(S, Δ(M_{1}) ⊗ Δ(M_{2})) = Tor𝔤̂ ; \frac{∞}{2} −•(M_{1}, M_{2})

where in the right hand side we have the Feigin (Lie algebra) semiinfinite homology.

As a corollary of Theorem 8.11 we get a description of local systems of conformal blocks in WZW models in genus zero (cf. [MS]) as natural subquotients of some semisimple local systems of geometric origin. In particular, these local systems are semisimple themselves.

1.2. We are grateful to G.Lusztig for the permission to use his unpublished results. Namely, Theorem 6.2.1 about braiding in the category C is due to G.Lusztig. Chapter 2 (semiinfinite homological algebra in C) is an exposition of the results due to S.Arkhipov (see [Ark]).

We are also grateful to A.Kirillov, Jr. who explained to us how to handle the conformal blocks of non simply laced Lie algebras.

1.3. Unless specified otherwise, we will keep the notations of parts I,II,III. For α = ∑_{i} a_{i}i ∈ ℕ[I] we will use the notation |α| := ∑_{i} a_{i}.

References to loc.cit. will look like Z.1.1 where Z=I, II or III.

We will keep assumptions of III.1.4 and III.16. In particular, a ”quantization” parameter ζ will be a primitive l-th root of unity where l is a fixed positive number prime to 2, 3.

Chapter 1. Gluing over ℙ^{1}

2. Cohesive local system

2.1. Notations. Let α ∈ ℕ[X], α = ∑ a_{μ}μ. We denote by suppα the subset of X consisting of all μ such that a_{μ} ≠ 0. Let π : J ⟶ X be an unfolding of α, that is a map of sets such that |π^{−1}(μ)| = a_{μ} for any μ ∈ X. As always, Σ_{π} denotes the group of automorphisms of J preserving the fibers of π.

ℙ^{1} will denote a complex projective line. The J-th cartesian power ℙ^{1J} will be denoted by P^{J}. The group Σ_{π} acts naturally on P^{J}, and the quotient space P^{J}/Σ_{π} will be denoted by P^{α}.

P^{oJ} (resp., P^{oα}) stands for the complement to diagonals in P^{J} (resp., in P^{α}).

T P^{J} stands for the total space of the tangent bundle to P^{J}; its points are couples ((x_{j}), (τ_{j})) where (x_{j}) ∈ P^{J} and τ_{j} is a tangent vector at x_{j}. An open subspace

T^{•} P^{J} ⊂ T P^{J}

consists of couples with τ_{j} ≠ 0 for all j. So, T^{•} P^{J} ⟶ P^{J} is a (ℂ^{∗})^{J}-torsor. We denote by T P^{oJ} its restriction to P^{oJ}. The group Σ_{π} acts freely on T P^{oJ}, and we denote the quotient T P^{oJ}/Σ_{π} by T P^{oα}.

The natural projection

T P^{oJ} ⟶ T P^{oα}

will be denoted by π, or sometimes by π_{J}.

2.2. Let ℙ_{st}^{1} (st for ”standard”) denote ”the” projective line with fixed coordinate z; D_{ϵ} ⊂ ℙ_{st}^{1} — the open disk of radius ϵ centered at z = 0; D := D_{1}. We will also use the notation D_{(ϵ,1)} for the open annulus D − \overline{D}_{ϵ} (bar means the closure).

The definitions of D^{J}, D^{α}, D^{oJ}, D^{oα}, T^{•} D^{J}, T D^{oJ}, T D^{oα}, etc., copy the above definitions, with D replacing ℙ^{1}.

2.3. Given a finite set K, let P̃^{K} denote the space of K-tuples (u_{k})_{k∈K} of algebraic isomorphisms ℙ_{st}^{1} ⟶^{∼} ℙ^{1} such that the images u_{k}(D) do not intersect.

Given a K-tuple α⃗ = (α_{k}) ∈ ℕ[X]^{K} such that α = ∑_{k} α_{k}, define a space

Pα⃗ := P̃^{K} × ∏_{k∈K} T^{•} D^{α_{k}}

and an open subspace

P^{oα⃗} := P̃^{K} × ∏_{k∈K} T D^{oα_{k}}.

We have an evident ”substitution” map

qα⃗ : Pα⃗ ⟶ T^{•} P^{α}

which restricts to qα⃗ : P^{oα⃗} ⟶ T P^{oα}. 2.3.1. In the same way we define the spaces T Dα⃗, T D^{oα⃗}. 2.3.2. Suppose that we have an epimorphism ξ : L ⟶ K, denote L_{k} := ξ^{−1}(k). Assume that each α_{k} is in turn decomposed as

α_{k} = ∑_{l∈L_{k}} α_{l}, α_{l} ∈ ℕ[X];

set α⃗_{k} = (α_{l}) ∈ ℕ[X]^{L_{k}}. Set α⃗_{L} = (α_{l}) ∈ ℕ[X]^{L}. Let us define spaces

P^{α⃗_{L};ξ} = P̃^{K} × ∏_{k∈K} T^{•} Dα⃗_{k}

and

P^{oα⃗_{L};ξ} = P̃^{K} × ∏_{k∈K} T D^{oα⃗_{k}}.

We have canonical substitution maps

q_{α⃗_{L};ξ}^{1} : P^{α⃗_{L};ξ} ⟶ Pα⃗

and

q_{α⃗_{L};ξ}^{2} : P^{α⃗_{L};ξ} ⟶ Pα⃗_{L}.

Obviously,

qα⃗ ∘ q_{α⃗_{L};ξ}^{1} = qα⃗_{L} ∘ q_{α⃗_{L};ξ}^{2}.

2.4. Balance function. Consider a function n : X ⟶ ℤ[\frac{1}{2 det A}] such that

n(μ + ν) = n(μ) + n(ν) + μ · ν

It is easy to see that n can be written in the following form:

n(μ) = \frac{1}{2}μ · μ + μ · ν_{0}

for some ν_{0} ∈ X. From now on we fix such a function n and hence the corresponding ν_{0}.

2.5. For an arbitrary α ∈ ℕ[X], let us define a one-dimensional local system I_{D}^{α} on T D^{oα}. We will proceed in the same way as in III.3.1. Pick an unfolding of α, π : J ⟶ X. Define a local system I_{D}^{J} on T D^{oJ} as follows: its stalk at each point ((τ_{j}), (x_{j})) where all x_{j} are real, and all the tangent vectors τ_{j} are real and directed to the right, is B. Monodromies are:

— x_{i} moves counterclockwise around x_{j}: monodromy is ζ^{−2π(i)·π(j)};

— τ_{j} makes a counterclockwise circle: monodromy is ζ^{−2n(π(j))}. This local system has an evident Σ_{π}-equivariant structure, and we define a local system I_{D}^{α} as

I_{D}^{α} = (π_{∗}I_{D}^{J})^{sgn}

where π : T D^{oJ} ⟶ T D^{oα} is the canonical projection, and (•)^{sgn} denotes the subsheaf of skew Σ_{π}-invariants.

2.6. We will denote the unique homomorphism

ℕ[X] ⟶ X

identical on X, as α ↦ α^{∼}.

2.6.1. Definition. An element α ∈ ℕ[X] is called admissible if α^{∼} ≡ −2ν_{0} mod lY.

2.7. We have a canonical ”1-jet at 0” map p_{K} : P̃^{K} ⟶ T P^{oK}

2.8. Definition. A cohesive local system (CLS) (over ℙ^{1}) is the following collection of data:

(i) for each admissible α ∈ ℕ[X] a one-dimensional local system I^{α} over T P^{oα}; (ii) for each decomposition α = ∑_{k∈K} α_{k}, α_{k} ∈ ℕ[X], a factorization isomorphism

ϕα⃗ : q∗ ; α⃗I^{α} ⟶^{∼} p_{K}^{∗}π_{K}^{∗}I^{α_{K}} ⊠ × _{k} I_{D}^{α_{k}^{∼}}

Here α_{K} := ∑_{k} α_{k}^{∼} ∈ ℕ[X] (note that α_{K} is obviously admissible); π_{K} : T P^{oK} ⟶ T P^{oα} is the symmetrization map. These isomorphisms must satisfy the following Associativity axiom. In the assumptions of 2.3.2 the equality

ϕ_{α⃗_{L};ξ} ∘ q_{α⃗_{L};ξ}^{1∗}(ϕα⃗) = q_{α⃗_{L};ξ}^{2∗}(ϕα⃗_{L})

should hold. Here ϕ_{α⃗_{L};ξ} is induced by the evident factorization isomorphisms for local systems on the disk Iα⃗_{k} ; D. Morphisms between CLS’s are defined in the obvious way.

2.9. Theorem. Cohesive local systems over ℙ^{1} exist. Every two CLS’s are isomorphic. The group of automorphisms of a CLS is B^{∗}.

This theorem is a particular case of a more general theorem, valid for curves of arbitrary genus, to be proved in Part V. We leave the proof in the case of ℙ^{1} to the interested reader.

3. Gluing

3.1. Let us define an element ρ ∈ X by the condition ⟨ρ, i⟩ = 1 for all i ∈ I. From now on we choose a balance function n, cf. 2.4, in the form

n(μ) = \frac{1}{2}μ · μ + μ · ρ.

It has the property that n(−i′) = 0 for all i ∈ I. Thus, in the notations of loc. cit. we set

ν_{0} = ρ.

We pick a corresponding CLS I = {I^{β}, β ∈ ℕ[X]}.

Given α = ∑ a_{i}i ∈ ℕ[I] and μ⃗ = (μ_{k}) ∈ X^{K}, we define an element

αμ⃗ = ∑ a_{i} · (−i′) + ∑_{k} μ_{k} ∈ ℕ[X]

where the sum in the right hand side is a formal one. We say that a pair (μ⃗, α) is admissible if αμ⃗ is admissible in the sense of the previous section, i.e.

∑_{k} μ_{k} − α ≡ −2ρ mod lY.

Note that given μ⃗, there exists α ∈ ℕ[I] such that (μ⃗, α) is admissible if and only if ∑_{k} μ_{k} ∈ Y; if this holds true, such elements α form an obvious countable set.

We will denote by

e : ℕ[I] ⟶ ℕ[X]

a unique homomorphism sending i ∈ I to −i′ ∈ X.

3.2. Let us consider the space T^{•} P^{K} × T^{•} P^{e(α)}; its points are quadruples ((z_{k}), (τ_{k}), (x_{j}), (ω_{j})) where (z_{k}) ∈ P^{K}, τ_{k} — a non-zero tangent vector to ℙ^{1} at z_{k}, (x_{j}) ∈ P^{e(α)}, ω_{j} — a non-zero tangent vector at x_{j}. To a point z_{k} is assigned a weight μ_{k}, and to x_{j} — a weight −π(j)′. Here π : J ⟶ I is an unfolding of α (implicit in the notation (x_{j}) = (x_{j})_{j∈J}).

We will be interested in some open subspaces:

T^{•} Pα ; μ⃗ := T P^{oK} × T^{•} P^{e(α)} ⊂ T^{•} P^{αμ⃗}

and

T Poα ; μ⃗ ⊂ T^{•} Pα ; μ⃗

whose points are quadruples ((z_{k}), (τ_{k}), (x_{j}), (ω_{j})) ∈ T^{•} Pα ; μ⃗ with all z_{k} ≠ x_{j}. We have an obvious symmetrization projection

pα ; μ⃗ : T Poα ; μ⃗ ⟶ T P^{oαμ⃗}.

Define a space

Pα ; μ⃗ = T P^{oK} × P^{e(α)};

its points are triples ((z_{k}), (τ_{k}), (x_{j})) where (z_{k}), (τ_{k}) and (x_{j}) are as above; and to z_{k} and x_{j} the weights as above are assigned. We have the canonical projection

T^{•} Pα ; μ⃗ ⟶ Pα ; μ⃗.

We define the open subspaces

Poα ; μ⃗ ⊂ P•α ; μ⃗ ⊂ Pα ; μ⃗.

Here the •-subspace (resp., o-subspace) consists of all ((z_{k}), (τ_{k}), (x_{j})) with z_{k} ≠ x_{j} for all k, j (resp., with all z_{k} and x_{j} distinct).

We define the principal stratification S of Pα ; μ⃗ as the stratification generated by subspaces z_{k} = x_{j} and x_{j} = x_{j′} with π(j) ≠ π(j′). Thus, Poα ; μ⃗ is the open stratum of S. As usually, we will denote by the same letter the induced stratifications on subspaces.

The above projection restricts to

T Poα ; μ⃗ ⟶ Poα ; μ⃗.

3.3. Factorization structure.

3.3.1. Suppose we are given α⃗ ∈ ℕ[I]^{K}, β ∈ ℕ[I]; set α := ∑_{k} α_{k}. Define a space

Pα⃗,β ; μ⃗ ⊂ P̃^{K} × ∏_{k} D^{α_{k}} × P^{e(β)}

consisting of all collections ((u_{k}), ((x_{j}^{(k)})_{k}), (y_{j})) where (u_{k}) ∈ P̃^{K}, (x_{j}^{(k)})_{k} ∈ D^{α_{k}}, (y_{j}) ∈ P^{e(β)}, such that

y_{j} ∈ ℙ^{1} − ⋃_{k∈K} \overline{u_{k}(D)}

for all j (the bar means closure).

We have canonical maps

q_{α⃗,β} : Pα⃗,β ; μ⃗ ⟶ Pα+β ; μ⃗,

assigning to ((u_{k}), ((x_{j}^{(k)})_{k}), (y_{j})) a configuration (u_{k}(0)), (u^{•}_{k}(τ)), (u_{k}(x_{j}^{(k)})), (y_{j})), where τ is the unit tangent vector to D at 0, and

p_{α⃗,β} : Pα⃗,β ; μ⃗ ⟶ ∏_{k} D^{α_{k}} × P_{μ⃗−α⃗}^{•β}

sending ((u_{k}), ((x_{j}^{(k)})_{k}), (y_{j})) to ((u_{k}(0)), (u^{•}_{k}(τ)), (y_{j})).

3.3.2. Suppose we are given α⃗, β⃗ ∈ ℕ[I]^{K}, γ ∈ ℕ[I]; set α := ∑_{k} α_{k}, β := ∑_{k} β_{k}.

Define a space D^{α,β} consisting of couples (D_{ϵ}, (x_{j})) where D_{ϵ} ⊂ D is some smaller disk (0 < ϵ < 1), and (x_{j}) ∈ D^{α+β} is a configuration such that α points dwell inside D_{ϵ}, and β points — outside \overline{D_{ϵ}}. We have an evident map

q_{α,β} : D^{α,β} ⟶ D^{α+β}.

Let us define a space

Pα⃗,β⃗,γ ; μ⃗ ⊂ P̃^{K} × ∏_{k∈K} D^{α_{k},β_{k}} × P^{e(γ)}

consisting of all triples ((u_{k}), x, (y_{j})) where (u_{k}) ∈ P̃^{K}, x ∈ ∏_{k} D^{α_{k},β_{k}}, (y_{j}) ∈ P^{e(γ)} such that

y_{j} ∈ ℙ^{1} − ⋃_{k} \overline{u_{k}(D)}.

We have obvious projections

q_{α⃗,β⃗,γ}^{1} : Pα⃗,β⃗,γ ; μ⃗ ⟶ Pα⃗+β⃗,γ ; μ⃗

and

q_{α⃗,β⃗,γ}^{2} : Pα⃗,β⃗,γ ; μ⃗ ⟶ Pα⃗,β+γ ; μ⃗

such that

q_{α⃗+β⃗,γ} ∘ q_{α⃗,β⃗,γ}^{1} = q_{α⃗,β+γ} ∘ q_{α⃗,β⃗,γ}^{2}.

We will denote the last composition by q_{α⃗,β⃗,γ}.

We have a natural projection

p_{α⃗,β⃗,γ} : Pα⃗,β⃗,γ ; μ⃗ ⟶ ∏_{k} Dα_{k}_{\frac{1}{2}} × ∏_{k} D_{( \frac{1}{2} ,1)}^{β_{k}} × P_{μ⃗−α⃗−β⃗}^{•γ}.

3.4. Let us consider a local system pα∗ ; μ⃗I^{αμ⃗} over T Poα ; μ⃗. By our choice of the balance function n, its monodromies with respect to the rotating of tangent vectors ω_{j} at points x_{j} corresponding to negative simple roots, are trivial. Therefore it descends to a unique local system over Poα ; μ⃗, to be denoted by I_{μ}^{α}.

We define a perverse sheaf

I•α ; μ⃗ := j_{!∗}Iα ; μ⃗[dim Pα ; μ⃗] ∈ M(P•α ; μ⃗; S).

3.5. Factorizable sheaves over ℙ^{1}. Suppose we are given a K-tuple of FFS’s {X_{k}}, X_{k} ∈ FS_{c_{k}}, k ∈ K, c_{k} ∈ X/Y, where ∑_{k} c_{k} = 0. Let us pick μ⃗ = (μ⃗_{k}) ≥ (λ(X_{k})).

Let us call a factorizable sheaf over ℙ^{1} obtained by gluing the sheaves X_{k} the following collection of data which we will denote by g({X_{k}}).

(i) For each α ∈ ℕ[I] such that (μ⃗, α) is admissible, a sheaf X α ; μ⃗ ∈ M(Pα ; μ⃗; S).

(ii) For each α⃗ = (α_{k}) ∈ ℕ[I]^{K}, β ∈ ℕ[I] such that (μ, α + β) is admissible (where α = ∑ α_{k}), a factorization isomorphism

ϕ_{α⃗,β} : q_{α⃗,β}^{∗}X α+β ; μ⃗ ⟶^{∼} p_{α⃗,β}^{∗}(( × _{k∈K} X_{μ_{k}}^{α_{k}}) ⊠ I_{μ⃗−α⃗}^{•β}).

These isomorphisms should satisfy

Associativity property. The following two isomorphisms

q_{α⃗,β⃗,γ}^{∗}X α+β+γ ; μ⃗ ⟶^{∼} p_{α⃗,β⃗,γ}^{∗}(( × _{k} X_{μ_{k}}^{α_{k}}) ⊠ ( × _{k} I_{μ_{k}−α_{k}}^{•β_{k}}) ⊠ I_{μ⃗−α⃗−β⃗}^{•γ})

are equal:

ψ_{β⃗,γ} ∘ q_{α⃗,β⃗,γ}^{2∗}(ϕ_{α⃗,β+γ}) = ϕ_{α⃗,β⃗} ∘ q_{α⃗,β⃗,γ}^{1∗}(ϕ_{α⃗+β⃗,γ}).

Here ψ_{β⃗,γ} is the factorization isomorphism for I^{•}, and ϕ_{α⃗,β⃗} is the tensor product of factorization isomorphisms for the sheaves X_{k}.

3.6. Theorem. There exists a unique up to a canonical isomorphism factorizable sheaf over ℙ^{1} obtained by gluing the sheaves {X_{k}}.

Proof is similar to III.10.3. □

Chapter 2. Semiinfinite cohomology

In this chapter we discuss, following essentially [Ark], the ”Semiinfinite homological algebra” in the category C.

4. Semiinfinite functors Ext and T or in C

4.1. Let us call an 𝔲-module 𝔲^{−}-induced (resp., 𝔲^{+}-induced) if it is induced from some 𝔲^{≥0} (resp., 𝔲^{≤0})-module.

4.1.1. Lemma. If M is a 𝔲^{−}-induced, and N is 𝔲^{+}-induced then M ⊗_{B} N is 𝔲-projective.

Proof. An induced module has a filtration whose factors are corresponding Verma modules. For Verma modules the claim is easy. □

4.2. Definition. Let M^{•} = ⊕_{λ∈X} M_{λ}^{•} be a complex (possibly unbounded) in C. We say that M^{•} is concave (resp. convex) if it satisfies the properties (a) and (b) below.

(a) There exists λ_{0} ∈ X such that for any λ ∈ X, if M_{λ}^{•} ≠ 0 then λ ≥ λ_{0} (resp. λ ≤ λ_{0}).

(b) For any μ ∈ X the subcomplex ⊕_{λ≤μ} M_{λ}^{•} (resp. ⊕_{λ≤μ} M_{λ}^{•}) is finite.

We will denote the category of concave (resp., convex) complexes by C^{↑} (resp., C^{↓}).

4.3. Let V ∈ C. We will say that a surjection ϕ : P ⟶ V is good if it satisfies the following properties:

(a) P is 𝔲^{−}-induced;

(b) Let μ ∈ suppP be an extremal point, that is, there is no λ ∈ suppP such that λ > μ. Then μ ∉ supp(ker ϕ).

For any V there exists a good surjection as above. Indeed, denote by p the projection p : M(0) ⟶ L(0), and take for ϕ the map p ⊗ id_{V}.

4.4. Iteratig, we can construct a 𝔲^{−}-induced convex left resolution of B = L(0). Let us pick such a resolution and denote it by P_{↙}^{•}:

… ⟶ P_{↙}^{−1} ⟶ P_{↙}^{0} ⟶ L(0) ⟶ 0

We will denote by

∗ : C ⟶ C^{opp}

the rigidity in C (see e.g. [AJS], 7.3). We denote by P_{↗}^{•} the complex (P_{↙}^{•})^{∗}. It is a 𝔲^{−}-induced concave right resolution of B. The fact that P_{↗}^{•} is 𝔲^{−}-induced follows since 𝔲^{−} is Frobenius (see e.g. [PW]).

4.5. In a similar manner, we can construct a 𝔲^{+}-induced concave left resolution of B. Let us pick such a resolution and denote it P_{↖}^{•}:

… ⟶ P_{↖}^{−1} ⟶ P_{↖}^{0} ⟶ L(0) ⟶ 0

We denote by P_{↘}^{•} the complex (P_{↖}^{•})^{∗}. It is a 𝔲^{+}-induced convex right resolution of B.

4.5.1. For M ∈ C we denote by P_{↙}^{•}(M) (resp., P_{↗}^{•}(M), P_{↖}^{•}(M), P_{↘}^{•}(M)) the resolution P_{↙}^{•} ⊗_{B} M (resp., P_{↗}^{•} ⊗_{B} M, P_{↖}^{•} ⊗_{B} M, P_{↘}^{•} ⊗_{B} M) of M.

4.6. We denote by C_{r} the category of X-graded right 𝔲-modules V = ⊕_{λ∈X}V_{λ} such that

K_{i}|_{V_{λ}} = ζ^{−⟨i,λ⟩}

(note the change of a sign!), the operators E_{i}, F_{i} acting as E_{i} : V_{λ} ⟶ V_{λ+i′}, F_{i} : V_{λ} ⟶ V_{λ−i′}.

4.6.1. Given M ∈ C, we define M^{∨} ∈ C_{r} as follows: (M^{∨})_{λ} = (M_{−λ})^{∗}, E_{i} : (M^{∨})_{λ} ⟶ (M^{∨})_{λ+i′} is the transpose of E_{i} : M_{−λ−i′} ⟶ M_{−λ}, similarly, F_{i} on M^{∨} is the transpose of F_{i} on M.

This way we get an equivalence

∨ : C^{opp} ⟶^{∼} C_{r}.

Similarly, one defines an equivalence ^{∨} : C_{r}^{opp} ⟶ C, and we have an obvious isomorphism ∨ ∘ ^{∨} ≅ Id.

4.6.2. Given M ∈ C, we define sM ∈ C_{r} as follows: (sM)_{λ} = M_{λ}; xg = (sg)x for x ∈ M, g ∈ 𝔲 where

s : 𝔲 ⟶ 𝔲^{opp}

is the antipode defined in [AJS], 7.2. This way we get an equivalence

s : C ⟶^{∼} C_{r}

One defines an equivalence s : C_{r} ⟶^{∼} C in a similar manner. The isomorphism of functors s ∘ s ≅ Id is constructed in loc. cit., 7.3.

Note that the rigidity ∗ is just the composition

∗ = s ∘ ^{∨}.

4.6.3. We define the categories C_{r}^{↑} and C_{r}^{↓} in the same way as in 4.2.

For V ∈ C_{r} we define P_{↙}^{•}(V) as

P_{↙}^{•}(V) = sP_{↙}^{•}(sV);

and P_{↗}^{•}(V), P_{↘}^{•}(V), P_{↖}^{•}(V) in a similar way.

4.7. Definition. (i) Let M, N ∈ C. We define

Ext_{C}^{\frac{∞}{2} +•}(M, N) := H^{•}(Hom_{C}(P_{↘}^{•}(M), P_{↗}^{•}(N))).

(ii) Let V ∈ C_{r}, N ∈ C. We define

Tor_{\frac{∞}{2} +•}^{C}(V, N) := H^{−•}(P_{↙}^{•}(V) ⊗_{C} P_{↘}^{•}(N)). □

Here we understand Hom_{C}(P_{↘}^{•}(M), P_{↗}^{•}(N) and P_{↙}^{•}(V) ⊗_{C} P_{↘}^{•}(N) as simple complexes associated with the corresponding double complexes. Note that due to our boundedness properties of weights of our resolutions, these double complexes are bounded. Therefore all Ext^{\frac{∞}{2} +i} and Tor_{\frac{∞}{2} +i} spaces are finite dimensional, and are non-zero only for finite number of i ∈ ℤ.

4.8. Lemma. For M, N ∈ C there exist canonical nondegenerate pairings

Ext_{C}^{\frac{∞}{2} +n}(M, N) ⊗ Tor_{\frac{∞}{2} +n}^{C}(N^{∨}, M) ⟶ B.

Proof. There is an evident non-degenenerate pairing

Hom_{C}(M, N) ⊗ (N^{∨} ⊗_{C} M) ⟶ B.

It follows that the complexes computing Ext and Tor are also canonically dual. □

4.9. Theorem. (i) Let M, N ∈ C. Let R_{↘}^{•}(M) be a 𝔲^{+}-induced convex right resolution of M, and R_{↗}^{•}(N) — a 𝔲^{−}-induced concave right resolution of N. Then there is a canonical isomorphism

Ext_{C}^{\frac{∞}{2} +•}(M, N) ≅ H^{•}(Hom_{C}(R_{↘}^{•}(M), R_{↗}^{•}(N))).

(ii) Let V ∈ C_{r}, N ∈ C. Let R_{↙}^{•}(V) be a 𝔲^{−}-induced convex left resolution of V, and R_{↘}^{•}(N) — a 𝔲^{+}-induced convex right resolution of N lying in C^{↓}. Then there is a canonical isomorphism

Tor_{\frac{∞}{2} +•}^{C}(V, N) ≅ H^{−•}(R_{↙}^{•}(V) ⊗_{C} R_{↘}^{•}(N)).

Proof will occupy the rest of the section.

4.10. Lemma. Let V ∈ C_{r}; let R_{i}^{•}, i = 1, 2, be two 𝔲^{−}-induced convex left resolutions of V. There exists a third 𝔲^{−}-induced convex left resolution R^{•} of V, together with two termwise surjective maps

R^{•} ⟶ R_{i}^{•}, i = 1, 2,

inducing identity on V.

Proof. We will construct R^{•} inductively, from right to left. Let

R_{i}^{•} : … ⟶^{d_{i}^{−2}} R_{i}^{−1} ⟶^{d_{i}^{−1}} R_{i}^{0} ⟶^{ϵ_{i}} V.

First, define L_{0} := R_{1}^{0} ×_{V} R_{2}^{0}. We denote by δ the canonical map L^{0} ⟶ V, and by q_{i}^{0} the projections L^{0} ⟶ R_{i}^{0}. Choose a good surjection ϕ_{0} : R^{0} ⟶ L^{0} and define p_{i}^{0} := q_{i}^{0} ∘ ϕ_{0} : R^{0} ⟶ R_{i}^{0}; ϵ := δ ∘ ϕ_{0} : R^{0} ⟶ V.

Set K_{i}^{−1} := ker ϵ_{i}; K := ker ϵ. The projections p_{i}^{0} induce surjections p_{i}^{0} : K^{−1} ⟶ K_{i}^{−1}. Let us define

L^{−1} := ker((d_{1}^{−1} − p_{1}^{0}, d_{2}^{−1} − p_{2}^{0}) : R_{1}^{−1} ⊕ K^{−1} ⊕ R_{2}^{−1} ⟶ K_{1}^{−1} ⊕ K_{2}^{−1}).

We have canonical projections q_{i}^{−1} : L^{−1} ⟶ R_{i}^{−1}, δ^{−1} : L^{−1} ⟶ K^{−1}. Choose a good surjection ϕ_{−1} : R^{−1} ⟶ L^{−1} and write d^{−1} : R^{−1} ⟶ R^{0} for δ^{−1} ∘ ϕ_{−1} composed with the inclusion K^{−1} ↪ R^{0}. We define p_{i}^{−1} := q_{i}^{−1} ∘ ϕ_{−1}.

We have just described an induction step, and we can proceed in the same manner. One sees directly that the left 𝔲^{−}-induced resolution R^{•} obtained this way actually lies in C_{r}^{↓}. □

4.11. Let N ∈ C, and let R_{↘}^{•} be a 𝔲^{+}-induced convex right resolution of N. For n ≥ 0 let b_{≥n}(R_{↘}^{•}) denote the stupid truncation:

0 ⟶ R_{↘}^{0} ⟶ … ⟶ R_{↘}^{n} ⟶ 0 ⟶ …

For m ≥ n we have evident truncation maps b_{≥m}(R_{↘}^{•}) ⟶ b_{≥n}(R_{↘}^{•}).

4.11.1. Lemma. Let R_{↙}^{•} be a 𝔲^{−}-induced left resolution of a module V ∈ C_{r}. We have

H^{•}(R_{↙}^{•} ⊗_{C} R_{↘}^{•}) = lim_{←}_{n} H^{•}(R_{↙}^{•} ⊗_{C} b_{≤n}R_{↘}^{•}).

For every i ∈ ℤ the inverse system

{H^{i}(R_{↙}^{•} ⊗_{C} b_{≤n}R_{↘}^{•})}

stabilizes.

Proof. All spaces H^{i}(R_{↙}^{•} ⊗_{C} b_{≤n}R_{↘}^{•}) and only finitely many weight components of R_{↙}^{•} and R_{↘}^{•} contribute to H^{i}. □

4.12. Proof of Theorem 4.9. Let us consider case (ii), and prove that H^{•}(R_{↙}^{•}(V) ⊗_{C} R_{↘}^{•}(N)) does not depend, up to a canonical isomorphism, on the choice of a resolution R_{↙}^{•}(V). The other independences are proved exactly in the same way.

Let R_{i}^{•}, i = 1, 2, be two left 𝔲^{−}-induced left convex resolutions of V. According to Lemma 4.10, there exists a third one, R^{•}, projecting onto R_{i}^{•}. Let us prove that the projections induce isomorphisms

H^{•}(R^{•} ⊗_{C} R_{↘}^{•}(N)) ⟶^{∼} H^{•}(R_{i}^{•} ⊗_{C} R_{↘}^{•}(N)).

By Lemma 4.11.1, it suffices to prove that

H^{•}(R^{•} ⊗_{C} b_{≤n}R_{↘}^{•}(N)) ⟶^{∼} H^{•}(R_{i}^{•} ⊗_{C} b_{≤n}R_{↘}^{•}(N)).

for all n. Let Q_{i}^{•} be a cone of R^{•} ⟶ R_{i}^{•}. It is an exact 𝔲^{−}-induced convex complex bounded from the right. It is enough to check that H^{•}(Q_{i}^{•} ⊗_{C} b_{≤n}R_{↘}^{•}(N)) = 0.

Note that for W ∈ C_{r}, M ∈ C we have canonically

W ⊗_{C} M = (W ⊗ sM) ⊗_{C} B.

Thus

H^{•}(Q_{i}^{•} ⊗_{C} b_{≤n}R_{↘}^{•}(N)) = H^{•}((Q_{i}^{•} ⊗ sb_{≤n}R_{↘}^{•}(N)) ⊗_{C} B) = 0,

since (Q_{i}^{•}⊗sb_{≤n}R_{↘}^{•}(N) is an exact bounded from the right complex, consisting of modules which are tensor products of 𝔲^{+}-induced and 𝔲^{−}-induced, hence 𝔲-projective modules (see Lemma 4.1.1).

4.12.1. It remains to show that if p′ and p′′ are two maps between 𝔲^{−}-induced convex resolutions of V, R_{1}^{•} ⟶ R_{2}^{•}, inducing identity on V, then the isomorphisms

H^{•}(R_{1}^{•} ⊗_{C} R_{↘}^{•}(N)) ⟶^{∼} H^{•}(R_{2}^{•} ⊗_{C} R_{↘}^{•}(N))

induced by p′ and p′′, coincide. Arguing as above, we see that it is enough to prove this with R_{↘}^{•}(N) replaced by b_{≤n}R_{↘}^{•}(N). This in turn is equivalent to showing that two isomorphisms

H^{•}((R_{1}^{•} ⊗ sb_{≤n}R_{↘}^{•}(N)) ⊗_{C} B) ⟶^{∼} H^{•}((R_{2}^{•} ⊗ sb_{≤n}R_{↘}^{•}(N)) ⊗_{C} B)

coincide. But R_{i}^{•}⊗sb_{≤n}R_{↘}^{•}(N) are complexes of projective 𝔲-modules, and the morphisms p′⊗id and p′′⊗id induce the same map on cohomology, hence they are homotopic; therefore they induce homotopic maps after tensor multiplication by B.

This completes the proof of the theorem. □

5. Some calculations

We will give a recipe for calculation of Tor_{\frac{∞}{2} +•}^{C} which will prove useful for the next chapter.

5.1. Recall that in III.13.2 the duality functor

D : C_{ζ^{−1}} ⟶ C^{opp}

has been defined (we identify C with C̃ as usually). We will denote objects of C_{ζ^{−1}} by letters with the subscript (•)_{ζ^{−1}}.

Note that DL(0)_{ζ^{−1}} = L(0).

Let us describe duals to Verma modules. For λ ∈ X let us denote by M^{+}(λ) the Verma module with respect to the subalgebra 𝔲^{+} with the lowest weight λ, that is

M^{+}(λ) := Ind_{𝔲^{≤0}}^{𝔲}χ_{λ}

where χ_{λ} is an evident one-dimensional representaion of 𝔲^{≤0} corresponding to the character λ.

5.1.1. Lemma. We have

DM(λ)_{ζ^{−1}} = M^{+}(λ − 2(l − 1)ρ).

Proof follows from [AJS], Lemma 4.10. □

5.2. Let us denote by K^{•} a two term complex in C

L(0) ⟶ DM(0)_{ζ^{−1}}

concentrated in degrees 0 and 1, the morphism being dual to the canonical projection M(0)_{ζ^{−1}} ⟶ L(0)_{ζ^{−1}}. For n ≥ 1 define a complex

K_{n}^{•} := b_{≥0}(K^{•⊗n}[1]);

it is concentrated in degrees from 0 to n − 1. For example, K_{1}^{•} = DM(0)_{ζ^{−1}}. For n ≥ 1 we will denote by

ξ : K_{n}^{•} ⟶ K_{n+1}^{•}

the map induced by the embedding L(0) ↪ DM(0)_{ζ^{−1}}. We will need the following evident properties of the system {K_{n}^{•}, ξ_{n}}:

(a) K_{n}^{•} is 𝔲^{+}-induced;

(b) K_{n}^{•} is exact off degrees 0 and n − 1; H^{0}(K_{n}^{•}) = B. ξ_{n} induces identity map between H^{0}(K_{n}^{•}) and H^{0}(K_{n+1}^{•}).

(c) For a fixed μ ∈ X there exists m ∈ ℕ such that for any n we have (b_{≥m}K_{n}^{•})_{≥μ} = 0. Here for V = ⊕_{λ∈X} ∈ C we set

V_{≥μ} := ⊕_{λ≥μ} V_{λ}.

5.3. Let V ∈ C_{r}; let R_{↙}^{•}(V) be a 𝔲^{−}-induced convex left resolution of V. Let N ∈ C.

5.3.1. Lemma. (i) For a fixed k ∈ ℤ the direct system {H^{k}(R_{↙}^{•}(V) ⊗_{C} (K_{n}^{•} ⊗ N)), ξ_{n}} stabilizes. (ii) We have a canonical isomorphism

Tor_{\frac{∞}{2} +•}^{C}(V, N) ≅ lim_{→}_{n} H^{−•}(R_{↙}^{•}(V) ⊗_{C} (K_{n}^{•} ⊗ N)).

Proof. (i) is similar to Lemma 4.11.1. (ii) By Theorem 4.9 we can use any 𝔲^{+}-induced right convex resolution of N to compute Tor_{\frac{∞}{2} +•}^{C}(V, N). Now extend K_{n}^{•} ⊗ N to a 𝔲^{+}- induced convex resolution of N and argue like in the proof of Lemma 4.11.1 again. □

5.4. Recall the notations of 4.4 and take for R_{↙}^{•}(V) the resolution P_{↙}^{•}(V) = P_{↙}^{•} ⊗ V. Then

H^{•}(P_{↙}^{•}(V) ⊗_{C} (K_{n}^{•} ⊗ N)) = H^{•}(V ⊗_{C} (P_{↙}^{•} ⊗ K_{n}^{•} ⊗ N)).

Note that P_{↙}^{•} ⊗ K_{n}^{•} ⊗ N is a right bounded complex quasi-isomorphic to K_{n}^{•} ⊗ N. The terms of P_{↙}^{•} ⊗ K_{n}^{•} are 𝔲-projective by Lemma 4.1.1, hence the terms of P_{↙}^{•} ⊗ K_{n}^{•} ⊗ N are projective by rigidity of C. Therefore,

H^{−•}(V ⊗_{C} (P_{↙}^{•} ⊗ K_{n}^{•} ⊗ N)) = Tor_{•}^{C}(V, K_{n}^{•} ⊗ N).

Here Tor_{•}^{C}(∗, ∗) stands for the zeroth weight component of Tor_{•}^{𝔲}(∗, ∗).

Putting all the above together, we get

5.5. Corollary. For a fixed k ∈ ℤ the direct system {Tor_{k}^{C}(V, K_{n}^{•} ⊗ N)} stabilizes.

We have

Tor_{\frac{∞}{2} +•}^{C}(V, N) = lim_{→}_{n} Tor_{•}^{C}(V, K_{n}^{•} ⊗ N). □

5.6. Dually, consider complexes DK_{n,ζ^{−1}}^{•}. They form a projective system

{… ⟶ DK_{n+1,ζ^{−1}}^{•} ⟶ DK_{n,ζ^{−1}}^{•} ⟶ … }

These complexes enjoy properties dual to (a) — (c) above.

5.7. Theorem. For every k ∈ ℤ we have canonical isomorphisms

Tor_{\frac{∞}{2} +k}^{C}(V, N) ≅ lim_{←}_{m}lim_{→}_{n} H^{−k}((V ⊗ sDK_{m,ζ^{−1}}^{•}) ⊗_{C} (K_{n}^{•} ⊗ N)).

Both the inverse and the direct systems actually stabilize.

Proof follows from Lemma 5.3.1. We leave details to the reader. □

Here is an example of calculation of Tor_{\frac{∞}{2} +•}^{C}.

5.8. Lemma. Tor_{\frac{∞}{2} +•}^{C}(B, L(2(l − 1)ρ)) = B in degree 0.

Proof. According to Lemma 4.8 it suffices to prove that Ext_{C}^{\frac{∞}{2} +•}(L(2(l−1)ρ), L(0)) = B. Choose a 𝔲^{+}-induced right convex resolution

L(2(l − 1)ρ) ⟶^{ϵ} R_{↘}^{•}

such that

R_{↘}^{0} = DM(2(l − 1)ρ)_{ζ^{−1}} = M^{+}(0),

and all the weights in R_{↘}^{≥1} are < 2(l − 1)ρ.

Similarly, choose a 𝔲^{−}-free right concave resolution

L(0) ⟶^{ϵ} R_{↗}^{•}

such that

R_{↗}^{0} = M(2(l − 1)ρ) = DM^{+}(0)_{ζ^{−1}},

and all the weights of R_{↗}^{≥1} are > 0. By Theorem 4.9 we have

Ext_{C}^{\frac{∞}{2} +•}(L(2(l − 1)ρ), L(0)) = H^{•}(Hom_{C}^{•}(R_{↘}^{•}, R_{↗}^{•})).

Therefore it is enough to prove that

(a) Hom_{C}(R_{↘}^{0}, R_{↗}^{0}) = B;

(b) Hom_{C}(R_{↘}^{m}, R_{↗}^{n}) = 0 for (m, n) ≠ (0, 0).

(a) is evident. Let us prove (b) for, say, n > 0. R_{↘}^{m} has a filtration with successive quotients of type M^{+}(λ), λ ≤ 0; similarly, R_{↘}^{n} has a filtration with successive quotients of type DM^{+}(μ)_{ζ^{−1}}, μ > 0. We have Hom_{C}(M^{+}(λ), DM^{+}(μ)_{ζ^{−1}}) = 0, therefore Hom_{C}(R_{↘}^{m}, R_{↗}^{n}) = 0. The proof for m > 0 is similar. Lemma is proven. □

CONFORMAL BLOCKS AND T or_{\frac{∞}{2} +•}^{C}

5.9. Let M ∈ C. We have a canonical embedding

Hom_{C}(B, M) ↪ M

which identifies Hom_{C}(B, M) with the maximal trivial subobject of M. Dually, we have a canonical epimorhism

M ⟶ Hom_{C}(M, B)^{∗}

which identifies Hom_{C}(M, B)^{∗} with the maximal trivial quotient of M. Let us denote by ⟨M⟩ the image of the composition

Hom_{C}(B, M) ⟶ M ⟶ Hom_{C}(M, B)^{∗}

Thus, ⟨M⟩ is canonically a subquotient of M.

One sees easily that if N ⊂ M is a trivial direct summand of M which is maximal, i.e. not contained in greater direct summand, then we have a canonical isomorphism ⟨M⟩ ⟶^{∼} N. By this reason, we will call ⟨M⟩ the maximal trivial direct summand of M.

5.10. Let

Δ = {λ ∈ X| ⟨i, λ + ρ⟩ > 0, for all i ∈ I; ⟨γ, λ + ρ⟩ < l}

denote the first alcove. Here γ ∈ R ⊂ Y is the highest coroot.

For λ_{1}, …, λ_{n} ∈ Δ the space of conformal blocks is defined as

⟨L(λ_{1}), …, L(λ_{n})⟩ := ⟨L(λ_{1}) ⊗ … ⊗ L(λ_{n})⟩

(see e.g. [A] and Lemma 9.3 below).

5.11. Corollary. The space of conformal blocks ⟨L(λ_{1}), …, L(λ_{n})⟩ is canonically a subquotient of Tor_{\frac{∞}{2} +0}^{C}(B, L(λ_{1}) ⊗ … ⊗ L(λ_{n}) ⊗ L(2(l − 1)ρ)).

Proof follows easily from the definition of ⟨•⟩ and Lemma 5.8. □

5.12. Let us consider an example showing that ⟨L(λ_{1}), …, L(λ_{n})⟩ is in general a proper subquotient of Tor_{\frac{∞}{2} +0}^{C}(B, L(λ_{1}) ⊗ … ⊗ L(λ_{n}) ⊗ L(2(l − 1)ρ)).

We leave the following to the reader.

5.12.1. Exercise. Let P (0) be the indecomposable projective cover of L(0). We have Tor_{\frac{∞}{2} +0}^{C}(B, P (0)) = B. □

We will construct an example featuring P (0) as a direct summand of L(λ_{1}) ⊗ … ⊗ L(λ_{n}).

Let us take a root datum of type sl(2); take l = 5, n = 4, λ_{1} = λ_{2} = 2, λ_{3} = λ_{4} = 3 (we have identified X with ℤ).

In our case ρ = 1, so 2(l − 1)ρ = 8. Note that P (0) has highest weight 8, and it is a unique indecomposable projective with the highest weight 8. So, if we are able to find a projective summand of highest weight 0 in V = L(2) ⊗ L(2) ⊗ L(3) ⊗ L(3) then V ⊗ L(8) will contain a projective summand of highest weight 8, i.e. P (0).

Let U_{B} denote the quantum group with divided powers over B (see [L2], 8.1). The algebra 𝔲 lies inside U_{B}. It is wellknown that all irreducibles L(λ), λ ∈ Δ, lift to simple U_{B}-modules { \atop L(λ)} and for λ_{1}, …, λ_{n} ∈ Δ the U_{B}-module { \atop L(λ_{1})} ⊗ … ⊗ { \atop L(λ_{n})} is a direct sum of irreducibles { \atop L(λ)}, λ ∈ Δ, and indecomposable projectives { \atop P (λ)}, λ ≥ 0 (see, e.g. [A]).

Thus L̃(2) ⊗ L̃(2) ⊗ L̃(3) ⊗ L̃(3) contains an indecomposable projective summand with the highest weight 10, i.e. { \atop P (8)}. One can check easily that when restricted to 𝔲, { \atop P (8)} remains projective and contains a summand P (−2). But the highest weight of P (−2) is zero.

We conclude that L(2) ⊗ L(2) ⊗ L(3) ⊗ L(3) ⊗ L(8) contains a projective summand P (0), whence

⟨L(2), L(2), L(3), L(3)⟩ ≠ Tor_{\frac{∞}{2} +0}^{C}(B, L(2) ⊗ L(2) ⊗ L(3) ⊗ L(3) ⊗ L(8)).

Chapter 3. Global sections

6. Braiding and balance in C and FS

6.1. Let U_{B} be the quantum group with divided powers, cf [L2], 8.1. Let _{R}C be the category of finite dimensional integrable U_{B}-modules defined in [KL]IV, §37. It is a rigid braided tensor category. The braiding, i.e. family of isomorphisms

R̃_{V,W} : V ⊗ W ⟶^{∼} W ⊗ V, V, W ∈ _{R}C,

satisfying the usual constraints, has been defined in [L1], Ch. 32.

6.2. As 𝔲 is a subalgebra of U_{B}, we have the restriction functor preserving X-grading

Υ : _{R}C ⟶ C.

The following theorem is due to G.Lusztig (private communication).

6.2.1. Theorem. (a) There is a unique braided structure (R_{V,W} , θ_{V}) on C such that the restriction functor Υ commutes with braiding.

(b) Let V = L(λ), and let μ be the highest weight of W ∈ C, i.e. W_{μ} ≠ 0 and W_{ν} ≠ 0 implies ν ≤ μ. Let x ∈ V, y ∈ W_{μ}. Then

R_{V,W} (x ⊗ y) = ζ^{λ·μ}y ⊗ x;

(c) Any braided structure on C enjoying the property (b) above coincides with that defined in (a). □

6.3. Recall that an automorphism θ̃ = {θ̃_{V} : V ⟶^{∼} V } of the identity functor of _{R}C is called balance if for any V, W ∈ _{R}C we have

R̃_{W,V} ∘ R̃_{V,W} = θ̃_{V ⊗W} ∘ (θ̃_{V} ⊗ θ̃_{W})^{−1}.

The following proposition is an easy application of the results of [L1], Chapter 32.

6.4. Proposition. The category _{R}C admits a unique balance θ̃ such that

— if L̃(λ) is an irreducible in _{R}C with the highest weight λ, then θ̃ acts on L̃(λ) as multiplication by ζ^{n(λ)}. □

Here n(λ) denotes the function introduced in 3.1.

Similarly to 6.2.1, one can prove 6.5. Theorem. (a) There is a unique balance θ on C such that Υ commutes with balance;

(b) θ_{L(λ)} = ζ^{n(λ)};

(c) If θ′ is a balance in C having property (b), then θ′ = θ. □

6.6. According to Deligne’s ideology, [D1], the gluing construction of 3.6 provides the category FS with the balance θ^{FS}. Recall that the braiding R^{FS} has been defined in III.11.11 (see also 11.4). It follows easily from the definitions that (Φ(R^{FS}), Φ(θ^{FS})) satisfy the properties (b)(i) and (ii) above. Therefore, we have (Φ(R^{FS}), Φ(θ^{FS})) = (R, θ), i.e. Φ is an equivalence of braided balanced categories.

7. Global sections over A(K)

7.1. Let K be a finite non-empty set, |K| = n, and let {X_{k}} be a K-tuple of finite gactorizable sheaves. Let λ_{k} := λ(X_{k}) and λ = ∑_{k} λ_{k}; let α ∈ ℕ[I]. Consider the sheaf X^{α}(K) over Aα ; λ⃗(K) obtained by gluing {X_{k}}, cf. III.10.3. Thus

X^{α}(K) = g_{K}({X_{k}})α ; λ⃗

in the notations of loc.cit.

We will denote by η, or sometimes by η^{α}, or η_{K}^{α} the projection A^{α}(K) ⟶ O(K). We are going to describe Rη_{∗}X^{α}(K)[−n]. Note that it is an element of D(O(K)) which is smooth, i.e. its cohomology sheaves are local systems.

7.2. Let V_{1}, …, V_{n} ∈ C. Recall (see II.3) that C_{𝔲^{−}}^{•}(V_{1} ⊗ … ⊗ V_{n}) denotes the Hochschild complex of the 𝔲^{−}-module V_{1} ⊗ … ⊗ V_{n}. It is naturally X-graded, and its λ-component is denoted by the subscript (•)_{λ} as usually.

Let us consider a homotopy point z = (z_{1}, …, z_{n}) ∈ O(K) where all z_{i} are real, z_{1} < … < z_{n}. Choose a bijection K ⟶^{∼} [n]. We want to describe a stalk Rη_{∗}X^{α}(K)z[−n]. The following theorem generalizes Theorem II.8.23. The proof is similar to loc. cit, cf. III.12.16, and will appear later.

7.3. Theorem. There is a canonical isomorphism, natural in X_{i},

Rη_{∗}X^{α}(K)z[−n] ≅ C_{𝔲^{−}}^{•}(Φ(X_{1}) ⊗ … ⊗ Φ(X_{n}))_{λ−α}. □

7.4. The group π_{1}(O(K); z) is generated by counterclockwise loops of z_{k+1} around z_{k}, σ_{i}, k = 1, …, n − 1. Let σ_{k} act on Φ(X_{1}) ⊗ … ⊗ Φ(X_{n}) as

id ⊗ … ⊗ R_{Φ(X_{k+1}),Φ(X_{k})} ∘ R_{Φ(X_{k}),Φ(X_{k+1})} ⊗ … ⊗ id.

This defines an action of π_{1}(O(K); z) on Φ(X_{1}) ⊗ … ⊗ Φ(X_{n}), whence we get an action of this group on C_{𝔲^{−}}^{•}(Φ(X_{1}) ⊗ … ⊗ Φ(X_{n})) respecting the X-grading. Therefore we get a complex of local systems over O(K); let us denote it C_{𝔲^{−}}^{•}(Φ(X_{1}) ⊗ … ⊗ Φ(X_{n}))^{♡}.

7.5. Theorem. There is a canonical isomorphism in D(O(K))

Rη_{∗}X^{α}(K)[−n] ⟶^{∼} C_{𝔲^{−}}^{•}(Φ(X_{1}) ⊗ … ⊗ Φ(X_{n}))_{λ−α}^{♡}.

Proof follows from 6.6 and Theorem 7.3. □

7.6. Corollary. Set λ_{∞} := α + 2(l − 1)ρ − λ. There is a canonical isomorphism in D(O(K))

Rη_{∗}X^{α}(K)[−n] ⟶^{∼} C_{𝔲}^{•}(Φ(X_{1}) ⊗ … ⊗ Φ(X_{n}) ⊗ DM(λ_{∞})_{ζ^{−1}})_{0}^{♡}.

Proof. By Shapiro’s lemma, we have a canonical morphism of complexes which is a quasiisomorphism

C_{𝔲^{−}}^{•}(Φ(X_{1}) ⊗ … ⊗ Φ(X_{n}))_{λ−α} ⟶ C_{𝔲}^{•}(Φ(X_{1}) ⊗ … ⊗ Φ(X_{n}) ⊗ M^{+}(α − λ))_{0}.

By Lemma 5.1.1, M^{+}(α − λ) = DM(λ_{∞})_{ζ^{−1}}. □

8. Global sections over P

8.1. Let J be a finite set, |J| = m, and {X_{j}} a J-tuple of finite factorizable sheaves. Set μ_{j} := λ(X_{j}), μ⃗ = (μ_{j}) ∈ X^{J}. Let α ∈ ℕ[I] be such that (μ⃗, α) is admissible, cf. 3.1. Let X α ; μ⃗ be the preverse sheaf on Pα ; μ⃗ obtained by gluing the sheaves X_{j}, cf. 3.5 and 3.6.

Note that the group PGL_{2}(ℂ) = Aut(ℙ^{1}) operates naturally on Pα ; μ⃗ and the sheaf X α ; μ⃗ is equivariant with respect to this action.

Let

η̄: Pα ; μ⃗ ⟶ T P^{oJ}

denote the natural projection; we will denote this map also by η̄_{J} or η̄_{J}^{α}. Note that η̄ commutes with the natural action of PGL_{2}(ℂ) on these spaces. Therefore Rη̄_{∗}X α ; μ⃗ is a smooth PGL_{2}(ℂ)-equivariant complex on T P^{oJ}. Our aim in this section will be to compute this complex algebraically.

Note that Rη̄_{∗}X α ; μ⃗ descends uniquely to the quotient

\underline{T P}^{oJ} := T P^{oJ}/PGL_{2}(ℂ)

8.2. Let us pick a bijection J ⟶^{∼} [m]. Let \underline{Z} be a contractible real submanifold of \underline{T P}^{oJ} defined in [KL]II, 13.1 (under the name \underline{V}_{0}). Its points are configurations (z_{1}, τ_{1}, …, z_{m}, τ_{m}) such that z_{j} ∈ ℙ^{1}(ℝ) = S ⊂ ℙ^{1}(ℂ); the points z_{j} lie on S in this cyclic order; they orient S in the same way as (0, 1, ∞) does; the tangent vectors τ_{j} are real and compatible with this orientation.

8.3. Definition. An m-tuple of weights μ⃗ ∈ X^{m} is called positive if

∑_{j=1}^{m} μ_{j} + (1 − l)2ρ ∈ ℕ[I] ⊂ X.

If this is so, we will denote

α(μ⃗) := ∑_{j=1}^{m} μ_{j} + (1 − l)2ρ □

8.4. Theorem. Let X_{1}, …, X_{m} ∈ FS. Let μ⃗ be a positive m-tuple of weights, μ_{j} ≥ λ(X_{j}), and let α = α(μ⃗). Let X α ; μ⃗ be the sheaf on Pα ; μ⃗ obtained by gluing the sheaves X_{j}. There is a canonical isomorphism

R^{•}η̄_{∗}X α ; μ⃗[−2m]_{Z} ≅ Tor_{\frac{∞}{2} −•}^{C}(B, Φ(X_{1}) ⊗ … ⊗ Φ(X_{m})).

Proof is sketched in the next few subsections.

8.5. Two-sided Čech resolutions. The idea of the construction below is inspired by [B1], p. 40.

Let P be a topological space, U = {U_{i}| i = 1, …, N} an open covering of P. Let j_{i_{0}i_{1}...i_{a}} denote the embedding

U_{i_{0}} ∩ … ∩ U_{i_{a}} ↪ P.

Given a sheaf F on P, we have a canonical morphism

F ⟶ Č^{•}(U; F)  (285)

where

Č^{a}(U; F) = ⊕_{i_{0}<i_{1}<...<i_{a}} j_{i_{0}i_{1}...i_{a}∗}j_{i_{0}i_{1}...i_{a}}^{∗}F,

the differential being the usual Čech one.

Dually, we define a morphism

Č_{•}(U; F) ⟶ F  (286)

where

Č_{•}(U; F) : 0 ⟶ Č_{N}(U; F) ⟶ Č_{N−1}(U; F) ⟶ … ⟶ Č_{0}(U; F) ⟶ 0

where

Č_{a}(U; F) = ⊕_{i_{0}<i_{1}<...<i_{a}} j_{i_{0}i_{1}...i_{a}!}j_{i_{0}i_{1}...i_{a}}^{∗}F.

If F is injective then the arrows (285) and (286) are quasiisomorphisms.

Suppose we have a second open covering of P, V = {V_{j}| j = 1, …, N}. Let us define sheaves

Č_{b}^{a}(U, V; F) := Č_{b}(V; Č^{a}(U; F));

they form a bicomplex. Let us consider the associated simple complex Č^{•}(U, V; F), i.e.

Č^{i}(U, V; F) = ⊕_{a−b=i} Č_{b}^{a}(U, V; F).

It is a complex concentrated in degrees from −N to N. We have canonical morphisms

F ⟶ Č^{•}(U; F) ⟵ Č^{•}(U, V; F)

If F is injective then both arrows are quasiisomorphisms, and the above functors are exact on injective sheaves. Therefore, they pass to derived categories, and we get a functor K ↦ Č^{•}(U, V; K) from the bounded derived category D^{b}(P) to the bounded filtered derived category DF (P). This implies

8.6. Lemma. Suppose that K ∈ D^{b}(P) is such that R^{i}Γ(P; Č_{b}^{a}(U, V; K)) = 0 for all a, b and all i ≠ 0. Then we have a canonical isomorphism in D^{b}(P),

RΓ(P; K) ⟶^{∼} R^{0}Γ(P; Č^{•}(U, V; K)). □

8.7. Returning to the assumptions of theorem 8.4, let us pick a point z = (z_{1}, τ_{1}, …, z_{m}, τ_{m}) ∈ T P^{oJ} such that z_{j} are real numbers z_{1} < … < z_{m} and tangent vectors are directed to the right.

By definition, we have canonically

Rη̄_{∗}(Pα ; μ⃗; X α ; μ⃗)\underline{Z} = RΓ(P^{α}; K)

where

K := X α ; μ⃗|_{η̄^{−1}(z)}[−2m].

Let us pick N ≥ |α| and reals p_{1}, …, p_{N}, q_{1}, …, q_{N} such that

p_{1} < … < p_{N} < z_{1} < … < z_{m} < q_{N} < … < q_{1}.

Let us define two open coverings U = {U_{i}| i = 1, …, N} and V = {V_{i}| i = 1, …, N} of the space P^{α} where

U_{i} = P^{α} − ⋃_{k} {t_{k} = p_{i}}; V_{i} = P^{α} − ⋃_{k} {t_{k} = q_{i}},

where t_{k} denote the standard coordinates.

8.8. Lemma. (i) We have

R^{i}Γ(P^{α}; Č_{b}^{a}(U, V; K)) = 0

for all a, b and all i ≠ 0.

(ii) We have canonical isomorphism

R^{0}Γ(P^{α}; Č^{•}(U, V; K)) ≅ sDK_{N,ζ^{−1}}^{•} ⊗_{C} (K_{N}^{•} ⊗ Φ(X_{1}) ⊗ … ⊗ Φ(X_{m})),

in the notations of 5.7.

Proof (sketch). We should regard the computation of RΓ(P^{α}; Č_{b}^{a}(U, V; K)) as the computation of global sections over P^{α} of a sheaf obtained by gluing X_{j} into points z_{j}, the Verma sheaves M(0) or irreducibles L(0) into the points p_{j}, and dual sheaves DM(0)_{ζ^{−1}} or DL(0)_{ζ^{−1}} into the points q_{j}.

Using PGL_{2}(ℝ)-invariance, we can move one of the points p_{j} to infinity. Then, the desired global sections are reduced to global sections over an affine space A^{α}, which are calculated by means of Theorem 7.3.

Note that in our situation all the sheaves Č_{b}^{a}(U, V; K) actually belong to the abelian category M(P^{α}) of perverse sheaves. So Č^{•}(U, V; K) is a resolution of K in M(P^{α}). □

8.9. The conclusion of 8.4 follow from the previous lemma and Theorem 5.7. □

8.10. The group π_{1}(\underline{T P}^{om}, Z) operates on the spaces Tor_{\frac{∞}{2} +•}^{C}(B, Φ(X_{1}) ⊗ … ⊗ Φ(X_{m})) via its action on the object Φ(X_{1}) ⊗ … ⊗ Φ(X_{m}) induced by the braiding and balance in C. Let us denote by

Tor_{\frac{∞}{2} +•}^{C}(B, Φ(X_{1}) ⊗ … ⊗ Φ(X_{m}))^{♡}

the corresponding local system on \underline{T P}^{om}.

8.11. Theorem. There is a canonical isomorphism of local systems on \underline{T P}^{om}:

R^{•−2m}η̄_{∗}X α ; μ⃗ ≅ Tor_{\frac{∞}{2} −•}^{C}(B, Φ(X_{1}) ⊗ … ⊗ Φ(X_{m}))^{♡}

Proof follows immediately from 6.6 and Theorem 8.4. □

9. Application to conformal blocks

9.1. In applications to conformal blocks we will encounter the roots of unity ζ of not necessarily odd degree l. So we have to generalize all the above considerations to the case of arbitrary l.

The definitions of the categories C and FS do not change (for the category C the reader may consult [AP], §3). The construction of the functor Φ : FS ⟶ C and the proof that Φ is an equivalence repeats the one in III word for word.

Here we list the only minor changes (say, in the definition of the Steinberg module) following [L1] and [AP].

9.1.1. So suppose ζ is a primitive root of unity of an even degree l.

We define ℓ := \frac{l}{2}. For the sake of unification of notations, in case l is odd we define ℓ := l. For i ∈ I we define ℓ_{i} := \frac{ℓ}{(ℓ,d_{i})} where (ℓ, d_{i}) stands for the greatest common divisor of ℓ and d_{i}.

For a coroot α ∈ R ∈ Y we can find an element w of the Weyl group W and a simple coroot i ∈ Y such that w(i) = α (notations of [L1], 2.3). We define ℓ_{α} := \frac{ℓ}{(ℓ,d_{i})}, and the result does not depend on a choice of i and w. We define γ_{0} ∈ R to be the highest coroot, and β_{0} ∈ R to be the coroot dual to the highest root. Note that γ_{0} = β_{0} iff our root datum is simply laced.

9.1.2. We define

Y_{ℓ}^{∗} := {λ ∈ X|λ · μ ∈ ℓℤ for any μ ∈ X}

One should replace the congruence modulo lY in the Definition 2.6.1 and in 3.1 by the congruence modulo Y_{ℓ}^{∗}. We define ρ_{ℓ} ∈ X as the unique element such that ⟨i, ρ_{ℓ}⟩ = ℓ_{i} − 1 for any i ∈ I. Then the Steinberg module L(ρ_{ℓ}) is irreducible projective in C (see [AP] 3.14). Note also that ρ_{ℓ} is the highest weight of 𝔲^{+}. One has to replace all the occurences of (l − 1)2ρ in the above sections by 2ρ_{ℓ}. In particular, the new formulations of the Definition 8.3 and the Theorem 8.4 force us to make the following changes in 3.1 and 3.4. In 3.1 we choose a balance function n in the form

n(μ) = \frac{1}{2}μ · μ − μ · ρ_{ℓ}

In other words, we set ν_{0} = −ρ_{ℓ}. This balance function does not necessarily have the property that n(−i′) ≡ 0 mod l. It is only true that n(−i′) ≡ 0 mod ℓ. We say that a pair (μ⃗, α) is admissible if ∑_{k} μ_{k} − α ≡ 2ρ_{ℓ} mod Y_{ℓ}^{∗}.

9.1.3. The last change concerns the definition of the first alcove in 5.10. The corrected definition reads as follows: if ℓ_{i} = ℓ for any i ∈ I, then

Δ_{l} = {λ ∈ X| ⟨i, λ + ρ⟩ > 0, for all i ∈ I; ⟨γ_{0}, λ + ρ⟩ < ℓ};

if not, then

Δ_{l} = {λ ∈ X| ⟨i, λ + ρ⟩ > 0, for all i ∈ I; ⟨β_{0}, λ + ρ⟩ < ℓ_{β_{0}}}

9.2. Let 𝔤̂ denote the affine Lie algebra associated with 𝔤:

0 ⟶ ℂ ⟶ 𝔤̂ ⟶ 𝔤((ϵ)) ⟶ 0.

Let Õ_{κ} be the category of integrable 𝔤̂-modules with the central charge κ − ȟ where ȟ stands for the dual Coxeter number of 𝔤. It is a semisimple balanced braided rigid tensor category (see e.g. [MS] or [F2]).

Let O_{−κ} be the category of 𝔤-integrable 𝔤̂-modules of finite length with the central charge −κ − ȟ. It is a balanced braided rigid tensor (bbrt) category (see [KL]). Let Õ_{−κ} be the semisimple subcategory of O_{−κ} formed by direct sums of simple 𝔤̂-modules with highest weights in the alcove ∇_{κ}:

∇_{κ} := {λ ∈ X| ⟨i, λ + ρ⟩ > 0, for all i ∈ I; ⟨β_{0}, λ + ρ⟩ < κ}

The bbrt structure on O_{−κ} induces the one on Õ_{−κ}, and one can construct an equivalence

Õ_{κ} ⟶^{∼} Õ_{−κ}

respecting bbrt structure (see [F2]). D.Kazhdan and G.Lusztig have constructed an equivalence

O_{−κ} ⟶^{∼} _{R}C_{ζ}

(notations of 6.1) respecting bbrt structure (see [KL] and [L3]). Here ζ = exp(\frac{π\sqrt{−1}}{dκ}) where d = max_{i∈I} d_{i}. Thus l = 2dκ, and ℓ = dκ.

Note that the alcoves ∇_{κ} and Δ_{l} (see 9.1.3) coincide.

The Kazhdan-Lusztig equivalence induces an equivalence

Õ_{−κ} ⟶^{∼} Õ_{ζ}

where Õ_{ζ} is the semisimple subcategory of _{R}C_{ζ} formed by direct sums of simple U_{B}- modules L̃(λ) with λ ∈ Δ (see [A] and [AP]). The bbrt structure on _{R}C_{ζ} induces the one on Õ_{ζ}, and the last equivalence respects bbrt structure. We denote the composition of the above equivalences by

ϕ : Õ_{κ} ⟶^{∼} Õ_{ζ}.

Given any bbrt category B and objects L_{1}, …, L_{m} ∈ B we obtain a local system Hom_{B}(1, L_{1} ⊗ … ⊗ L_{m})^{♡} on T P^{om} with monodromies induced by the action of braiding and balance on L_{1} ⊗ … ⊗ L_{m}.

Here and below we write a superscript X^{♡} to denote a local system over T P^{om} with the fiber at a standard real point z_{1} < … < z_{m} with tangent vectors looking to the right, equal to X.

Thus, given L_{1}, …, L_{m} ∈ Õ_{κ}, the local system

HomÕ_{κ}(1, L_{1} ˜⊗ … ˜⊗L_{m})^{♡}

called local system of conformal blocks is isomorphic to the local system HomÕ_{ζ}(1, ϕ(L_{1}) ˜⊗ … ˜⊗ϕ(L_{m}))^{♡}. Here ˜⊗ will denote the tensor product in ”tilded” categories.

To unburden the notations we leave out the subscript ζ in _{R}C_{ζ} from now on.

For an object X ∈ _{R}C let us define a vector space ⟨X⟩_{_{R}C} in the same manner as in 5.9, i.e. as an image of the canonical map from the maximal trivial subobject of X to the maximal trivial quotient of X. Given X_{1}, …, X_{m} ∈ _{R}C, we denote

⟨X_{1}, … X_{m}⟩ := ⟨X_{1} ⊗ … ⊗ X_{m}⟩_{_{R}C}.

9.2.1. Lemma. We have an isomorphism of local systems

HomÕ_{ζ}(1, ϕ(L_{1}) ˜⊗ … ˜⊗ϕ(L_{m}))^{♡} ≅ ⟨ϕ(L_{1}), …, ϕ(L_{m})⟩_{_{R}C}^{♡}

Proof. Follows from [A]. □

9.3. Lemma. The restriction functor Υ : _{R}C ⟶ C (cf. 6.2) induces isomorphism

⟨ϕ(L_{1}), …, ϕ(L_{m})⟩_{_{R}C} ⟶^{∼} ⟨Υϕ(L_{1}), …, Υϕ(L_{m})⟩_{C}.

Proof. We must prove that if λ_{1}, …, λ_{m} ∈ Δ, { \atop L(λ_{1})}, …, { \atop L(λ_{m})} are corresponding simples in _{R}C, and L(λ_{i}) = Υ{ \atop L(λ_{i})} — the corresponding simples in C, then the maximal trivial direct summand of { \atop L(λ_{1})} ⊗ … ⊗ { \atop L(λ_{m})} in _{R}C maps isomorphically to the maximal trivial direct summand of L(λ_{1}) ⊗ … ⊗ L(λ_{m}) in C.

According to [A], [AP], { \atop L(λ_{1})} ⊗ … ⊗ { \atop L(λ_{m})} is a direct sum of a module { \atop L(λ_{1})} ˜⊗ … ˜⊗{ \atop L(λ_{m})} ∈ O_{ζ} and a negligible module N ∈ _{R}C. Here negligible means that any endomorphism of N has quantum trace zero (see loc. cit.). Moreover, it is proven in loc cit. that N is a direct summand of W ⊗ M for some M ∈ _{R}C where W = ⊕_{ω∈Ω} { \atop L(ω)},

Ω = {ω ∈ X| ⟨i, ω + ρ⟩ > 0 for all i ∈ I; ⟨β_{0}, ω + ρ⟩ = κ}

being the affine wall of the first alcove. By loc. cit., W is negligible. Since Υ{ \atop L(ω)} = L(ω), ω ∈ Ω and since Υ commutes with braiding, balance and rigidity, we see that the modules L(ω) are negligible in C. Hence ΥW is negligible, and ΥW ⊗ΥM is negligible, and finally ΥN is negligible. This implies that ΥN cannot have trivial summands (since L(0)) is not negligible).

We conclude that

⟨Υ({ \atop L(λ_{1})} ⊗ … ⊗ { \atop L(λ_{m})})⟩_{C} = ⟨Υ{ \atop L(λ_{1})} ˜⊗ … ˜⊗Υ{ \atop L(λ_{m})}⟩_{C} =

⟨{ \atop L(λ_{1})} ˜⊗ … ˜⊗{ \atop L(λ_{m})}⟩_{_{R}C} = ⟨{ \atop L(λ_{1})} ⊗ … ⊗ { \atop L(λ_{m})}⟩_{_{R}C} □

9.4. Corollary 5.11 implies that the local system

⟨Υϕ(L_{1}), …, Υϕ(L_{m})⟩_{C}^{♡}

is canonically a subquotient of the local system

Tor_{\frac{∞}{2} +0}^{C}(B, Υϕ(L_{1}) ⊗ … ⊗ Υϕ(L_{m}) ⊗ L(2ρ_{ℓ})^{♡}

(the action of monodromy being induced by braiding and balance on the first m factors).

9.5. Let us fix a point ∞ ∈ ℙ^{1} and a nonzero tangent vector v ∈ T_{∞}ℙ^{1}. This defines an open subset

T A^{om} ⊂ T P^{om}

and the locally closed embedding

ξ : T A^{om} ↪ T P^{om+1}.

Given λ_{1}, …, λ_{m} ∈ Δ, we consider the integrable 𝔤̂-modules L̂(λ_{1}), …, L̂(λ_{m}) of central charge κ − ȟ.

Suppose that

λ_{1} + … + λ_{m} = α ∈ ℕ[I] ⊂ X.

We define λ_{∞} := 2ρ_{ℓ}, and λ⃗ := (λ_{1}, …, λ_{m}, λ_{∞}). Note that λ⃗ is positive and α = α(λ⃗), in the notations of 8.3.

Denote by X α ; λ⃗ the sheaf on Pα ; λ⃗ obtained by gluing L(λ_{1}), …, L(λ_{m}), L(λ_{∞}). Note that

X α ; λ⃗ = j_{!∗}Iα ; λ⃗

where j : Poα ; λ⃗ ↪ Pα ; λ⃗.

Consider the local system of conformal blocks

HomÕ_{κ}(1, L̂(λ_{1}) ˜⊗ … ˜⊗L̂(λ_{m}))^{♡}.

If ∑_{i=1}^{m} λ_{i} ∉ ℕ[I] ⊂ X then it vanishes by the above comparison with its ”quantum group” incarnation.

9.6. Theorem. Suppose that ∑_{i=1}^{m} λ_{i} = α ∈ ℕ[I]. Then the local system of conformal blocks restricted to T A^{om} is isomorphic to a canonical subquotient of a ”geometric” local system

ξ^{∗}R^{−2m−2}η̄_{m+1∗}^{α}j_{!∗}Iα ; λ⃗.

Proof. This follows from Theorem 8.11 and the previous discussion. □

9.7. Corollary. The above local system of conformal blocks is semisimple. It is a direct summand of the geometric local system above.

Proof. The geometric system is semisimple by Decomposition theorem, [BBD], Théorème 6.2.5. □

9.8. Example 5.12 shows that in general a local system of conformal blocks is a proper direct summand of the corresponding geometric system.

Part V. MODULAR STRUCTURE

ON THE CATEGORY FS

1. Introduction

1.1. Let C ⟶ S be a smooth proper morphism of relative dimension 1. Let x⃗ = (x_{1}, …, x_{m}) be an m-tuple of disjoint sections x_{k} : S ⟶ C. In this part we will show how to localize 𝔲-modules to the sections x⃗. To this end we will need a version of cohesive local system on the space of (relative) configurations on C.

The main difference from the case C = 𝔸^{1} is that the local systems are in general no more abelian one-dimensional. In fact, the monodromy in these local systems factors through the finite Heisenberg group. To stress the difference we will call them the Heisenberg local systems.

These local systems are constructed in Chapter 1.

1.2. Given a K-tuple of 𝔲-modules (or, equivalently, factorizable sheaves) {X_{k}} we study the sheaf g({X_{k}}) on C obtained by gluing the sheaves {X_{k}}. Namely, we study its behaviour when the curve C degenerates into a stable curve \underline{C} with nodes.

It appears that the sheaf g(\{X_{k}\}) degenerates into a sheaf \underline{g}(\{X_{k}\}, \{R_{j}\}) obtained by gluing the sheaves \{X_{k}\} and a few copies of the sheaf R: one for each node of \underline{C} (Theorem 17.3).

The sheaf R is not an object of FS, but rather of FS^{⊗2} (or, strictly speaking, of IndFS ⊗ IndFS). It corresponds to the regular 𝔲-bimodule R under the equivalence Φ.

The Theorem 17.3 is the central result of this part. Its proof occupies Chapters 2–5. We study the degeneration away from the nodes in Chapter 2. We study the degeneration near the nodes in Chapter 4, after we collect the necessary information about the regular bimodule R in Chapter 3.

As a byproduct of geometric construction of the regular bimodule we derive the hermitian autoduality of R and the adjoint representation ad.

1.3. In Chapter 5 we investigate the global sections of the sheaf g({X_{k}}). They form a local system on the moduli space of curves with K marked points and nonzero tangent vectors at these points (strictly speaking, the local system lives on the punctured determinant line bundle over this space). The collection of all such local systems equips the category FS with the fusion, or modular, structure in the terminology of [BFM] (Theorem 18.2).

Historically, first examples of modular categories appeared in the conformal field theory (WZW models), see e.g. [MS] and [TUY]. Namely, the category Õ_{κ} of integrable 𝔤̂- modules of central charge κ − ȟ has a natural modular structure.

As far as we know, the category FS is the first example of nonsemisimple modular category.

1.4. In Chapter 6 we study the connection between modular categories Õ_{κ} and C_{ζ} for ζ = exp(\frac{π\sqrt{−1}}{dκ}).

It appears that the modular structure on the former category can be reconstructed in terms of the modular structure on the latter one.

As a corollary we get a description of local systems of conformal blocks in WZW models in arbitrary genus as natural subquotients of some semisimple local systems of geometric origin (Theorem 19.8).

The geometric local systems are equipped with natural hermitian nondegenerate fiberwise scalar product (being direct images of perverse sheaves which are Verdier-autodual up to the replacement ζ ↦ ζ^{−1}) which gives rise to a hermitian nondegenerate scalar product on conformal blocks in WZW models.

1.5. Our work on this Part began 4 years ago as an attempt to understand the remarkable paper [CFW]. In fact, the key ingredients — Heisenberg local system, and adjoint representation — were already present in this paper.

V.Ginzburg has drawn our attention to this paper. D.Kazhdan’s interest to our work proved extremely stimulating.

During these years we benefited a lot from discussions with many people. The idea of Chapter 4 is due to P.Deligne. The idea to study the degeneration of Heisenberg local system is due to B.Feigin. We are grateful to R.Hain, J.Harris and T.Pantev who took pain of answering our numerous questions about different line bundles on the moduli spaces. The second author is obliged to V.Ostrik for useful discussions of adjoint representation.

Chapter 1. Heisenberg local system

2. Notations and statement of the main result

2.1. Let α ∈ ℕ[X], α = ∑ a_{μ}μ. We denote by suppα the subset of X consisting of all μ s.t. a_{μ} ≠ 0. Let π : J ⟶ X be an unfolding of α, that is ♯π^{−1}(μ) = a_{μ} for any μ ∈ X. As always, Σ_{π} denotes the group of automorphisms of J preserving the fibers of π.

2.1.1. The fibered product C ×_{S} … ×_{S} C (J times) will be denoted by C^{J}. The group Σ_{π} acts naturally on C^{J}, and the quotient space C^{J}/Σ_{π} will be denoted by C^{α}.

C^{∘}^{J} (resp. C^{∘}^{α}) stands for the complement to diagonals in C^{J} (resp. in C^{α}).

2.1.2. T C^{J} stands for the complement to the zero sections in the relative (over S) tangent

bundle. So T C^{J} ⟶ C^{J} is a (ℂ^{∗})^{J}-torsor. We denote by T C^{∘}^{J} its restriction to C^{∘}^{J}. The

group Σ_{π} acts freely on T C^{∘}^{J}, and we denote the quotient T C^{∘}^{J}/Σ_{π} by T C^{∘}^{α}.

The natural projection T C^{∘}^{J} ⟶ T C^{∘}^{α} will be denoted by π, or sometimes π_{J}.

2.1.3. Given j ∈ J we consider the relative tangent bundle on C^{J} along the j-th coordinate T_{j}. It is a line bundle on C^{J}. For μ ∈ suppα we consider the line bundle ⊗_{j∈π^{−1}(μ)}T_{j}

on C^{J}. It has a natural Σ_{π}-equivariant structure and can be descended to the line bundle T_{μ} on C^{α}.

2.1.4. Given ε > 0 we denote by D_{ε} the standard disk of radius ε. If there is no danger of confusion we will omit ε from our notations and will denote D_{ε} simply by D.

The definitions of D^{J}, D^{α}, D^{∘}^{J}, D^{∘}^{α}, T D^{∘}^{J}, T D^{∘}^{α} simply copy the above definitions, and we do not reproduce them.

2.1.5. Given a surjection τ : J ⟶ K we consider the map π_{K} : K ⟶ X, k ↦ ∑_{j∈τ^{−1}(k)} π(j). We will use the notation α_{K} for ∑_{μ∈X} ♯π_{K}^{−1}(μ)μ ∈ ℕ[X].

We consider the following (infinite dimensional) manifold

T C^{∘}^{τ} = \begin{matrix} \\ T C^{∘} & K \end{matrix} × ∏_{k∈K} T D^{∘}^{τ^{−1}(k)}

where \begin{matrix} \\ T C^{∘} & K \end{matrix} is the space of analytic open embeddings S_{K}×D ↪ C such that the restriction to S_{K} × 0 is just a K-tuple of sections S ⟶ C. Here S_{K} denotes the disjoint union of K copies of S.

We have an evident projection p_{K} : \begin{matrix} \\ T C^{∘} & K \end{matrix} ⟶ T C^{∘}^{K} taking the first jet. Note that p_{K} is a homotopy equivalence.

We denote by q_{τ} the natural substitution map T C^{∘}^{τ} ⟶ T C^{∘}^{J}.

2.1.6. Recall (see IV.9.1) that ℓ stands for \frac{l}{2} in case l is even, and for l in case l is odd. Recall (see loc. cit.) that

Y_{ℓ}^{∗} := {λ ∈ X | λ · μ ∈ ℓℤ ∀μ ∈ X}

We denote by d the cardinality of X/Y, and by d_{ℓ} the determinant of the form \frac{1}{ℓ}?·? restricted to the sublattice Y_{ℓ}^{∗} ⊂ X Note that if ℓ is divisible by d := max_{i∈I} d_{i} then d_{ℓ} = ♯(X/Y_{ℓ}^{∗}). This will be the case in our applications to conformal blocks.

To handle the general case we need to introduce some new characters. We define

X_{ℓ}^{∗} := {μ ∈ X ⊗ ℚ | μ · Y_{ℓ}^{∗} ∈ ℓℤ}

Evidently, Y_{ℓ}^{∗} ⊂ X ⊂ X_{ℓ}^{∗}, and X_{ℓ}^{∗} is generated by X and \{\frac{ℓ}{d_{i}} i′, i ∈ I\}. So if d|ℓ then X_{ℓ}^{∗} = X but in general this is not necessarily the case.

Note that d_{ℓ} = ♯(X_{ℓ}^{∗}/Y_{ℓ}^{∗}).

To study the modular properties of the Heisenberg local system and the category FS (cf. especially the Theorem 7.6(b)) we will have to modify slightly the definition of the latter one, and, correspondingly, of the category C. We start with the category C.

Consider the subalgebra 𝔲′ ⊂ 𝔲 (see II.12.3) generated by θ_{i}, ϵ_{i}, K̃_{i}^{±1}, i ∈ I (notations of loc. cit.).

We define C′ to be a category of finite dimensional X_{ℓ}^{∗}-graded vector spaces V = ⊕V_{λ} equipped with a structure of a left 𝔲′-module compatible with X_{ℓ}^{∗}-gradings and such that

K̃_{i}x = ζ^{⟨d_{i}i,λ⟩}x

for x ∈ V_{λ}, i ∈ I.

This is well defined since ⟨d_{i}i, X_{ℓ}^{∗}⟩ ∈ ℤ for any i ∈ I.

We have a natural inclusion of a full subcategory C ↪ C′.

We define the category FS′ exactly as in III.5.2, just replacing all occurences of X by X_{ℓ}^{∗}.

We have a natural inclusion of a full bbrt subcategory FS ↪ FS′, and the equivalence Φ : FS ⟶ C extends to the same named equivalence Φ : FS′ ⟶ C′. The proof is the same as in III; one only has to replace X by X_{ℓ}^{∗} everywhere. Recall that in case d|ℓ (the case of interest for applications to conformal blocks) we have X_{ℓ}^{∗} = X, FS′ = FS, C′ = C. From now on we will restrict ourselves to the study of the categories FS′, C′. However, in order not to scare the reader away by a bunch of new notations we will denote them by FS, C. The interested reader will readily perform substitutions in the text below.

2.1.7. Consider a function n : X_{ℓ}^{∗} ⟶ ℤ[\frac{1}{2d}] such that n(μ + ν) = n(μ) + n(ν) + μ · ν. We will choose n of the following form:

n(μ) = \frac{1}{2}μ · μ + μ · ν_{0}

for some ν_{0} ∈ X_{ℓ}^{∗}. From now on we fix such a function n and the corresponding ν_{0}. Let g be the genus of our relative curve C ⟶ S.

2.1.8. Definition. (a) We will denote the unique homomorphism

ℕ[X_{ℓ}^{∗}] ⟶ X_{ℓ}^{∗}

identical on X_{ℓ}^{∗} by α ↦ α^{∼};

(b) α ∈ ℕ[X_{ℓ}^{∗}] is called g-admissible if α^{∼} = (2g − 2)ν_{0}.

2.1.9. From now on we assume that ζ is a primitive root of unity of degree l. Then ζ = exp(2π\sqrt{−1}\frac{k}{l}) for some integer k prime to l. We fix k, and for a rational number q we define ζ^{q} := exp(2π\sqrt{−1}q \frac{k}{l}).

2.1.10. Given α = ∑ a_{μ}μ ∈ ℕ[Y] and its unfolding π : J ⟶ Y, we consider the following one-dimensional local system I^{J} on T D^{∘}^{J}: by definition, its monodromies are as follows: around diagonals: ζ^{2π(j_{1})·π(j_{2})}; around zero sections of tangent bundle: ζ^{2n(π(j))}. We define the one-dimensional local system I^{α} on T D^{∘}^{α} as I^{α} := (π_{J∗}I^{J})^{Σ_{π},−} (cf. III(46)).

2.1.11. For a line bundle L we denote the corresponding ℂ^{∗}-torsor by L̇.

2.2. Statement of the main result.

2.2.1. Definition. The Heisenberg local system H is the following collection of data:

1) A local system H^{α} on T C^{∘}^{α} for each admissible α ∈ ℕ[X_{ℓ}^{∗}];

2) Factorization isomorphisms: for each α ∈ ℕ[X_{ℓ}^{∗}], unfolding π : J ⟶ X_{ℓ}^{∗}, surjection τ : J ⟶ K, the following isomorphisms are given:

ϕ_{τ} : q_{τ}^{∗}π_{J}^{∗}H^{α} ⟶^{∼} p_{K}^{∗}π_{K}^{∗}H^{α_{K}} ⊠ × _{k∈K} I^{τ^{−1}(k)}

satisfying the usual associativity constraints.

2.2.2. Theorem. Let δ ⟶ S denote the determinant line bundle of the family C ⟶ S (see e.g. [KM]). Then after the base change

\begin{matrix}C_{δ} & ⟶ & C \\ ↓ & & ↓ \\ δ̇ & ⟶ & S \end{matrix}

there exists a Heisenberg Local System H.

The dimension of H is equal to d_{ℓ}^{g}, and the monodromy around the zero section of δ is equal to (−1)^{rkX_{ℓ}^{∗}}ζ^{12ν_{0}·ν_{0}}.

2.2.3. Remark. In the case g = 1 the line bundle δ^{12} is known to be trivial. It is easy to see that there exists a one-dimensional local system on δ̇ with any given monodromy around the zero section.

We will construct the Heisenberg local system H over S (as opposed to δ̇). Lifting it to δ̇ and twisting by the above one-dimensional systems we can obtain a Heisenberg local system with any given scalar monodromy around the zero section.

The construction of the desired Heisenberg System will be given in the rest of this Chapter.

3. The scheme of construction

3.1. First note that it suffices to construct the desired local system H for ζ = exp(\frac{π\sqrt{−1}}{ℓ}). If k is prime to l, and ζ′ = exp(2π\sqrt{−1}\frac{k}{l}), then H_{ζ′} is obtained from H_{ζ} just by application of a Galois automorphism of our field B. So till the end of construction we will assume that ζ = exp(\frac{π\sqrt{−1}}{ℓ}).

3.2. In what follows everything is relative over the base S. To unburden the notations we will pretend though that C is an absolute curve. Thus H^{1}(C) stands, say, for the local system of ℤ-modules of rank 2g over S.

For α ∈ ℕ[X_{ℓ}^{∗}] we introduce the following divisor D^{α} on C^{α}:

D^{α} = \frac{d_{ℓ}^{g}}{ℓ} (∑_{μ≠ν} μ · νΔ_{μν} + \frac{1}{2} ∑_{μ} μ · μΔ_{μμ})

where Δ_{μν}, μ, ν ∈ suppα, stands for the corresponding diagonal in C^{α}. Note that for g ≥ 2 all the coefficients of the above sum are integers. To simplify the exposition we will assume that g ≥ 2. For the case of g = 1 see 3.6.

Given an unfolding π : J ⟶ X_{ℓ}^{∗} we denote the pullback of D^{α} under π_{J} by D^{J}; this is a divisor on C^{J}.

We consider the (relative) Picard scheme Pic(C)⊗X_{ℓ}^{∗}. The group of its connected components is naturally isomorphic to X_{ℓ}^{∗}. Each component carries a canonical polarization ω which we presently describe. It is a skew-symmetric bilinear form on H_{1}(Pic^{0}(C)⊗X_{ℓ}^{∗}) = H_{1}(C) ⊗ X_{ℓ}^{∗} equal to the tensor product of the canonical cup-product form on H_{1}(C) and the symmetric bilinear form \frac{d_{ℓ}^{g}}{ℓ} (?·?) on X_{ℓ}^{∗}. Note that \frac{d_{ℓ}^{g}}{ℓ} (?·?) is positive definite, so ω is (relatively) ample.

We denote by AJ_{α} : C^{α} ⟶ Pic(C)⊗X_{ℓ}^{∗} the Abel-Jacobi map AJ_{α}(∑ μx_{μ}) = ∑(x_{μ})⊗μ.

The admissibility condition implies that the Abel-Jacobi map lands into the connected component (Pic(C) ⊗ X_{ℓ}^{∗})_{(2g−2)ν_{0}} to be denoted by A. Note that the projection A ⟶ S has a canonical section Ω ⊗ ν_{0}, so A is a genuine abelian variety, not just a torsor over one. Here Ω denotes the (relative) canonical line bundle on C.

3.2.1. Definition. We define the following line bundle L_{α} on C^{α}:

L_{α} = ⊗_{μ∈supp(α)}T_{μ}^{⊗^{\frac{d{g \atop ℓ}}{ℓ}} n(μ)} ⊗ O(D^{α})

The desired construction is an easy consequence of the following Propositions:

3.3. Proposition. There is a unique line bundle L on A such that for any g-admissible α we have L_{α} = AJ_{α}^{∗}(L). The first Chern class c_{1}(L) = −[ω].

3.4. Proposition. There exists a local system ℌ on L̇ ×_{S} δ̇ such that dim ℌ = d_{ℓ}^{g}

(see 2.1.6); the monodromy around the zero section of L is equal to ζ^{\frac{2/ell}{d{g \atop ℓ}}}; and the monodromy around the zero section of δ is equal to (−1)^{rkX_{ℓ}^{∗}}ζ^{12ν_{0}·ν_{0}}.

3.5. In the remainder of this section we derive the desired construction from the above Propositions.

We fix a g-admissible α. We denote by ȦJ_{α} the natural map between the total spaces of the corresponding ℂ^{∗}-torsors:

ȦJ_{α} : (AJ_{α}^{∗}(L))˙ ⟶ L̇

By the Proposition 3.3 we have an isomorphism

AJ_{α}^{∗}(L) ⟶^{∼} L_{α}  (287)

It is clear from the definition of L_{α} that the pullback of L_{α} to T C^{α} has the canonical meromorphic section s_{α}. The restriction of this section to T C^{∘}^{α} does not have poles nor zeros, and hence it defines the same named section s_{α} of the ℂ^{∗}-torsor (AJ_{α}^{∗}(L))˙ .

We change the base to δ̇, and preserve the notations AJ_{α}, s_{α} for the base change of the same named morphism and section. By the Proposition 3.4 we have the local system ℌ on L̇ ×_{S} δ̇.

We define H^{α} to be s_{α}^{∗}ȦJ_{α}^{∗}ℌ twisted by the one-dimensional sign local system.

The proof of the above Propositions and the construction of factorization isomorphisms will be given in the following sections.

3.6. The above construction does not work as stated in the case of elliptic curves: the line bundle O(D^{α}) in the Definition 3.2.1 does not make sense since the coefficients \frac{d_{ℓ}}{2ℓ}μ·μ of the divisor D^{α} apriori may be halfintegers.

We will indicate how to carry out the construction in this case.

For any N ∈ ℤ such that N \frac{d_{ℓ}}{2ℓ}μ · μ, N \frac{d_{ℓ}}{ℓ} n(μ) ∈ ℤ ∀μ ∈ X_{ℓ}^{∗} we define

L_{α}^{N} := ⨂_{μ∈suppα} T^{N \frac{d_{ℓ}}{ℓ} n(μ)} ⊗ O(ND^{α}).

We will prove the following versions of the above Propositions.

3.6.1. Proposition. There is a unique line bundle L^{N} on A such that for any gadmissible α ∈ ℕ[X_{ℓ}^{∗}] we have L_{α}^{N} = AJ_{α}^{∗}L^{N}. The first Chern class c_{1}(L^{N}) = −N[ω].

3.6.2. Proposition. There exists a local system ℌ^{N} on (L^{N})˙ such that dim ℌ^{N} = d_{ℓ}. The monodromy of ℌ^{N} around the zero section of L^{N} is equal to ζ^{\frac{2}{Nd_{ℓ}}}.

Moreover, for any N|N′ we have ℌ^{N} = [\frac{N′}{N}]^{∗}ℌ^{N′} where [\frac{N′}{N}] : L^{N} ⟶ L^{N′} is the map of raising to the (\frac{N′}{N})-the power.

3.6.3. Now H^{α} is defined as s_{α,N}^{∗} ȦJ^{∗}ℌ^{N} for any N as above exactly as in 3.5. In fact, it is enough to take N = 2.

4. The universal line bundle

In this section we will give a proof of the Propositions 3.3 and 3.6.1.

4.1. First we formulate a certain generalization. Suppose given a free ℤ-module Λ of finite rank with an even symmetric bilinear pairing (,) : Λ × Λ ⟶ ℤ. We fix an element ν ∈ Λ, and a function b : Λ ⟶ ℤ, b(λ) := \frac{1}{2}(λ, λ) + (λ, ν).

4.1.1. For α ∈ ℕ[Λ] we introduce the following divisor D^{α} on C^{α}:

D^{α} = ∑_{μ≠λ}(μ, λ)Δ_{μλ} + \frac{1}{2} ∑_{μ}(μ, μ)Δ_{μμ}

where Δ_{μλ}, μ, λ ∈ suppα, stands for the corresponding diagonal in C^{α}. Note that all the coefficients of the above sum are integers.

Given an unfolding π : J ⟶ Λ we denote the pullback of D^{α} under π_{J} by D^{J}; this is a divisor on C^{J}.

We consider the (relative) Picard scheme Pic(C) ⊗ Λ. The group of its connected components is naturally isomorphic to Λ. Each component carries a canonical polarization ω which we presently describe. It is a skew-symmetric bilinear form on H_{1}(Pic^{0}(C) ⊗ Λ) = H_{1}(C) ⊗ Λ equal to the tensor product of the canonical cup-product form on H_{1}(C) and the symmetric bilinear form (,) on Λ.

We denote by AJ_{α} : C^{α} ⟶ Pic(C) ⊗ Λ the Abel-Jacobi map AJ_{α}(∑ μx_{μ}) = ∑(x_{μ}) ⊗ μ.

The admissibility condition implies that the Abel-Jacobi map lands into the connected component (Pic(C) ⊗ Λ)_{(2g−2)ν} to be denoted by A_{Λ}.

Note that the projection A_{Λ} ⟶ S has a canonical section Ω ⊗ ν, so A_{Λ} is a genuine abelian variety, not just a torsor over one. Here Ω denotes the (relative) canonical line bundle on C.

4.1.2. Definition. We define the following line bundle L_{α} on C^{α}:

L_{α} = ⊗_{μ∈supp(α)}T_{μ}^{⊗b(μ)} ⊗ O(D^{α})

An element α = ∑ a_{λ}λ ∈ ℕ[Λ] (formal sum) is called g-admissible if α^{∼} = (2g − 2)ν.

Now we are able to formulate the Proposition generalizing 3.3.

4.2. Proposition. There is a unique line bundle L(Λ, (,), ν) on A_{Λ} such that for any g-admissible α we have L_{α} = AJ_{α}^{∗}(L(Λ, (,), ν)). The first Chern class c_{1}(L(Λ, (,), ν)) = −[ω].

4.3. We start the proof of the Proposition 4.2 with the following Lemma. Let π_{J} : J ⟶ Λ be an unfolding of α. Let τ : J ⟶ K be a surjection, and π_{K} : K ⟶ Λ be an unfolding of α_{K} as in 2.1.5. To simplify the notations we will denote α_{K} by β.

Let σ_{τ} denote the natural (“diagonal”) embedding C^{K} ↪ C^{J}.

4.3.1. Lemma. There is a canonical isomorphism

σ_{τ}^{∗}π_{J}^{∗}(L_{α}) = π_{K}^{∗}(L_{β})

Proof. It is enough to prove the Lemma in the case |J| = |K| + 1. So we fix i, j ∈ J such that τ(i) = τ(j) = k, and we denote π_{J}(i) by μ_{i}, and π_{J}(j) by μ_{j}.

We have π_{J}^{∗}(D^{α}) = μ_{i} · μ_{j}Δ_{ij} + D′ where Δ_{ij} ⊄ supp(D′).

Moreover, it is clear that D′ ∩ Δ_{ij} = σ_{τ}(π_{K}^{∗}(D^{β})), and hence

σ_{τ}^{∗}π_{τ}^{∗}(O(D′)) = π_{K}^{∗}(O(D^{β}))

On the other hand, for any smooth divisor D we have a canonical isomorphism O(D)|_{D} = N_{D} (the normal bundle).

In particular, we have σ_{τ}^{∗}(O(Δ_{ij})) = σ_{τ}^{∗}(T_{i}) = σ_{τ}^{∗}(T_{j}) = T_{k}.

Thus σ_{τ}^{∗}(T_{i}^{b(μ_{i})} ⊗ T_{j}^{b(μ_{j})} ⊗ O((μ_{i}, μ_{j})Δ_{ij})) = T_{k}^{b(μ_{i})+b(μ_{j})+(μ_{i},μ_{j})} = T_{k}^{b(μ_{i}+μ_{j})} = T_{k}^{b(π(k))}.

Finally, if m ≠ i, j, then evidently σ_{τ}^{∗}(T_{m}) = T_{τ(m)}.

Putting all this together we obtain the statement of the Lemma. □

4.4. To prove the Proposition we have to check a necessary condition: that the first Chern class of L_{α} is a pullback of some cohomology class on A_{Λ} under the Abel-Jacobi map AJ_{α}. This is the subject of the following Lemma.

4.4.1. Lemma. If α is admissible, then c_{1}(L_{α}) = AJ_{α}^{∗}(−[ω]).

Proof. Let us choose an unfolding π : J ⟶ Λ. We denote by π_{J} the corresponding projection C^{J} ⟶ C^{α}.

It is enough to prove that the pullback of the both sides to C^{J} under π_{J} coincide.

We introduce the following family of 2-cycles in C^{J}. For 1-cycles a, b in C, and i ≠ j ∈ J, we denote by a_{i} × b_{j} the following product cycle: the i-th coordinate runs along the cycle a, the j-th coordinate runs along the cycle b, all the rest coordinates are fixed. The homology class of a_{i} × b_{j} depends only on i, j and the classes of a and b.

We denote by f_{i} the following 2-cycle: the i-th coordinate runs along the fundamental cycle of C, all the rest coordinates are fixed. The homology class of f_{i} depends on i only.

It suffices to check that the pairings of both sides of 4.4.1 with this family of cycles coincide.

We have:

⟨−ω, a_{i} × b_{j}⟩ = (π(i), π(j))a ∩ b = (a_{i} × b_{j}) ∩ ((π(i), π(j))Δ_{ij}) = ⟨c_{1}(L_{α}), a_{i} × b_{j}⟩; (288)

⟨−ω, f_{i}⟩ = −g(π(i), π(j));  (289)

⟨c_{1}(L_{α}), f_{i}⟩ = (2 − 2g)b(π(i)) + f_{i} ∩ π^{∗}(D^{α}) = (2 − 2g)b(π(i)) + (π(i), (∑_{j∈J} π(j) − π(i))). (290)

To assure the equality of (289) and (290) we must have

(1 − g)(π(i), π(i)) = (2 − 2g)b(π(i)) + (π(i), ∑_{j∈J} π(j)),  (291)

that is,

(1 − g)((π(i), π(i)) − 2b(π(i))) = (π(i), ∑_{j∈J} π(j))  (292)

which is precisely the admissibility condition. □

4.5. Let us choose a basis I of Λ. Using the Lemma 4.3.1 we see that it suffices to prove the Proposition for α of a particular kind, namely

α = ∑_{i∈I} a_{i}i + ∑_{i∈I} a_{−i}(−i)

where all the positive integers a_{i}, a_{−i} are big enough.

4.5.1. We define α_{+} := ∑_{i∈I} a_{i}i, and α_{−} := ∑_{i∈I} a_{−i}(−i). We consider the following Abel-Jacobi maps:

AJ_{+} : C^{α_{+}} ⟶ (Pic(C) ⊗ Λ)_{α_{+}} =: A_{+},

and

AJ_{−} : C^{α_{−}} ⟶ (Pic(C) ⊗ Λ)_{α_{−}} =: A_{−}.

We have

AJ_{α} = m ∘ (AJ_{+} × AJ_{−})

where m : A_{+} ×_{S} A_{−} ⟶ A_{Λ} is the addition map. If all the a_{±i} are bigger than g, then the map

(AJ_{+} × AJ_{−})^{∗} : Pic^{0}(A_{+} ×_{S} A_{−}) ⟶ Pic^{0}(C^{α_{+}} ×_{S} C^{α_{−}})

is an isomorphism, and the induced map on the whole Picard groups is an inclusion. According to the Lemma 4.4.1, c_{1}(L_{α}) = AJ_{α}^{∗}(−[ω]) = (AJ_{+} × AJ_{−})^{∗}(m^{∗}(−[ω])). So we deduce that there exists a unique line bundle L′ on A_{+} ×_{S} A_{−} such that L_{α} = (AJ_{+} × AJ_{−})^{∗}L′. It remains to show that L′ = m^{∗}L for some line bundle L on A_{Λ} (necessarily uniquely defined). To this end it is enough to verify that the restrictions of L′ to the fibers of m are trivial line bundles.

4.5.2. We choose an unfolding J of α. We choose a surjection τ : J ⟶ K with the following property: K = K_{0} ⊔ K_{1}; τ is one-to-one over K_{1}, and for any k ∈ K_{0} we have τ^{−1}(k) = {i, −i} for some i ∈ I.

Moreover, we assume that α is big enough so that for each i ∈ I both i and −i appear at least g times in τ^{−1} of both K_{0} and K_{1}.

Recall that σ_{τ} stands for the diagonal embedding C^{K} ↪ C^{J}.

It is clear that C^{K} = C^{K_{0}} ×_{S} C^{K_{1}}, and the Abel-Jacobi map AJ_{J} ∘ σ_{τ} : C^{K} ⟶ A_{Λ} factors through the projection onto the second factor.

Fix a point a ∈ A_{Λ}. Let us choose p ∈ C^{K_{1}} such that AJ_{J}(σ_{τ}(C^{K_{0}} × p)) = a. Then (AJ_{+} × AJ_{−}) ∘ σ_{τ} maps C^{K_{0}} × p to the fiber m^{−1}(a).

It is easy to see that the induced maps on the Picard groups ((AJ_{+} × AJ_{−}) ∘ σ_{τ})^{∗} : Pic(m^{−1}(a)) ⟶ Pic(C^{K_{0}}) is injective.

Thus we only have to check that

((AJ_{+} × AJ_{−}) ∘ σ_{τ})^{∗}L′|_{C^{K_{0}}×p} = (σ_{τ} ∘ π_{J})^{∗}L_{α}|_{C^{K_{0}}×p}

is trivial.

According to the Lemma 4.3.1, this line bundle is equal to π_{K}^{∗}(L_{α_{K}}). It is clear from the definition that L_{α_{K}} is lifted from the projection to the factors carrying nonzero weights. In particular, the restriction of L_{α_{K}} to a fiber of this projection is trivial.

This completes the proof of the Proposition 4.2.

4.6. To prove the Proposition 3.3 it suffices to apply Proposition 4.2 to the case Λ = X_{ℓ}^{∗}; (?,?) = \frac{d_{ℓ}^{g}}{ℓ}?·?; ν = ν_{0}; b(λ) = \frac{d_{ℓ}^{g}}{ℓ} n(λ).

To prove the Proposition 3.6.1 we take Λ = X_{ℓ}^{∗}; (?,?) = N \frac{d_{ℓ}^{g}}{ℓ}?·?; ν = ν_{0}; b(λ) = N \frac{d_{ℓ}^{g}}{ℓ} n(λ). □

5. The universal local system

In this section we will give a proof of the Propositions 3.4 and 3.6.2.

5.1. Recall the notations of 4.1. Let us take Λ = X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}; ((x_{1}, y_{1}), (x_{2}, y_{2})) = −\frac{1}{ℓ}(y_{1} · x_{2} + y_{2} · x_{1} + y_{1} · y_{2}); ν = (ν_{0}, 0).

We denote A_{Λ} by A′. We have an obvious projection pr_{1} : A′ ⟶ A. We denote the line bundle L(Λ, (,), ν) (notations of 4.2) by L′.

5.2. Theorem. (a) L′ is relatively ample with respect to pr_{1}.

(b) The direct image E := pr_{1∗}L′ is a locally free sheaf of rank d_{ℓ}^{g}.

(c) In the situation of 3.4 (that is, g > 1) we have det(E) = L ⊗ (p^{∗}δ)^{d_{ℓ}^{g}(− \frac{1}{2} rkX_{ℓ}^{∗}+6 \frac{ν_{0}·ν_{0}}{ℓ})}, where p stands for the projection A ⟶ S.

(d) Assume g = 1. Then in the notations of 3.6 we have for any N as in loc. cit. (det(E))^{⊗N} = L^{N} ⊗ p^{∗}δ^{⊗iN} for some i ∈ ℤ.

5.2.1. Let us construct the desired local system assuming the Theorem 5.2. By the virtue of [BK], Corollary 3.4 and Corollary 4.2, E is naturaly equipped with the flat projective connection. Hence its lifting to (det(E))˙ carries a flat connection with scalar monodromy around the zero section equal to exp(\frac{2π\sqrt{−1}}{d_{ℓ}^{g}}). Isomorphism 5.2.(c) yields the

map m : L̇ ×_{S} δ̇ ⟶ (det(E))˙ , where m(λ, t) = λ ⊗ t^{d_{ℓ}^{g}(− \frac{1}{2} rkX_{ℓ}^{∗}+6 \frac{ν_{0}·ν_{0}}{ℓ})}. It is clear that m^{∗}E is a locally free sheaf with flat connection, whose monodromy around the 0-section of L

is equal to exp(\frac{2π\sqrt{−1}}{d_{ℓ}^{g}}) = ζ^{\frac{2ℓ}{d{g \atop ℓ}}}, and monodromy around the zero section of δ is equal to (−1)^{rkX_{ℓ}^{∗}}ζ^{12ν_{0}·ν_{0}}. This proves the Proposition 3.4.

The proof of the Proposition 3.6.2 is even simpler. Note that for any M the lifting of E to ((det(E))^{⊗M})˙ carries a flat connection with scalar monodromy around the zero section equal to exp(\frac{2π\sqrt{−1}}{Md_{ℓ}}).

Now 5.2(d) implies the isomorphism (det(E))^{⊗12N} = (L^{N})^{⊗12} for any N as in 3.6 (recall that δ^{⊗12} is trivial).

So for any N we can define ℌ^{N} to be m^{∗}E where m : L^{N} ⟶ (det(E))^{⊗12N} is the composition of raising to the 12-th power and the above isomorphism. □

5.3. We now proceed with the proof of the Theorem 5.2. The statements (a) and (b) follow immediately from the Proposition 4.2 and the Riemann-Roch formula for abelian varieties (see e.g. [M1]).

To prove (c) we need two auxilliary Propositions.

As usually, it is more convenient to work in greater generality. Thus suppose given Λ, (,), ν as in 4.1.

We will denote the canonical section Ω ⊗ ν of A_{Λ} = (Pic(C) ⊗ Λ)_{(2g−2)ν} by s. This section identifies A_{Λ} with Pic^{0}(C) ⊗ Λ, and we will use this identification in what follows.

5.4. Proposition. s^{∗}(L(Λ, (,), ν) = δ^{−6(ν,ν)} (notations of 4.2).

5.5. Corollary. L(Λ, (,), ν) = L(Λ, (,), 0) ⊗ p^{∗}δ^{−6(ν,ν)}.

Proof. Let first g = 1. Then note that the condition of g-admissibility reads α^{∼} = 0 independently of ν. To stress the dependence of L^{α} on ν we will include ν as the subindex for a moment. We have T_{λ} = p^{∗}(δ^{−1}) for any λ ∈ Λ. For any 1-admissible α, and any ν we have

L_{ν}^{α} = L_{0}^{α} ⊗ ⨂_{λ∈suppα} T_{λ}^{(λ,ν)} = L_{0}^{α} ⊗ p^{∗}(δ^{−(α^{∼},ν)}) = L_{0}^{α}

It follows that L(Λ, (,), ν) = L(Λ, (,), 0) for any ν. On the other hand 6(ν, ν) ∈ 12ℤ, and δ^{⊗12} is known to be trivial. This takes care of the case g = 1. Assume now g > 1. In view of the Proposition 5.4 it is enough to show that L(Λ, (,), ν) = L(Λ, (,), 0) ⊗ p^{∗}Ξ for some line bundle Ξ on S. So we can assume that S is a point. Let us choose a nonzero holomorphic form σ with zero divisor (σ) on C. Then, for any α ∈ ℕ[Λ], we have an embedding

ι_{σ} : C^{α} ↪ C^{α+(2g−2)ν}; x ↦ x + (σ) ⊗ ν.

Recall the Definition 4.1.2 of g-admissibility. Since g is fixed, while ν varies, till the end of the proof we will replace this term by ν-admissibility. Note that if α is 0-admissible, then α + (2g − 2)ν is ν-admissible. It is clear that for 0-admissible α we have

ι_{σ}^{∗}L^{α+(2g−2)ν} = ⨂_{μ∈suppα} T_{μ}^{b_{ν}(μ)} ⊗ O(D^{α}) ⊗ ⨂_{μ∈suppα} T_{μ}^{−(μ,ν)} =

= ⨂_{μ∈suppα} T_{μ}^{\frac{1}{2} (μ,μ)} ⊗ O(D^{α}) = ⨂_{μ∈suppα} T_{μ}^{b_{0}(μ)} ⊗ O(D^{α}) = L^{α}

(notations of 4.1). Now the Corollary follows from the definition (and uniqueness) of L(Λ, (,), ν) (Proposition 4.2). □

5.6. Proposition. Assume g > 1. Then [Rp_{Λ∗}(L(Λ, (,), 0))]^{1} = δ^{− \frac{1}{2} d^{g}rkΛ} where d stands for det(−(,) ), while p_{Λ} denotes the projection A_{Λ} ⟶ S, and [Rp_{Λ∗}(?)]^{1} denotes the determinant of the complex Rp_{Λ∗}(?) (see e.g. [KM]). Note that d equals the Euler characteristic of Rp_{Λ∗}(L(Λ, (,), 0)).

5.7. Let us derive the Theorem 5.2(c) assuming the above Propositions. Note that the scalar product ((ν_{0}, 0), (ν_{0}, 0)) = 0, and hence the Corollary 5.5 implies that L′ does not depend on ν_{0}. It follows from loc. cit. that the RHS of 5.2(c) does not depend on ν_{0} either. We will assume that ν_{0} = 0 until the end of the proof. For M ∈ ℤ and an abelian group G let [M] : G ⟶ G denote the multiplication by M. We choose M such that MX_{ℓ}^{∗} ⊂ Y_{ℓ}^{∗}.

5.7.1. Lemma. [M]^{∗} det(E) = [M]^{∗}(L ⊗ (p^{∗}δ)^{d_{ℓ}^{g}(− \frac{1}{2} rkX_{ℓ}^{∗}+6 \frac{ν_{0}·ν_{0}}{ℓ})}). Proof. Let us define +_{M} : X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} ⟶ Y_{ℓ}^{∗} as +_{M}(x, y) = Mx + y. We have an equality of quadratic forms

([M] × id)^{∗}(,) = (+_{M})^{∗}(−\frac{1}{ℓ}?·?) + pr_{1}^{∗}(\frac{M^{2}}{ℓ}?·?)

(note that all the quadratic forms involved are integer valued and even). It implies the equality of line bundles:

([M] × id)^{∗}L′ = (+_{M} ⊗ id)^{∗}(L(Y_{ℓ}^{∗}, −\frac{1}{ℓ}?·?, 0) ⊗ pr_{1}^{∗}L(X_{ℓ}^{∗}, \frac{M^{2}}{ℓ}?·?, 0))

Note that (pr_{1}, +_{M}) : X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} ⟶ X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} is an automorphism. Hence

[M]^{∗} det(E) = det(pr_{1∗}(pr_{2}^{∗}L(Y_{ℓ}^{∗}, −\frac{1}{ℓ}?·?, 0) ⊗ pr_{1}^{∗}L(X_{ℓ}^{∗}, \frac{M^{2}}{ℓ}?·?, 0)) =

= det(pr_{1∗}pr_{2}^{∗}L(Y_{ℓ}^{∗}, −\frac{1}{ℓ}?·?, 0)) ⊗ L(X_{ℓ}^{∗}, \frac{M^{2}}{ℓ}?·?, 0)^{d{g \atop ℓ}} (293)

by the projection formula (note that d_{ℓ}^{g} is the Euler characteristic of pr_{1∗}pr_{2}^{∗}L(Y_{ℓ}^{∗}, −\frac{1}{ℓ}?·?, 0)). The first tensor multiple is equal to

p_{A}^{∗} det((p_{Pic^{0}(C)⊗Y_{ℓ}^{∗}})_{∗}L(Y_{ℓ}^{∗}, −\frac{1}{ℓ}?·?, 0)) = p_{A}^{∗}δ^{− \frac{1}{2} d_{ℓ}^{g}rkY_{ℓ}^{∗}}

by the Proposition 5.6. The second tensor multiple is identified with

L(X_{ℓ}^{∗}, \frac{d_{ℓ}^{g}M^{2}}{ℓ}?·?, 0) = [M]^{∗}L(X_{ℓ}^{∗}, \frac{d_{ℓ}^{g}}{ℓ}?·?, 0)

This completes the proof of the Lemma. □

5.7.2. Now the difference of the LHS and RHS of 5.2(c) is a line bundle Ξ on Pic^{0}(C)⊗X_{ℓ}^{∗} which is defined functorially with respect to S. Moreover, we have seen that [M]^{∗}Ξ is trivial. This implies that Ξ is trivial itself.

In effect, it defines a section of Pic^{0}(C)_{M} ⊗ X_{ℓ}^{∗} (where the subscript _{M} stands for Mtorsion).

But Pic^{0}(C)_{M} ⊗ X_{ℓ}^{∗} = R^{1}p_{C∗}(\underline{ℤ/Mℤ}) ⊗ X_{ℓ}^{∗}, and it is well known that R^{1}p_{C∗}(\underline{ℤ/Mℤ}) does not have nonzero functorial sections. Hence the above section vanishes, so the difference of the LHS and RHS is a line bundle lifted from the base S.

But we know that the restrictions of the LHS and RHS to the zero section of Pic^{0}(C)⊗X_{ℓ}^{∗} coincide.

This completes the proof of 5.2(c).

5.7.3. Now we will prove 5.2(d).

According to the proof of Corollary 5.5 both sides are independent of ν, so we put ν = 0.

The formula (293) and the argument just above it applies to the case of elliptic curve as well provided N|M. The first tensor multiple of (293) is obviously lifted from the base. Since the Picard group of the moduli space M_{1} is generated by δ, loc. cit. implies that

[M]^{∗} det(E) = p^{∗}(δ^{i}) ⊗ L(X_{ℓ}^{∗}, \frac{d_{ℓ}M^{2}}{ℓ}?·?, 0)

for some i. Hence

[M]^{∗}(det(E))^{⊗N} = p^{∗}(δ^{iN}) ⊗ L(X_{ℓ}^{∗}, \frac{Nd_{ℓ}M^{2}}{ℓ}?·?, 0) = [M]^{∗}(p^{∗}(δ^{iN}) ⊗ L^{N})

Exactly as in 5.7.2 this implies that

(det(E))^{⊗N} = p^{∗}(δ^{iN}) ⊗ L^{N}

This completes the proof of 5.2(d). □

5.8. We start with the proof of Proposition 5.4.

If g = 1 then both sides are trivial. Indeed, the RHS is trivial since 6(ν, ν) ∈ 12ℤ, and δ^{⊗12} is trivial.

To see that the LHS is trivial it suffices to consider the case α = {0} (neutral element of Λ with multiplicity one).

¿From now on we assume that g > 1.

Put P := ℙ((p_{C})_{∗}(Ω_{C/S})) where p_{C} denotes the projection C ⟶ S. We have the natural inclusion i : P ↪ C^{(2g−2)}. Let α ∈ ℕ[Λ] be the multiset consisting of (2g − 2) copies of ν. We will identify C^{(2g−2)} with C^{α}. Of course AJ_{α} ∘ i maps P to the image of s. So it suffices to prove that

(AJ_{α} ∘ i)^{∗}(L(Λ, (,), ν)) = p_{P}^{∗} (δ^{−6(ν,ν)})  (294)

where p_{P} denotes the projection P ⟶ S.

Using the fact that b(ν) = \frac{3}{2}(ν, ν) we deduce

(AJ_{α} ∘ i)^{∗}(L(Λ, (,), ν)) = i^{∗}(L_{α}) = i^{∗}(S^{2g−2}(Ω^{− \frac{3}{2} (ν,ν)})(\frac{1}{2}(ν, ν)Δ)).  (295)

Here Δ ⊂ C^{(n)} is the diagonal divisor, and for an invertible sheaf F on C we denote by S^{n}(F) the invertible sheaf π_{∗}(F^{⊠n})^{Σ_{n}} on C^{(n)} (where π : C^{n} → C^{(n)} is the projection).

Let us introduce some more notation. Let I ⊂ C × C^{(n)} be the incidence divisor. We will denote the projection of C × C^{(n)} to the i-th factor by pr_{i} till the further notice. This will not cause any confusion with our previous use of the notation pr_{1}. For an invertible sheaf F on C we define a rank n locally free sheaf F^{(n)} on C^{(n)} by: F^{(n)} := pr_{2∗}(O_{I} ⊗ pr_{1}^{∗}(F)).

5.8.1. Lemma. (a) The map F ↦ S^{n}(F) defines a homomorphism from Pic(C) to Pic(C^{(n)});

(b) For any line bundle F on C we have ((π_{∗}(F^{⊠n}))^{Σ_{n},−})^{⊗2} = S^{n}(F^{⊗2})(−Δ);

(c) det(F^{(n)}) = (π_{∗}(F^{⊠n}))^{Σ_{n},−}.

Proof. (a) is clear.

(b) We obviously have a morphism from the LHS to S^{n}(F^{2}). We have to check that the cokernel is supported on Δ, and its stalk at the generic point of Δ is 1-dimensional. For this we can assume that C = 𝔸^{1}, and F = O, and check the statement directly.

(c) Let us first construct a morphism from the LHS to the RHS. Let Ĩ ⊂ C^{n+1} be the union of diagonals x_{1} = x_{i}; let p̃r_{2} (resp. p̃r_{1}) be the projection of C^{n+1} onto the last n (resp. first) coordinates. Then

π^{∗}(det(F^{(n)})) = det(p̃r_{2∗}(p̃r_{1}^{∗}F ⊗ OĨ))  (296)

because the square is Cartesian.

We also have the arrow: OĨ → ⊕_{i=2,..,n+1}O_{{x_{1}=x_{i}}} which yields the morphism

det(p̃r_{2∗}(p̃r_{1}^{∗}F ⊗ OĨ) ⟶ det(p̃r_{2∗}(p̃r_{1}^{∗}F ⊗ (⊕O_{{x_{1}=x_{i}}}))) = F^{⊠n}  (297)

It is clear that the morphism (297) anticommutes with the action of Σ_{n}, hence, by (296), it defines the desired arrow.

It is easy to see that the image of the monomorphism (297) is F^{⊠n}(−D) ⊂ F^{⊠n}, where D is the union of all diagonals. But the latter inclusion induces isomorphism

(π_{∗}(F^{⊠n}(−D)))^{Σ_{n},−} = (π_{∗}(F^{⊠n}))^{Σ_{n},−}

This completes the proof of the Lemma. □

5.8.2. We return to the proof of Proposition 5.4. It suffices to prove it assuming that the line bundle Ω^{1/2} exists over S. Indeed, let M_{g} denote the moduli space of curves of genus g. Then it is known that Pic(M_{g}) is torsion free (see e.g. [M2], Lemma 5.14). Hence it injects into the Picard group of the moduli space of curves with θ-characteristics.

We have

i^{∗}(L_{α}) = i^{∗}(S^{2g−2}(Ω^{− \frac{3}{2} (ν,ν)})(\frac{1}{2}(ν, ν)Δ)) = i^{∗}(S^{2g−2}(Ω^{3})(−Δ))^{− \frac{1}{2} (ν,ν)} =

i^{∗}(det^{⊗2}((Ω^{\frac{3}{2}} )^{(2g−2)}))^{− \frac{1}{2} (ν,ν)} = i^{∗}(det((Ω^{\frac{3}{2}} )^{(2g−2)}))^{−(ν,ν)}

So the Proposition follows from

5.8.3. Lemma. det(i^{∗}(Ω^{\frac{3}{2}} )^{(2g−2)}) = p_{P}^{∗} δ^{6}.

Proof. Consider the exact sequence of sheaves on C × C^{(n)}:

0 ⟶ F ⟶ pr_{1}^{∗}Ω^{\frac{3}{2}} ⟶ pr_{1}^{∗}Ω^{\frac{3}{2}} ⊗ O_{I} ⟶ 0

where F denotes the kernel. Let now (until the end of the proof) pr_{i} denote the projections of C × P to the respective factors. We see that

(id × i)^{∗}(F) = pr_{1}^{∗}Ω^{\frac{1}{2}} ⊗ pr_{2}^{∗}(?)

We apply the functor Rpr_{2∗}∘(id×i)^{∗} to this exact sequence (note that [Rpr_{2∗}(Ω^{\frac{1}{2}} ⊗pr_{2}^{∗}(?))]^{1} does not depend on ? since the Euler characteristic of Ω^{\frac{1}{2}} is 0).

Using Mumford’s formula (see [M2], Theorem 5.10) we get

det(i^{∗}(Ω^{\frac{3}{2}} )^{(2g−2)}) = [Rpr_{2∗}(pr_{1}^{∗}Ω^{\frac{3}{2}} ⊗ i^{∗}(O_{I}))]^{1} =

= [Rpr_{2∗}(pr_{1}^{∗}Ω^{\frac{3}{2}} )]^{1} − [Rpr_{2∗}(pr_{1}^{∗}Ω^{\frac{1}{2}} )]^{1} =

= [6((\frac{3}{2})^{2} − \frac{3}{2}) + 1 − (6((\frac{1}{2})^{2} − \frac{1}{2}) + 1)]p_{P}^{∗} δ = 6p_{P}^{∗} δ

This completes the proof of the Lemma along with the Proposition 5.4.

5.9. We start the proof of the Proposition 5.6.

5.9.1. Lemma. Let 𝔄 → S be a family of abelian varieties over a smooth base S. Let 𝔞 : 𝔄′ ⟶ 𝔄 be an isogeny. Let L be a line bundle over 𝔄. Then deg(𝔞)[Rp_{𝔄∗}L]^{1}−[Rp_{𝔄′∗}𝔞^{∗}L]^{1} lies in the torsion of Pic(S).

Proof. By the relative Riemann-Roch theorem (see e.g. [Fu] Theorem 15.2) we have the equalities in the Chow ring A_{ℚ}^{S}

ch[R(p_{𝔄′∗}𝔞^{∗}L] = p_{𝔄′∗}(ch(𝔞^{∗}L)td_{𝔄′/S}) = p_{𝔄∗}𝔞_{∗}𝔞^{∗}[ch(L)td_{𝔄/S}] =
= deg(𝔞)p_{𝔄∗}[ch(L)td_{𝔄/S}] = deg(𝔞)ch[Rp_{𝔄∗}(L)]

Taking the components of degree 1 we get the Lemma. □

5.9.2. Remark. We will apply the Lemma to the situation 𝔄 = Pic^{0}(C) ⊗ Λ. In this case the hypothesis of smoothness of S is redundant. Indeed, it is enough to take for S the moduli space of genus g curves M_{g}. It is well known that there exists a smooth covering π : M̃_{g} ⟶ M_{g} inducing injection on Picard groups. One can take for M̃_{g} the moduli space of curves with a basis in H^{1}(C, ℤ/3ℤ).

We continue the proof of the Proposition. We define the integer function m(Λ, (,) ) by the requirement

[Rp_{Λ∗}(L(Λ, (,), 0))]^{1} = δ^{m(Λ, (,) )}

We will need one more Lemma.

5.9.3. Lemma. The function m satisfies the following properties:

(a) m(Λ_{1} ⊕ Λ_{2}, (,)_{1} + (,)_{2}) = m(Λ_{1}, (,)_{1})d_{2}^{g} + m(Λ_{2}, (,)_{2})d_{1}^{g} (for the notation d see Proposition 5.6);

(b) Let ι : Λ′ ↪ Λ be an embedding with finite cokernel. Then

m(Λ′, ι^{∗}(,) ) = #(Λ/Λ′)^{g}m(Λ, σ);

(c) If Λ = ℤ and (1, 1) = −16 then m(Λ, (,) ) = −\frac{16^{g}}{2} .

5.9.4. We will prove the Lemma below, and now we derive the Proposition from the Lemma.

We call 2 lattices isogenic if they contain isomorphic sublattices of maximal rank. From (a) it follows that the Proposition holds for the sum of 2 lattices if it holds for each of them. From (b) it follows that the Proposition holds for a lattice if it holds for an isogenic one. Thus it is enough to prove it for (Λ, (,) ) = (ℤ, n).

Note that the statement for the lattice (ℤ, n) is equivalent to the statement for the lattice (ℤ⊕ℤ, n⊕(−n)) since m(ℤ⊕ℤ, n⊕(−n)) = 2(−n)^{g} ·m(ℤ, n). But the lattices (ℤ⊕ℤ, n⊕ (−n)) are isogenic for various n, and (c) says that the Proposition is true for n = −16.

The Proposition is proved. This completes the proof of the Theorem 5.2(c). □

5.10. It remains to prove the Lemma 5.9.3.

(a) is easy.

(b) follows from the Lemma 5.9.1.

To prove (c) we will need the following Lemma.

Let C ⟶ S be a family such that a square root Ω^{\frac{1}{2}} exists globally over C. It gives the section ϱ : S ⟶ Pic^{g−1}(C). On the other hand, Pic^{g−1}(C) posesses the canonical theta line bundle O(θ).

5.10.1. Lemma. ϱ^{∗}(O(θ)) = δ^{\frac{1}{2}}

Proof. (J.Harris and T.Pantev) Consider a relative curve C over base B, and a line bundle L over C of relative degree g − 1. This gives the section from B to Pic^{g−1}(C), and we can restrict O(θ) to B. Evidently, the class of O(θ)|_{B} equals the class of divisor D ⊂ B : b ∈ D iff h^{0}(L_{b}) > 0.

On the other hand, one checks readily that O(−D) = [Rp_{C∗}L]^{1}.

We apply this equality to the case B = S, L = Ω^{\frac{1}{2}} . Then [Rp_{C∗}L]^{1} is computed by Mumford formula (see [M2], Theorem 5.10). Namely, we get [Rp_{C∗}Ω^{\frac{1}{2}} ]^{1} = δ^{k} where k = 6(\frac{1}{2})^{2} − 6(\frac{1}{2}) + 1 = −\frac{1}{2}.

We conclude that ϱ^{∗}(O(θ)) = δ^{\frac{1}{2}} . □

5.10.2. Now we are able to prove the Lemma 5.9.3(c).

We can assume that Ω^{\frac{1}{2}} exists globally over C. The corresponding section ϱ : S ⟶ Pic^{g−1}(C) identifies Pic^{g−1}(C) with Pic^{0}(C). We will use this identification, and we will denote the projection Pic^{0}(C) ⟶ S by p.

We have Rp_{∗}(O(θ)) = O and ϱ^{∗}(O(θ)) = δ^{\frac{1}{2}} by the Lemma 5.10.1. We define L_{1} := O(θ) ⊗ p^{∗}δ^{− \frac{1}{2}} .

Then (4)^{∗}L_{1} = L(ℤ, −16). Indeed, the θ-divisor, and hence L_{1} is (−1)-invariant. Thus (2)^{∗}L_{1} and L(ℤ, −4) are the line bundles with the same Chern class, and both are (−1)- invariant. So fiberwise they can differ only by a 2-torsion element. We conclude that L(ℤ, −16) = (2)^{∗}L(ℤ, −4) = (4)^{∗}L_{1} fiberwise. But the restriction of both these line bundles to the 0-section is trivial. Hence they are isomorphic.

Now (c) follows from the Lemma 5.9.1 since obviously p_{∗}L_{1} = δ^{− \frac{1}{2}} . □

5.11. We finish this section with a proof of one simple property of the Heisenberg local system H^{α}, α ∈ ℕ[X_{ℓ}^{∗}]. It states, roughly speaking, that H^{α} depends only on the class of α modulo ℕ[Y_{ℓ}^{∗}]. More precisely, the following Lemma holds.

5.11.1. Lemma. Let α = ∑_{k∈K} a_{k}μ_{k}, α′ = ∑_{k∈K} a_{k}λ_{k} ∈ ℕ[X_{ℓ}^{∗}]. Suppose λ_{k} − μ_{k} ∈ Y_{ℓ}^{∗} for every k ∈ K. Then the natural identification C^{α} = C^{α′} lifts to a canonical isomorphism H^{α} = H^{α′}.

Proof. Let γ ∈ ℕ[Y_{ℓ}^{∗}], γ^{∼} = 0. We define β ∈ ℕ[X_{ℓ}^{∗}] as β = α + γ. We have an obvious projection pr : C^{β} ⟶ C^{α}. It is enough to construct a canonical isomorphism

H^{β} ⟶^{∼} H^{α}.

5.11.2. Consider the addition homomorphism of abelian varieties

a : (Pic(C) ⊗ X_{ℓ}^{∗})_{(2g−2)ν_{0}} × (Pic(C) ⊗ Y_{ℓ}^{∗})_{0} ⟶ (Pic(C) ⊗ X_{ℓ}^{∗})_{(2g−2)ν_{0}}

We will compute the inverse image a^{∗}E.

To this end we introduce the following bilinear form σ on X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}:

σ((x, y_{1}, y_{2}), (x′, y′_{1}, y′_{2})) := −\frac{1}{ℓ}(x · y′_{2} + x′ · y_{2} + y_{1} · y′_{2} + y′_{1} · y_{2} + y_{2} · y′_{2}).

The automorphism (x, y_{1}, y_{2}) ↦ (x, y_{1}, y_{1} + y_{2}) of X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} carries this form into σ′, where

σ′((x, y_{1}, y_{2}), (x′, y′_{1}, y′_{2})) := −\frac{1}{ℓ}(x · y′_{2} + x′ · y_{2} + y_{2} · y′_{2} − y′_{1} · y_{1} − x · y′_{1} − x′ · y_{1}).

Note that σ′ = σ_{1} + σ_{2} where

σ_{1}((x, y_{1}, y_{2}), (x′, y′_{1}, y′_{2})) := −\frac{1}{ℓ}(x · y′_{2} + x′ · y_{2} + y′_{2} · y_{2})

and

σ_{2}((x, y_{1}, y_{2}), (x′, y′_{1}, y′_{2})) := −\frac{1}{ℓ}(−y′_{1} · y_{1} − x · y′_{1} − x′ · y_{1})

5.11.3. We will denote by pr_{12} the projection of (Pic(C)⊗X_{ℓ}^{∗})_{(2g−2)ν_{0}} ×(Pic(C)⊗Y_{ℓ}^{∗})_{0} × (Pic(C)⊗Y_{ℓ}^{∗})_{0} to (Pic(C)⊗X_{ℓ}^{∗})_{(2g−2)ν_{0}} ×(Pic(C)⊗Y_{ℓ}^{∗})_{0} (the product of first two factors).

We also denote by pr_{1} the projection of (Pic(C)⊗X_{ℓ}^{∗})_{(2g−2)ν_{0}} ×(Pic(C)⊗Y_{ℓ}^{∗})_{0} to (Pic(C)⊗ X_{ℓ}^{∗})_{(2g−2)ν_{0}}.

Then we have

a^{∗}E = pr_{12∗}L(X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}, σ, (ν_{0}, 0, 0)) = pr_{12∗}L(X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}, σ′, (ν_{0}, 0, 0)) =
= pr_{12∗}(L(X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}, σ_{1}, (ν_{0}, 0, 0)) ⊗ L(X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}, σ_{2}, (ν_{0}, 0, 0))) =
= pr_{1}^{∗}E ⊗ L(X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}, σ_{2}, (ν_{0}, 0))

In the last line we view σ_{2} as a bilinear form on X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} since anyway it factors through the projection of X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗} to the sum of the first two summands. The last equality follows from the projection formula.

5.11.4. Let us denote the line bundle L(X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}, σ_{2}, (ν_{0}, 0)) by 𝔏. Then we have

det(a^{∗}E) = det(pr_{1}^{∗}E) ⊗ 𝔏^{d{𝔤 \atop ℓ}}

Hence we can define a map

m : (det(pr_{1}^{∗}E))˙ × 𝔏̇ ⟶ (det(a^{∗}E))˙ , (t, u) ↦ t ⊗ u^{d{g \atop ℓ}}.

Recall that ℌ is the universal local system on (det E)˙ . We obviously have the equality

m^{∗}a^{∗}ℌ = pr_{1}^{∗}ℌ

(To be more precise, one could put the exterior tensor product of pr_{1}^{∗}ℌ with the constant sheaf on 𝔏̇ in the RHS).

5.11.5. We are ready to finish the proof of the Lemma. Recall that H^{β} is defined as s_{β}^{∗}AJ_{β}^{∗}ℌ (notations of 3.5). We consider also the Abel-Jacobi map AJ_{α,γ} : C^{β} ⟶ (Pic(C) ⊗ X_{ℓ}^{∗})_{(2g−2)ν_{0}} × (Pic(C) ⊗ Y_{ℓ}^{∗})_{0}. As in loc. cit. we have the canonical section ς of AJ_{α,γ}^{∗}(det(pr_{1}^{∗}E)) and the canonical section s of AJ_{α,γ}^{∗}(det(a^{∗}E)). We have

H^{β} = s^{∗}AJ_{α,γ}^{∗}a^{∗}ℌ

and

pr^{∗}H^{α} = ς^{∗}AJ_{α,γ}^{∗}pr_{1}^{∗}ℌ

It is easy to see that there exists a section 𝔰 of AJ_{α,γ}^{∗}𝔏 such that

m(ς, 𝔰) = s.

Hence H^{β} = s^{∗}AJ_{α,γ}^{∗}a^{∗}ℌ = (ς, 𝔰)^{∗}AJ_{α,γ}^{∗}pr_{1}^{∗}ℌ = ς^{∗}AJ_{α,γ}^{∗}pr_{1}^{∗}ℌ = pr^{∗}H^{α}. This completes the proof of the Lemma. □

6. Factorization isomorphisms

6.1. We need to introduce some more notation. Recall the setup of 2.1. First, we will need the space T C^{τ} containing T C^{∘}^{τ} as an open subset. It is defined in the same way as T C^{∘}^{τ}, only we do not throw away nor the diagonals, neither the zero sections. The space T C^{τ} decomposes into the direct product

T C^{τ} = { \atop T C^{K}} × ∏_{k∈K} T D^{τ^{−1}(k)}

The projection to the first (resp. second) factor will be denoted by pr_{1} (resp. pr_{2}).

We have the natural substitution map T C^{τ} ⟶ T C^{J}. We will denote it by q_{τ} when there

is no risk of confusing it with the same named map T C^{∘}^{τ} ⟶ T C^{∘}^{J}.

The evident projection map T C^{J} ⟶ C^{J} will be denoted by pr. We will use the same

notation for the projections T C^{∘}^{J} ⟶ C^{∘}^{J}, and T C^{∘}^{α} ⟶ C^{∘}^{α} when there is no risk of confusion.

The open embedding C^{∘}^{J} ↪ C^{J} will be denoted by j. We will keep the same notation for the open embedding C^{∘}^{α} ↪ C^{α}.

6.2. Recall that we made a choice in the definition of H^{α}: the isomorphism (287) was defined up to a scalar multiple.

Let us explain how to make a consistent family of choices.

Given a surjection τ : J ⟶ K we denote α_{K} by β for short.

6.2.1. Definition. The isomorphisms θ_{α} : L_{α} ⟶^{∼} AJ_{α}^{∗}(L), and θ_{β} : L_{β} ⟶^{∼} AJ_{β}^{∗}(L) are called compatible if the following diagram commutes:

(π_{J} ∘ σ_{τ})^{∗}L_{α} ⟶^{∼}^{π_{J}^{∗}(θ_{α})} (AJ_{α} ∘ σ_{τ})^{∗}L
∥  ∥
π_{K}^{∗}L_{β}  ⟶^{∼}^{π_{K}^{∗}(θ_{β})}  AJ_{β}^{∗}L

Here the left vertical equality is just the Lemma 4.3.1, and the right vertical equality is tautological since AJ_{β} = AJ_{α} ∘ σ_{τ}.

Clearly, we can choose the isomorphisms (287) for all J in a compatible way. We will assume such a choice made once and for all.

This gives rise to the local system H̃^{α} on L̇_{α}.

6.3. Now we are ready for the construction of factorization isomorphisms.

6.3.1. Let us start with the case of disk.

Given an unfolding π : J ⟶ X_{ℓ}^{∗} we define the meromorphic function F_{J} on T D^{J} as follows. We fix a total order < on J. The standard coordinates on T D^{J} are denoted by (x_{j}, ξ_{j}; j ∈ J). We define

F_{J}(x_{j}, ξ_{j}) = ∏_{π(i)≠π(j),i<j}(x_{i} − x_{j})^{2π(i)·π(j)} × ∏_{π(i)=π(j)}(x_{i} − x_{j})^{π(i)·π(j)} × ∏_{j∈J} ξ_{j}^{2n(π(j))} (298)

The function does not depend on the order on J.

Recall the setup of 2.1.5 for the case C = D. We have the following two functions on T D^{τ}:

F_{J}(q_{τ}), and F_{K}(p_{K}) × ∏_{k∈K} F_{τ^{−1}(k)}  (299)

It is easy to see that the quotient of these two functions T D^{τ} ⟶ ℂ actually lands to ℂ^{∗}; moreover the quotient function T D^{τ} ⟶ ℂ^{∗} is “canonically” homotopic to the constant map T D^{τ} ⟶ 1.

Let Q denote the one dimensional local system on ℂ^{∗} with monodromy ζ around the origin and with fiber at 1 trivialized.

We define I^{J} := F_{J}^{∗}(Q).

Let m : ℂ^{∗}×ℂ^{∗} ⟶ ℂ^{∗} denote the multiplication map. Then m^{∗}(Q) ≅ Q⊠Q canonically.

Hence the homotopy between the functions 299 yields the factorization isomorphisms for the local systems I^{J}.

The associativity of these factorization isomorphisms follows from associativity of the above mentioned homotopies between functions 299.

6.3.2. We return to the case of an arbitrary curve. We denote the “zero section” C^{K} ⟶ T C^{τ} by z.

Consider the following line bundle on T C^{τ}:

L_{τ} = (q_{τ} ∘ pr)^{∗}L_{α}

L̇_{τ} carries the local system H^{τ} := (q_{τ} ∘ pr)^{∗}H̃^{α}.

Since z ∘ q_{τ} ∘ pr = σ_{τ}, we have z^{∗}L_{τ} = L_{β}.

The latter line bundle over C^{K} has a meromorphic section π_{K}^{∗}s_{β} constructed in 3.5. We will extend this section to the whole T C^{τ}.

We do have a meromorphic section s′_{τ} : T C^{τ} ⟶ L_{τ}, s′_{τ} = q_{τ}^{∗}π_{J}^{∗}(s_{α}), where s_{α} : C^{α} ⟶ L_{α} was defined in 3.5.

Recall that in 6.3.1 we have defined the meromorphic function F_{M} on T D^{M} for any unfolding M ⟶ X_{ℓ}^{∗}.

We have

div(s′_{τ}) = pr_{1}^{−1}(div(p_{K}^{∗}π_{K}^{∗}s_{β})) + pr_{2}^{−1}(div( ∏_{k∈K} F_{τ^{−1}(k)}))

We define the meromorphic section

s_{τ} = s′_{τ} ∏_{k∈K} F_{τ^{−1}(k)}^{−1}  (300)

Then div(s_{τ}) = pr_{1}^{−1}(div(p_{K}^{∗}π_{K}^{∗}s_{β})), and it is easy to see that z^{∗}s_{τ} = π_{K}^{∗}s_{β}.

246 ROMAN BEZRUKAVNIKOV, MICHAEL FINKELBERG, AND VADIM SCHECHTMAN

Since pr_{1} is a projection with contractible fibers it follows that restricting to T C^{∘}^{τ} we have

s_{τ}^{∗}(H^{τ}) ≃ pr_{1}^{∗}p_{K}^{∗}π_{K}^{∗}s_{β}^{∗}H̃^{β} = pr_{1}^{∗}p_{K}^{∗}π_{K}^{∗}H^{β}  (301)

6.3.3. Let m : ℂ^{∗} × L_{τ} ⟶ L_{τ} denote the multiplication map. Then m^{∗}H^{τ} = Q ⊠ H^{τ}. So putting all the above together, we get

q_{τ}^{∗}π_{J}^{∗}H^{α} = (s′_{τ})^{∗}H^{τ} = (s_{τ} ∏_{k∈K} F_{τ^{−1}(k)})^{∗}H^{τ} =

= ( ∏_{k∈K} F_{τ^{−1}(k)}; s_{τ})^{∗}m^{∗}H^{τ} = ( ∏_{k∈K} F_{τ^{−1}(k)}; s_{τ})^{∗}(Q ⊠ H^{τ}) =

= ( ∏_{k∈K} F_{τ^{−1}(k)})^{∗}Q ⊠ s_{τ}^{∗}H^{τ} = ( ∏_{k∈K} F_{τ^{−1}(k)})^{∗}Q ⊠ s_{β}^{∗}H^{β} =

= p_{K}^{∗}π_{K}^{∗}H^{α_{K}} ⊠ × _{k∈K} I^{τ^{−1}(k)}

which completes the construction of the factorization isomorphisms. 6.4. The compatibility of the constructed factorization isomorphism for H with the ones for I is immediate.

Chapter 2. The modular property of the Heisenberg local system

7. Degeneration of curves: recollections and notations

7.1. Let M_{g} be the moduli space of curves of genus g. Let 𝔇 be a smooth locus of an irreducible component of the divisor at infinity in the compactification \overline{M_{g}}. It has one of the following types.

(a) Either we have a decomposition g = g_{1} +g_{2}, g_{1}, g_{2} > 0. Then 𝔇 is the moduli space of the following objects: curves C_{1}, C_{2} of genera g_{1}, g_{2} respectively; points x_{1} ∈ C_{1}, x_{2} ∈ C_{2}. Thus, C_{g} degenerates into C_{1} ⊔ C_{2}/(x_{1} = x_{2}).

(b) Or we put g_{0} = g − 1, and then 𝔇 is the moduli space of the following objects: curve C_{0} of genus g_{0} with two distinct points x_{1} ≠ x_{2} ∈ C_{0}. Thus, C_{g} degenerates into C_{0}/(x_{1} = x_{2}).

We denote by C_{𝔇} the universal family over 𝔇.

7.2. Let 𝔇̃ ⟶ 𝔇 denote the moduli space of objects as above plus nonzero tangent vectors v_{1} at x_{1}, and v_{2} at x_{2}. Let T^{•}_{𝔇}M_{g} denote the normal bundle to 𝔇 in \overline{M_{g}} with the

zero section removed. There is the canonical map ℘ : 𝔇̃ → T^{•}_{𝔇}\overline{M_{g}} (see e.g. [BFM], §4).

7.3. Let \overline{C} be the universal curve over \overline{M_{g}}. Let α ∈ ℕ[X_{ℓ}^{∗}].

Consider T^{∘} \overline{C}^{α} → \overline{M_{g}} (see 2.1.2). Note that \overline{C} has singularities (nodes) over \overline{M_{g}} − M_{g}.

To simplify the exposition we just throw these nodal points away. That is, T^{∘} \overline{C}^{α} is formed by configurations of nonsingular points with nonzero tangent vectors.

Let us describe the preimage of 𝔇 in T^{∘} \overline{C}^{α}, to be denoted by T C_{𝔇}^{α}.

In case 7.1(a) T C_{𝔇}^{α} is a union of connected components numbered by the decompositions α = α_{1} + α_{2}. The connected component T C_{𝔇}^{α_{1}α_{2}} is the product C̃_{1}^{α_{1}} × C̃_{2}^{α_{2}}. Here C̃_{r}^{α_{r}} denotes the configurations of α_{r} distinct points with nonzero tangent vectors on C_{r} − x_{r}, r = 1, 2.

In case 7.1(b) T C_{𝔇}^{α} is the space of configurations of α distinct points with nonzero tangent vectors on C_{0} − {x_{1}, x_{2}}.

7.4. Thus we obtain the map ℘_{α} : 𝔇̃ ×_{𝔇} T C_{𝔇}^{α} ⟶ T^{•}_{T C_{𝔇}^{α}} T^{∘} \overline{C}^{α} — the normal bundle to

T C_{𝔇}^{α} in T^{∘} \overline{C}^{α} with the zero section removed.

Note that if we prescribe a weight μ_{1} to the point x_{1}, and a weight μ_{2} to the point x_{2} then in case 7.1(a) we can identify 𝔇̃ ×_{𝔇} T C_{𝔇}^{α_{1}α_{2}} with T^{∘} C_{1}^{α_{1}+μ_{1}} × T^{∘} C_{2}^{α_{2}+μ_{2}} (notations of 2.1.2); in case 7.1(b) we can identify 𝔇̃ ×_{𝔇} T C_{𝔇}^{α} with T^{∘} C_{0}^{α+μ_{1}+μ_{2}} (the exponents are formal sums, and we do not perform symmetrization over μ_{1} and μ_{2} if they happen to be equal or contained in supp α).

7.5. It is known that the determinant line bundle δ_{g} over M_{g} extends to the line bundle \overline{δ_{g}} over \overline{M_{g}}. In case 7.1(a) let ϖ_{i}, i = 1, 2, denote the natural projection from 𝔇 to M_{g_{i}}. In case 7.1(b) let ϖ_{0} denote the natural projection from 𝔇 to M_{g_{0}}. Then \overline{δ_{g}}|_{𝔇} = ϖ_{1}^{∗}δ_{g_{1}} ⊗ ϖ_{2}^{∗}δ_{g_{2}} (resp. \overline{δ_{g}}|_{𝔇} = ϖ_{0}^{∗}δ_{g_{0}}) (see [BFM]). Summing up we obtain the map also denoted by ℘_{α}: in case 7.1(a):

(T^{∘} C_{1}^{α_{1}+μ_{1}} ×_{M_{g_{1}}} δ_{g_{1}}) × (T^{∘} C_{2}^{α_{2}+μ_{2}} ×_{M_{g_{2}}} δ_{g_{2}}) ⟶ T^{•}_{T C_{𝔇}^{α}×\overline{Mg} \overline{δ_{g}}}(T^{∘} \overline{C}^{α} ×\overline{M_{g}} \overline{δ_{g}}); (302)

in case 7.1(b):

T^{∘} C_{0}^{α+μ_{1}+μ_{2}} ×_{M_{g_{0}}} δ_{g_{0}} ⟶ T^{•}_{T C_{𝔇}^{α}×\overline{Mg} \overline{δ_{g}}}(T^{∘} \overline{C}^{α} ×\overline{M_{g}} \overline{δ_{g}}).  (303)

7.6. Consider the specialization SpH_{g}^{α} of the Heisenberg local system along the boundary component T C_{𝔇}^{α} ×\overline{M_{g}} \overline{δ_{g}}. It is a sheaf on T^{•}_{T C_{𝔇}^{α}×\overline{Mg} \overline{δ_{g}}}(T^{∘} \overline{C}^{α} ×\overline{M_{g}} \overline{δ_{g}}). We will describe its inverse image under the map ℘_{α}. Theorem. (a) In case 7.1(a) we have

℘_{α}^{∗}SpH_{g}^{α}|_{(T^{∘} C_{1}^{α_{1}+μ_{1}}×_{Mg_{1}} δ_{g_{1}})×(T^{∘} C_{2}^{α_{2}+μ_{2}}×_{Mg_{2}} δ_{g_{2}})} = H_{g_{1}}^{α_{1}+μ_{1}} ⊠ H_{g_{2}}^{α_{2}+μ_{2}}

for μ_{1} = (2g_{1} − 2)ν_{0} − α_{1}^{∼}, and μ_{2} = (2g_{2} − 2)ν_{0} − α_{2}^{∼} (notations of 2.1.8);

(b) In case 7.1(b) we have

℘_{α}^{∗}SpH_{g}^{α} = ⊕_{μ_{1}+μ_{2}=−2ν_{0}}H_{g_{0}}^{α+μ_{1}+μ_{2}}

Here the sum is taken over the set X_{ℓ}^{∗}/Y_{ℓ}^{∗} (see IV.9.1.2 and 2.1.6). The statement makes sense by the virtue of the Lemma 5.11.1. Thus there are exactly d_{ℓ} summands.

The proof occupies the rest of this Chapter.

8. Proof of Theorem 7.6(a)

We are in the situation of 7.1(a). Let us denote M_{g} ∪ 𝔇 by \overline{M}^{∘} ⊂ \overline{M_{g}}. In this section we will extend the constructions of sections 4 and 5 to the universal family of curves over \overline{M}^{∘}.

8.1. It is well known that Pic(C/M_{g}) extends to the family of algebraic groups over \overline{M}^{∘} which will be denoted by \overline{Pic}(\overline{C}/\overline{M}^{∘}).

Restricting to 𝔇 we get

\overline{Pic}(\overline{C}/\overline{M}^{∘})|_{𝔇} = Pic(C_{1}/𝔇) ×_{𝔇} Pic(C_{2}/𝔇)/⟨(x_{1}) − (x_{2})⟩,  (304)

and Pic(\overline{C}/𝔇) = Pic(C_{1}/𝔇) ×_{𝔇} Pic(C_{2}/𝔇).

In particular, \overline{Pic}(\overline{C}/𝔇) is an extension of ℤ by an abelian scheme.

8.1.1. Recall the notations of 4.1. Given Λ, (,), ν as in 4.1 one constructs the linear bundle L(Λ, (,), ν) over A_{Λ} (see 4.2). To unburden the notations we will denote this line bundle simply by L when it does not cause confusion.

\overline{A}_{Λ} will denote (\overline{Pic}(\overline{C}/\overline{M}^{∘}) ⊗ Λ)_{(2g−2)ν}, and p will denote the projection \overline{A}_{Λ} ⟶ \overline{M}^{∘}.

We have an isomorphism

℧ : A_{Λ}(C_{1}/𝔇) ×_{𝔇} A_{Λ}(C_{2}/𝔇) ⟶^{∼} \overline{A}_{Λ}|_{𝔇}

where

℧(a, b) := (a + x_{1} ⊗ ν, b + x_{2} ⊗ ν) ∈ (Pic(C_{1}/𝔇) ⊗ Λ)_{(2g_{1}−1)ν} ×_{𝔇} (Pic(C_{2}/𝔇) ⊗ Λ)_{(2g_{2}−1)ν} =

= \overline{Pic}(\overline{C}/𝔇) ⊗ Λ)_{(2g−2)ν}

(the latter isomorphism is obtained from (304)).

8.1.2. Lemma. There exists a line bundle \overline{L} on \overline{A}_{Λ} such that \overline{L}|_{p^{−1}(M_{g})} = L, and

℧^{∗}(\overline{L}|_{𝔇}) = L_{C_{1}}^{d_{ℓ}^{g_{2}}} ⊠ L_{C_{2}}^{d_{ℓ}^{g_{1}}}.

Proof. Note that the line bundle L_{α} (see 4.1.2) defined over C^{α} extends to C^{α} ∪ (C_{1} − x_{1})^{α_{1}} × (C_{2} − x_{2})^{α_{2}} ⊂ \overline{C}^{α} by the same formula 4.1.2. Moreover, L_{α} automatically extends to C^{α} ∪ C_{1}^{α_{1}} × C_{2}^{α_{2}} since the complement to C^{α} ∪ (C_{1} − x_{1})^{α_{1}} × (C_{2} − x_{2})^{α_{2}} ⊂ \overline{C}^{α} in C^{α} ∪ C_{1}^{α_{1}} × C_{2}^{α_{2}} has codimension two.

We define the following line bundle on C^{α} ∪ C_{1}^{α_{1}} × C_{2}^{α_{2}}:

\overline{L}_{α} := L_{α}(b(μ_{1})(C_{1}^{α_{1}} × C_{2}^{α_{2}}))  (305)

Here C_{1}^{α_{1}} × C_{2}^{α_{2}} is viewed as a divisor on C^{α} ∪ C_{1}^{α_{1}} × C_{2}^{α_{2}}. Note also that b(μ_{1}) = b(μ_{2}) since μ_{1} + μ_{2} = −2ν.

The line bundle L on A_{Λ} extends uniquely to a line bundle \overline{L} on \overline{A}_{Λ} in such a way that

AJ^{∗}\overline{L} = \overline{L}_{α}.  (306)

In effect, any two extensions of L to \overline{A}_{Λ} differ by the twist by some power of O(p^{−1}𝔇) — the line bundle lifted from \overline{M}^{∘}. But evidently, AJ^{∗}O(p^{−1}𝔇) = O(C_{1}^{α_{1}} × C_{2}^{α_{2}}), so we can choose our twist uniquely to satisfy (306).

8.1.3. Let us choose a basis I of Λ. Let us fix α_{1}, α_{2} ∈ ℕ[Λ]; α_{1} = ∑_{i∈I} b_{i}i + ∑_{i∈I} b_{−i}(−i), α_{2} = ∑_{i∈I} c_{i}i + ∑_{i∈I} c_{−i}(−i) such that b_{i}, b_{−i} > g_{1}; c_{i}, c_{−i} > g_{2}.

We have the closed embeddings σ_{1} : C_{1}^{α_{1}} ↪ C_{1}^{α_{1}+μ_{1}} (resp. σ_{2} : C_{2}^{α_{2}} ↪ C_{2}^{α_{2}+μ_{2}}) adding to a configuration on C_{1} (resp. C_{2}) μ_{1} copies of x_{1} (resp. μ_{2} copies of x_{2}).

It is clear that AJ_{1} ∘σ_{1} (resp. AJ_{2} ∘σ_{2}) induces injection from Pic(A_{1Λ}/𝔇) to Pic(C_{1}^{α_{1}}/𝔇) (resp. from Pic(A_{2Λ}/𝔇) to Pic(C_{2}^{α_{2}}/𝔇)).

Hence it is enough to check that

(℧ ∘ (AJ_{1} × AJ_{2}) ∘ (σ_{1} × σ_{2}))^{∗}\overline{L}|_{p^{−1}𝔇} = σ_{1}^{∗}L_{α_{1}+μ_{1}}^{d_{ℓ}^{g_{2}}} ⊠ σ_{2}^{∗}L_{α_{2}+μ_{2}}^{d_{ℓ}^{g_{1}}}  (307)

Consider unfoldings π_{1} : J_{1} ⟶ Λ of α_{1}, and π_{2} : J_{2} ⟶ Λ of α_{2}. It is enough to check that

(π_{1} × π_{2})^{∗}(℧ ∘ (AJ_{1} × AJ_{2}) ∘ (σ_{1} × σ_{2}))^{∗}\overline{L}|_{p^{−1}𝔇} = (π_{1} × π_{2})^{∗}(σ_{1}^{∗}L_{α_{1}+μ_{1}}^{d_{ℓ}^{g_{2}}} ⊠ σ_{2}^{∗}L_{α_{2}+μ_{2}}^{d_{ℓ}^{g_{1}}}) (308)

One checks readily that the isomorphism (308) holds after restriction to (C_{1} − x_{1})^{J_{1}} × (C_{2} − x_{2})^{J_{2}}.

The necessary condition to extend the isomorphism across the complement divisor is the equality of (relative) first Chern classes:

c_{1}((℧ ∘ (AJ_{1} × AJ_{2}) ∘ (σ_{1} × σ_{2}))^{∗}\overline{L}|_{p^{−1}𝔇}) = c_{1}(σ_{1}^{∗}L_{α_{1}+μ_{1}}^{d_{ℓ}^{g_{2}}} ⊠ σ_{2}^{∗}L_{α_{2}+μ_{2}}^{d_{ℓ}^{g_{1}}})

This is immediate since c_{1}(℧^{∗}(\overline{L}_{𝔇})) = c_{1}(L_{C_{1}}^{d_{ℓ}^{g_{2}}}) ⊠ c_{1}(L_{C_{2}}^{d_{ℓ}^{g_{1}}}). If the complement divisor were irreducible, this condition would be sufficient as well. If the divisor is reducible, one has to check that the fundamental classes of different irreducible components are linearly independent. All the irreducible components of our divisor are diagonals z_{j} = x_{r}, r = 1, 2; j ∈ J_{r}. Their fundamental classes are linearly independent in H^{2}(C_{1}^{J_{1}} × C_{2}^{J_{2}}). This completes the proof of the Lemma. □

8.2. We proceed with the proof of the Theorem 7.6(a). Let us consider the line bundle \overline{L} := \overline{L}(X_{ℓ}^{∗}, \frac{d_{ℓ}^{g}}{ℓ}?·?, ν_{0}) on \overline{A} := \overline{A}_{X_{ℓ}^{∗}}. On the other hand, let us consider the triple Λ, (,), ν defined in 5.1, and the corresponding line bundle \overline{L}′ := \overline{L}(Λ, (,), ν) on \overline{A}′ := \overline{A}_{X_{ℓ}^{∗}⊕Y_{ℓ}^{∗}}. We denote by 𝔭 (resp. 𝔭_{r}, r = 1, 2) the obvious projection \overline{A}′ ⟶ \overline{A} (resp. A′(C_{r}) ⟶ A(C_{r})). We define \overline{E} := 𝔭_{∗}\overline{L}′, E_{C_{r}} := 𝔭_{r∗}L′_{C_{r}}. It is obvious that

\overline{E}|_{p^{−1}(𝔇)} = (℧^{−1})^{∗}(E_{C_{1}} ⊠ E_{C_{2}})  (309)

and hence

det(\overline{E})|_{p^{−1}(𝔇)} = (℧^{−1})^{∗}((det(E_{C_{1}}))^{rk(E_{C_{2}})} ⊠ (det(E_{C_{2}}))^{rk(E_{C_{1}})}) = = (℧^{−1})^{∗}((det(E_{C_{1}}))^{d{g_{2} \atop ℓ}} ⊠ (det(E_{C_{2}}))^{d{g_{1} \atop ℓ}}) (310)

Recall that the line bundle δ_{g} on M_{g} extends to the line bundle \overline{δ_{g}} on \overline{M_{g}}, and \overline{δ_{g}}|_{𝔇} = δ_{g_{1}} ⊠ δ_{g_{2}}. Proposition. det(\overline{E}) = \overline{L}⊗(p^{∗}\overline{δ_{g}})^{d_{ℓ}^{g}(− \frac{1}{2} rkX_{ℓ}^{∗}+6k \frac{ν_{0}·ν_{0}}{ℓ})} where p denotes the projection \overline{A} ⟶ \overline{M}^{∘}. Proof. If we restrict both sides to p^{−1}M_{g} ⊂ p^{−1}\overline{M}^{∘} then we just have the Theorem 5.2. On the other hand, it is easy to see that the normal line bundle to 𝔇 in \overline{M}^{∘} is nontrivial. This means that two extensions of a line bundle from p^{−1}M_{g} to p^{−1}\overline{M}^{∘} coincide iff their restrictions to p^{−1}𝔇 coincide. So it remains to compare the LHS and RHS restricted to p^{−1}𝔇. By the virtue of (310) above we have

det(\overline{E})|_{p^{−1}(𝔇)} = (℧^{−1})^{∗}((det(E_{C_{1}}))^{d{g_{2} \atop ℓ}} ⊠ (det(E_{C_{2}}))^{d{g_{1} \atop ℓ}}) =

= (℧^{−1})^{∗}((L_{C_{1}}^{d_{ℓ}^{g_{2}}} ⊠ L_{C_{2}}^{d_{ℓ}^{g_{1}}}) ⊗ (δ_{g_{1}}^{d_{ℓ}^{g}(− \frac{1}{2} rkX_{ℓ}^{∗}+6 \frac{ν_{0}·ν_{0}}{ℓ})} ⊠ δ_{g_{2}}^{d_{ℓ}^{g}(− \frac{1}{2} rkX_{ℓ}^{∗}+6 \frac{ν_{0}·ν_{0}}{ℓ})})) =
= \overline{L}|_{p^{−1}(𝔇)} ⊗ p^{∗}\overline{δ}^{d_{ℓ}^{g}(− \frac{1}{2} rkX_{ℓ}^{∗}+6 \frac{ν_{0}·ν_{0}}{ℓ})}

Here the second equality follows from the Theorem 5.2, and the third one follows from the Lemma 8.1.2. This completes the proof of the Proposition. □

252 ROMAN BEZRUKAVNIKOV, MICHAEL FINKELBERG, AND VADIM SCHECHTMAN

8.3. We are ready to finish the proof of the Theorem 7.6(a). 8.3.1. Let N be the standard deformation of T^{∘} \overline{C}^{α}×\overline{M_{g}} \overline{δ_{g}} to the normal cone of T C_{𝔇}^{α}×\overline{M_{g}} \overline{δ_{g}} (see e.g. [Fu]). So we have

T_{T C_{𝔇}^{α}×\overline{Mg} \overline{δ_{g}}}(T^{∘} \overline{C}^{α} ×\overline{M_{g}} \overline{δ_{g}}) ↪ N ↩ T^{∘} \overline{C}^{α} ×\overline{M_{g}} \overline{δ_{g}} × ℂ^{∗}

Let t : N ⟶ ℂ; pr : N ⟶ T^{∘} \overline{C}^{α} ×\overline{M_{g}} \overline{δ_{g}} denote the canonical projections. We define an open subset U ⊂ N as

U := (T^{∘} C^{α} ×_{M_{g}} δ̇) × ℂ^{∗} ⊔ T^{∘}_{T C_{𝔇}^{α}×\overline{Mg} \overline{δ_{g}}}(T^{∘} \overline{C}^{α} ×\overline{M_{g}} (\overline{δ_{g}})˙ )

To finish the proof it is enough to construct a local system H_{N} on U such that

H_{N} |_{(T^{∘} C^{α}×_{Mg} δ̇_{g})×ℝ^{>0}} = H_{g}^{α};  (311)
℘_{α}^{∗}(H_{N} |_{(T^{∘} C_{1}^{α_{1}+μ_{1}}×_{𝔇} δ̇_{g_{1}})×(T^{∘} C_{2}^{α_{2}+μ_{2}}×_{𝔇} δ̇_{g_{2}})}) = H_{g_{1}}^{α_{1}+μ_{1}} ⊠ H_{g_{2}}^{α_{2}+μ_{2}}.  (312)

Recall the notations of 3.5. The canonical meromorphic section s_{α} of AJ_{α}^{∗}(det(E)) extends to the same named meromorphic section of AJ_{α}^{∗}(det(\overline{E})) over T \overline{C}^{α} ×\overline{M_{g}} (\overline{δ_{g}})˙ . 8.3.2. Lemma. The section s′_{α} := t^{−^{\frac{d{g \atop ℓ}}{ℓ}} n(μ_{1})}pr^{∗}s_{α} is regular on U, and we have

℘_{α}^{∗}(s′_{α}|_{p^{−1}(𝔇)}) = s_{α_{1}}^{d_{ℓ}^{g_{2}}} ⊗ s_{α_{2}}^{d_{ℓ}^{g_{1}}}

(see (310)) Proof. Immediate from the Proposition 8.2 and the construction of \overline{L} in 8.1.2. Note also that n(μ_{1}) = n(μ_{2}). □ 8.3.3. Remark. If one of g_{1}, g_{2} is equal to 1, then the following remark is in order. Suppose, say, g_{1} = 1. Then N = d_{ℓ}^{g_{2}} satisfies the assumptions of 3.6, and s_{α_{1}}^{d_{ℓ}^{g_{2}}} should be understood as the canonical section of AJ_{α_{1}}^{∗}(det(E_{C_{1}}))^{⊗d_{ℓ}^{g_{2}}} arising from the Theorem 5.2(d).

8.3.4. We construct the local system \overline{ℌ} on (det(E))˙ as in 5.2.1. Now we define

H_{N} := (s′_{α})^{∗}\overline{ℌ}

The property (311) is clear. Let us prove the property (312). It is clear that the isomorphism (309) is the isomorphism of bundles with flat projective connections. Suppose g_{1}, g_{2} > 1. Let us define a map

˜℧ : (det(E_{C_{1}}))˙ ×_{𝔇} (det(E_{C_{1}}))˙ ⟶ (det(\overline{E}|_{𝔇}))˙

as a composition of

(det(E_{C_{1}}))˙ ×_{𝔇} (det(E_{C_{1}}))˙ ⟶ (det(E_{C_{1}})^{⊗d_{ℓ}^{g_{2}}})˙ ⊗ (det(E_{C_{1}})^{⊗d_{ℓ}^{g_{1}}})˙ , (λ_{1}, λ_{2}) ↦ λ_{1}^{d_{ℓ}^{g_{2}}} ⊗ λ_{2}^{d_{ℓ}^{g_{1}}}, the isomorphism (310), and the obvious map (℧^{∗}(det(\overline{E}|_{𝔇})))˙ ⟶ (det(\overline{E}|_{𝔇}))˙ . Then we see that

˜℧^{∗}(\overline{ℌ}|_{𝔇}) = ℌ_{C_{1}} ⊠ ℌ_{C_{2}}

The property (312) follows. This completes the proof of 7.6 in case g_{1}, g_{2} > 1. The minor changes in case one of g_{1}, g_{2} equals 1 are safely left to the interested reader. □

9. Proof of Theorem 7.6(b)

We start the proof with the following weaker statement.

9.1. Proposition. ℘_{α}^{∗}SpH_{g}^{α} = ⊕_{μ_{1}+μ_{2}=−2ν_{0}}H_{g_{0}}^{α+μ_{1}+μ_{2}} ⊗ P_{μ_{1}} for some one-dimensional local systems P_{μ_{1}}.

9.2. To prove the proposition, let us introduce some notations. We denote M_{g} ∪ 𝔇 by \overline{M}^{∘} ⊂ \overline{M_{g}}. It is well known that Pic(C/M_{g}) extends to the family of algebraic groups Pic(\overline{C}/\overline{M}^{∘}) over \overline{M}^{∘}. Restricting to 𝔇 we get the projection

π : Pic(\overline{C}/\overline{M}^{∘})|_{𝔇} ⟶ Pic(C_{0}/𝔇)

which identifies Pic(\overline{C}/\overline{M}^{∘})|_{𝔇} with a ℂ^{∗}-torsor over Pic(C_{0}/𝔇). The class of this torsor is equal to (x_{1}) − (x_{2}) ∈ Pic^{0}(C_{0}/𝔇) = Pic^{0}(Pic^{n}(C_{0}/𝔇)) for any n ∈ ℤ. For any μ_{1}, μ_{2} ∈ X_{ℓ}^{∗} such that μ_{1} + μ_{2} = −2ν_{0} we have a subtraction map r_{μ_{1},μ_{2}} : (Pic(C_{0}/𝔇) ⊗ X_{ℓ}^{∗})_{(2g−2)ν_{0}} ⟶ (Pic(C_{0}/𝔇) ⊗ X_{ℓ}^{∗})_{(2g_{0}−2)ν_{0}},

x ↦ x − μ_{1} ⊗ x_{1} − μ_{2} ⊗ x_{2} (313)

Recall that in 5.2 we have defined for any curve C of genus g a vector bundle with flat projective connection E on (Pic(C) ⊗ X_{ℓ}^{∗})_{(2g−2)ν_{0}}. For the curve C_{0} we will denote this bundle by E_{C_{0}}.

We define the vector bundle E_{μ_{1}} := r_{μ_{1},μ_{2}}^{∗}E_{C_{0}} on (Pic(C_{0}/𝔇) ⊗ X_{ℓ}^{∗})_{(2g−2)ν_{0}}. It carries a flat projective connection.

If μ′_{1} − μ_{1} ∈ Y_{ℓ}^{∗} then the vector bundles E_{μ_{1}} and E_{μ′_{1}} differ by a twist by a line bundle, and the corresponding flat projective connections are identified by the Lemma 5.11.1. In particular, the locally constant sheaf of algebras \underline{End}(E_{μ_{1}}) canonically depends only on the class of μ_{1} modulo Y_{ℓ}^{∗}. We will use the notation \underline{End}(E_{μ}) for μ ∈ X_{ℓ}^{∗}/Y_{ℓ}^{∗} from now on.

9.3. The plan of the proof is as follows.

Consider the sheaf of algebras \underline{End}(E) on (Pic(C) ⊗ X_{ℓ}^{∗})_{(2g−2)ν_{0}}. The data of a flat projective connection on E is equivalent to the data of a flat genuine connection on \underline{End}(E) preserving the algebra structure.

Let U be a small (punctured) neighbourhood (in classical topology) of p^{−1}(𝔇) in (Pic(C)⊗ X_{ℓ}^{∗})_{(2g−2)ν_{0}}. The monodromy around p^{−1}(𝔇) acts on ℌ, or, equivalently, on E, semisimply. It acts on the sheaf of algebras \underline{End}(E)|_{U} by the algebra automorphism

𝔪 : \underline{End}(E) ⟶ \underline{End}(E)

We will construct a locally constant sheaf of algebras 𝔐 ⊂ \underline{End}(E) with the following properties.

9.3.1. (i) 𝔐 can be extended to a locally constant sheaf of algebras \overline{𝔐} on (Pic(\overline{C}/\overline{M}^{∘})⊗ X_{ℓ}^{∗})_{(2g−2)ν_{0}}.

(ii) The sheaf E|_{U} decomposes into direct sum of d_{ℓ} eigensheaves of 𝔪 : E|_{U} = ⊕_{λ}E_{λ}; and 𝔐 = ⊕_{λ}\underline{End}(E_{λ}).

(iii) \overline{𝔐}|_{p^{−1}(𝔇)} = ⊕_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} \underline{End}(E_{μ}).

9.3.2. The construction and the proof of the above properties will occupy the rest of this section. Let us derive the Proposition 9.1 from these properties.

The decomposition (ii) of E|_{U} induces a decomposition: Spℌ_{g} = ⊕_{μ}ℌ_{μ}. It follows from (ii),(iii) that there is a bijection between the set of summands and X_{ℓ}^{∗}/Y_{ℓ}^{∗} such that \overline{𝔐}|_{p^{−1}(𝔇)} = ⊕_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} \underline{End}(ℌ_{μ}). Moreover, \underline{End}(ℌ_{μ}) = \underline{End}(E_{μ}).

Now for any g-admissible α ∈ X_{ℓ}^{∗} we have an isomorphism of local systems

⨁_{μ_{1}+μ_{2}=−2ν_{0}}^{μ_{1},μ_{2}∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} \underline{End}(H_{g_{0}}^{α+μ_{1}+μ_{2}}) = ⨁_{μ_{1}+μ_{2}=−2ν_{0}}^{μ_{1},μ_{2}∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} s_{α+μ_{1}+μ_{2}}^{∗}AJ_{α+μ_{1}+μ_{2}}^{∗}\underline{End}(E_{μ_{1}}) =

= ⨁_{μ_{1}+μ_{2}=−2ν_{0}}^{μ_{1},μ_{2}∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} s_{α+μ_{1}+μ_{2}}^{∗}AJ_{α+μ_{1}+μ_{2}}^{∗}\underline{End}(ℌ_{μ_{1}})

Hence s_{α+μ_{1}+μ_{2}}^{∗}AJ_{α+μ_{1}+μ_{2}}^{∗}ℌ_{μ_{1}} = H_{g_{0}}^{α+μ_{1}+μ_{2}} ⊗ P_{μ_{1}} for some one-dimensional local system P_{μ_{1}}.

On the other hand, obviously,

⨁_{μ_{1}+μ_{2}=−2ν_{0}}^{μ_{1},μ_{2}∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} s_{α+μ_{1}+μ_{2}}^{∗}AJ_{α+μ_{1}+μ_{2}}^{∗}ℌ_{μ_{1}} = s_{α}^{∗}AJ_{α}^{∗}Spℌ_{g} = SpH_{g}^{α}

This completes the proof of the Proposition 9.1. □

9.4. Before we prove the properties 9.3.1(i)–(iii) we need to introduce the following general construction.

Let p_{𝔄} : 𝔄 ⟶ S be an abelian scheme over a base S. Let L be a relatively ample line bundle on 𝔄. It defines a morphism ϕ_{L} from 𝔄 to the dual abelian variety 𝔄̌ : a ↦ T_{a}L ⊗ L^{−1} where T_{a} denotes the translation by a. Let i_{G} : G ↪ 𝔄 be the kernel of this homomorphism. Then p_{G} : G ⟶ S is an etale cover.

Let L_{♡} denote the line bundle L ⊗ p_{𝔄}^{∗}0^{∗}L^{−1} where 0 stands for the zero section S ⟶ 𝔄. Recall that L̇ denotes the ℂ^{∗}-torsor corresponding to the line bundle L.

9.4.1. The sheaf p_{G∗}i_{G}^{∗}L̇_{♡} has a natural structure of a group scheme acting on the sheaf p_{𝔄∗}L. There is an exact sequence

0 ⟶ O^{∗} ⟶ p_{G∗}i_{G}^{∗}L̇_{♡} ⟶^{𝔭} G ⟶ 0

The subsheaf O^{∗} acts on p_{𝔄∗}L by multiplications with functions. The action of an element h ∈ p_{G∗}i_{G}^{∗}L̇_{♡} covers the translation by 𝔭(h) ∈ G ⊂ 𝔄.

The group scheme p_{G∗}i_{G}^{∗}L̇_{♡} carries a unique flat connection compatible with multiplication. In effect, the elements of finite order form a Zariski dense union of etale group subschemes.

9.4.2. The sheaf p_{G∗}i_{G}^{∗}L_{♡} has a natural structure of algebra acting on the sheaf p_{𝔄∗}L. The natural inclusion p_{G∗}i_{G}^{∗}L̇_{♡} ↪ p_{G∗}i_{G}^{∗}L_{♡} is multiplicative and compatible with the action on p_{𝔄∗}L. The sheaf of algebras p_{G∗}i_{G}^{∗}L_{♡} carries a unique flat connection compatible with the one on p_{G∗}i_{G}^{∗}L̇_{♡}.

The map

p_{G∗}i_{G}^{∗}L_{♡} ⟶^{∼} \underline{End}(p_{𝔄∗}L)  (314)

is an isomorphism.

9.4.3. Remark. Note that the above construction gives an alternative way to define the flat projective connection on p_{𝔄∗}L — without referring to [BK].

9.5. We apply the above construction to the following situation. We take S = (Pic(C/U) ⊗ X_{ℓ}^{∗})_{(2g−2)ν_{0}}; 𝔄 = (Pic(C/U) ⊗ (X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}))_{(2g−2)ν_{0},0}; π = pr_{1}; L = L′ as in 5.1.

Then G = p_{(Pic(C/U)⊗X_{ℓ}^{∗})_{(2g−2)ν_{0}}}^{∗}R^{1}p_{C∗}(\underline{X_{ℓ}^{∗}/Y_{ℓ}^{∗}}).

Recall that by the Picard-Lefschetz theory the monodromy 𝔪 acts on R^{1}p_{C∗}(\underline{ℤ}) by the formula

y ↦ y ± ⟨y, x⟩x

where x is the vanishing cycle. In particular, the invariants of 𝔪 in R^{1}p_{C∗}(\underline{ℤ}) form a sublattice of codimension 1, namely, the orthogonal complement of the vanishing cycle.

We define 𝔐 ⊂ \underline{End}(E) = 𝔭_{𝔊∗}𝔦_{𝔊}^{∗}L′_{♡} as follows:

𝔐 := 𝔭_{𝔊^{𝔪}∗}𝔦_{𝔊^{𝔪}}^{∗}L′_{♡}

where G^{𝔪} stands for the invariants of 𝔪 on G.

9.6. Let us check the properties 9.3.1.

The sheaf G^{𝔪} = (X_{ℓ}^{∗}/Y_{ℓ}^{∗}) ⊗ (R^{1}p_{C∗}(\underline{ℤ}))^{𝔪} extends to a sheaf \overline{G} on \overline{M}^{∘}. Moreover, we have an embedding

i\overline{G} : \overline{G} ↪ (Pic(\overline{C}/\overline{M}^{∘}) ⊗ Y_{ℓ}^{∗})_{0}

Let us choose some extension \overline{L′} of L′ to (Pic(\overline{C}/\overline{M}^{∘}) ⊗ (X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}))_{(2g−2)ν_{0},0}. Recall that \overline{L′}_{♡} denotes \overline{L′} ⊗ p^{∗}0^{∗}\overline{L′}^{−1}. We define

\overline{𝔐} := p_{\overline{G}∗}i∗ ; \overline{G}\overline{L′}_{♡}

Evidently, \overline{𝔐}|_{U} = 𝔐.

It is easy to see that the algebra structure on 𝔐 extends uniquely to \overline{𝔐}. This proves (i).

Property (i) implies that 𝔐 ⊂ (\underline{End}(E))^{𝔪}. On the other hand one can check that there exists a unique decomposition of E in the sum of d_{ℓ} summands of equal dimension: E = ⊕E_{λ} such that 𝔐 = ⊕\underline{End}(E_{λ}) ⊂ \underline{End}(E). Since 𝔪 commutes with 𝔐 it follows that each E_{λ} is an eigenspace of 𝔪. (ii) is proved.

9.7. It remains to check 9.3.1(iii). Recall that for μ_{1}, μ_{2} ∈ X_{ℓ}^{∗} such that μ_{1} + μ_{2} = −2ν_{0} we have introduced the substraction map r_{μ_{1},μ_{2}} in (313).

Let us denote the line bundle introduced in 5.1 for the curve C_{0} by L′_{C_{0}}.

9.7.1. Lemma. For any μ_{1}, μ_{2} ∈ X_{ℓ}^{∗} such that μ_{1} + μ_{2} = −2ν_{0} we have a canonical isomorphism

\overline{L′}_{♡}|_{p^{−1}(𝔇)} = (r_{μ_{1},μ_{2}} × id)^{∗}(L′_{C_{0}})_{♡}

Proof. Just as in the proof of the Lemma 8.1.2 we define the extension of the line bundle L_{α} on C^{α} to the line bundle \overline{L}_{α} on C^{α} ∪ (C_{0} − x_{1} − x_{2})^{α} for any α ∈ ℕ[X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}]. The extension is given by the same formula (305).

We necessarily have

AJ_{α}^{∗}\overline{L′} = \overline{L}_{α}(n(C_{0} − x_{1} − x_{2})^{α})

for some n ∈ ℤ where (C_{0} − x_{1} − x_{2})^{α} is viewed as a divisor in \overline{C}^{α}.

On the other hand,

\overline{L}_{α}|_{p^{−1}(𝔇)} = AJ_{α}^{∗}π^{∗}r_{μ_{1},μ_{2}}^{∗}(L′_{C_{0}}) ⊗ N^{m}

for some m ∈ ℤ where N denotes the normal line bundle to (C_{0} − x_{1} − x_{2})^{α} in C^{α} ∪ (C_{0} − x_{1} − x_{2})^{α}.

The line bundle N is lifted from the base 𝔇. Hence (L ⊗ N)_{♡} = L_{♡} for any line bundle L on (Pic(C_{0}/𝔇) ⊗ (X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}))_{(2g−2)ν_{0},0}.

Finally, two line bundles on (Pic(C_{0}/𝔇) ⊗ (X_{ℓ}^{∗} ⊕ Y_{ℓ}^{∗}))_{(2g−2)ν_{0},0} are isomorphic iff their inverse images with respect to AJ_{α} are isomorphic for any α.

This completes the proof of the Lemma. □

9.8. Now we can use the above Lemma together with (314) to define for any μ ∈ X_{ℓ}^{∗}/Y_{ℓ}^{∗} the morphism of algebras

φ_{μ} : \overline{𝔐}|_{p^{−1}(𝔇)} ⟶ \underline{End}(E_{μ})

It remains to show that the morphism

φ := ∑_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} φ_{μ} : \overline{𝔐}|_{p^{−1}(𝔇)} ⟶ ⨁_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} \underline{End}(E_{μ})

is an isomorphism. It is enough to prove that φ is surjective.

Since each morphism φ_{μ} is obviously surjective, it suffices to check that all the idempotents of ⨁_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} \underline{End}(E_{μ}) lie in the image of φ.

Consider the subgroup Z ⊂ \overline{G}|_{p^{−1}(𝔇)},

Z := Ker π ∩ \overline{G}|_{p^{−1}(𝔇)}

We have Z ≃ \underline{X_{ℓ}^{∗}/Y_{ℓ}^{∗}}.

Clearly, the morphism φ_{μ} maps p_{Z∗}i_{Z}^{∗} \overline{L′}_{♡} ⊂ p_{\overline{G}∗}i∗ ; \overline{G}\overline{L′}_{♡} to the one-dimensional subspace of \underline{End}(E_{μ}) generated by id. Let us denote the corresponding linear functional on p_{Z∗}i_{Z}^{∗} \overline{L′}_{♡} by ℵ_{μ}.

It is enough to prove that the set of functionals {ℵ_{μ}, μ ∈ X_{ℓ}^{∗}/Y_{ℓ}^{∗}} is linearly independent.

Note that the quadratic form ?·? on X_{ℓ}^{∗} induces a perfect pairing (X_{ℓ}^{∗}/Y_{ℓ}^{∗})×(X_{ℓ}^{∗}/Y_{ℓ}^{∗}) ⟶ ℚ/lℤ, and thus identifies the finite abelian group X_{ℓ}^{∗}/Y_{ℓ}^{∗} with its dual (X_{ℓ}^{∗}/Y_{ℓ}^{∗})^{∨}.

Since p_{Z∗}i_{Z}^{∗} \overline{L′}_{♡} is X_{ℓ}^{∗}/Y_{ℓ}^{∗}-graded, the group (X_{ℓ}^{∗}/Y_{ℓ}^{∗})^{∨} acts on it by multiplication, and the corresponding module is isomorphic to the dual regular representation B[X_{ℓ}^{∗}/Y_{ℓ}^{∗}].

Consider μ = 0 ∈ X_{ℓ}^{∗}/Y_{ℓ}^{∗}. The functional ℵ_{0} does not vanish on any graded component of p_{Z∗}i_{Z}^{∗} \overline{L′}_{♡}. Hence ℵ_{0} generates (p_{Z∗}i_{Z}^{∗} \overline{L′}_{♡})^{∨} as a (X_{ℓ}^{∗}/Y_{ℓ}^{∗})^{∨}-module. On the other hand, it is easy to see that for any λ ∈ (X_{ℓ}^{∗}/Y_{ℓ}^{∗})^{∨} we have λ(ℵ_{0}) = ℵ_{λ}.

This completes the proof of the property 9.3.1(iii) along with the statement the Proposition 9.1. □

9.9. Now we will derive the Theorem 7.6(b) from the Proposition 9.1.

Given μ ∈ X_{ℓ}^{∗}/Y_{ℓ}^{∗} we will denote −2ν_{0} − μ by μ′.

We have proved that SpE = ⊕_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}}E_{μ}, and the flat projective connections on E, E_{μ} agree (note that the sum of bundles with flat projective connections does not have a natural projective connection).

Recall that in order to define the local system H^{α+μ+μ′} we have used the canonical section s_{α+μ+μ′} of the line bundle AJ_{α+μ+μ′}^{∗} det(E_{μ}).

The Theorem follows immediately from the next two statements.

9.10. Lemma. The canonical section s_{α} of AJ_{α}^{∗} det(E) can be extended to a section \overline{s}_{α} over \overline{M}^{∘} in such way that

\overline{s}_{α}|_{p^{−1}(𝔇)} = ∏_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} s_{α+μ+μ′}

The proof is completely parallel to the one of 8.3.2. □

9.11. We consider ⊕_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}}E_{μ} as a projectively flat bundle via the isomorphism with SpE. Then ⊕_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} det(E_{μ}) is a direct summand of an exterior power of ⊕_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}}E_{μ} and thus inherits a flat projective connection.

Proposition. The collection (s_{α+μ+μ′}, μ ∈ X_{ℓ}^{∗}/Y_{ℓ}^{∗}) forms a projectively flat section of ⊕_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} det(E_{μ}).

9.12. Proof. The flat projective connection on ⊕_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}} det(E_{μ}) induces a flat genuine connection on det(E_{μ}) ⊗ det(E_{λ})^{−1} for any μ, λ ∈ X_{ℓ}^{∗}/Y_{ℓ}^{∗}. We have to prove that s_{μ}s_{λ}^{−1} is a flat section of det(E_{μ}) ⊗ det(E_{λ})^{−1}.

It suffices to prove that s_{μ}^{N}s_{λ}^{−N} is a flat section of (det(E_{μ})⊗det(E_{λ})^{−1})^{N} for some positive integer N.

This follows immediately from the next two Lemmas.

9.12.1. Lemma. There exists N > 0 such that the one-dimensional local system (det(E_{μ})⊗ det(E_{λ})^{−1})^{N} is trivial for any μ, λ ∈ X_{ℓ}^{∗}/Y_{ℓ}^{∗}.

9.12.2. Lemma. Any invertible function on T^{∘} C_{0}^{α+μ+λ} ×_{M_{g_{0}}} δ̇ is constant if g_{0} > 1.

9.12.3. We will prove the Lemmas in the following subsections. Let us derive the Proposition now. First of all, if g_{0} = 0 there is nothing to prove. The case g_{0} = 1 is left to the reader. Otherwise, we conclude that s_{μ}^{N}s_{λ}^{−N} being invertible over T^{∘} C_{0}^{α+μ+λ} ×_{M_{g_{0}}} δ̇ is a constant function, and thus a flat section of a trivial local system (det(E_{μ})⊗det(E_{λ})^{−1})^{N}. □

9.13. To prove the Lemma 9.12.1 it is enough to show that the image of the monodromy representation of the fundamental group π_{1}(T^{•}_{T C_{𝔇}^{α}×\overline{Mg} \overline{δ_{g}}}(T^{∘} \overline{C}^{α} ×\overline{M_{g}} \overline{δ_{g}})) (see (303))

in P GL(⊕_{μ∈X_{ℓ}^{∗}/Y_{ℓ}^{∗}}E_{μ}) is finite.

To this end it suffices to show that the image of π_{1}(T^{∘} C^{α}) in P GL(E) is finite.

Recall the notations of 9.4, 9.5. Let us choose a finite subgroup G̃ ⊂ p_{G∗}i_{G}^{∗}L̇′_{♡} projecting surjectively onto G and such that the kernel of this projection contains at least three elements.

Then it follows from the discussion in 9.4 that the action of π_{1}(T^{∘} C^{α}) on E factors through the group Aut(G̃).

This proves the Lemma 9.12.1. □

9.14. Let us prove the Lemma 9.12.2. To unburden the notation we will omit the subindex _{0}. Assume there is a nonconstant invertible function. We can lift it to the nonsymmetrized Cartesian power and obtain a nonconstant invertible function on (T^{•} C)^{J} ×_{M_{g}} δ̇ for some set J of positive cardinality n.

Let us choose a subset K = J − j of cardinality n − 1 and consider the projection

pr : (T^{•} C)^{J} ×_{M_{g}} δ̇ ⟶ (T^{•} C)^{K} ×_{M_{g}} δ̇

9.14.1. Claim. pr_{∗}O^{∗} = O^{∗}. Proof. Consider a projection

pr′ : (T^{•} C)^{K} ×_{M_{g}} δ̇ ×_{M_{g}} C_{j} ⟶ (T^{•} C)^{K} ×_{M_{g_{0}}} δ̇

Then we have

pr_{∗}O^{∗} = ⨁_{i∈ℤ}pr′_{∗}(Ω̇^{i}(C_{j}/M_{g}))

But for generic (C, z_{k} ∈ C, k ∈ K) the subgroup of Pic(C) generated by {Ω, (z_{k}), k ∈ K} is free. That is, for any i ∈ ℤ there are no meromorphic sections of Ω^{i} invertible on C − {z_{k}, k ∈ K}. This completes the proof of the Claim. □

9.14.2. Using the Claim inductively we conclude that

Γ((T^{•} C)^{J} ×_{M_{g}} δ̇, O^{∗}) = Γ(δ̇, O^{∗}) = ⨁_{i∈ℤ}Γ(M_{g}, δ̇^{i})

Since g > 1, we know that for i ≠ 0 the line bundle δ^{i} is nontrivial, and hence does not have any invertible sections. For i = 0 we know that that the only invertible functions on M_{g} are constants (see e.g. [BM] 2.3). This completes the proof of Lemma 9.12.2 along with the Theorem 7.6. □

Chapter 3. Regular Representation

10. A characterization of the Regular bimodule

This section belongs really to the Chapter 2 of IV.

10.1. Consider the algebra 𝔲̃ defined in [AJS], Remark 1.4. It is an infinite dimensional B-algebra containing 𝔲 as a subalgebra. We have 𝔲̃ = 𝔲^{−} ⊗ 𝔲̃^{0} ⊗ 𝔲^{+} where 𝔲̃^{0} is formed by some divided power expressions in K_{i}.

We have C = 𝔲̃ − mod, and 𝔲̃ itself viewed as a left regular 𝔲-module is isomorphic to ⊕_{λ∈Y} 𝔲 ⊗ L(lλ) where 𝔲 is considered as a left regular 𝔲-module.

Thus 𝔲̃ does not belong to C being infinite dimensional, but belongs to IndC, see [D4], §4. If we take into account the right regular action of 𝔲 an 𝔲̃ as well, then we can regard 𝔲̃ as an object of IndC ⊗IndC_{r}, that is, an infinite dimensional 𝔲-bimodule. Finally, composing the right action with the antipode, we may view 𝔲̃ as an object of IndC ⊗ IndC. We will denote this object by R.

10.2. Given V, W ∈ C we define a functor F_{V ⊗W} : C ⟶ C,

F_{V ⊗W} (M) = (sM ⊗_{C} V) ⊗ W.

The functor C × C ⟶ F(C, C), (V, W) ↦ F_{V ⊗W}, is exact in W and right exact in V. Therefore, by the universal property it extends to a functor

C ⊗ C ⟶ F(C, C), O ↦ F_{O}.

This in turn extends to a functor

IndC ⊗ IndC ⟶ F(IndC, IndC), O ↦ F_{O}.

We have FR = Id.

In concrete terms, C ⊗ C is equivalent to X_{ℓ}^{∗} × X_{ℓ}^{∗}-graded 𝔲-bimodules. This equivalence sends V ⊗ W to V ⊗ sW with an evident bimodule structure (the latter tensor product is understood as a product of vector spaces). For a bimodule O and M ∈ C we have

F_{O}(M) = s(sM ⊗_{C} O).

Here sM ⊗_{C} O has a right 𝔲-module structure and X_{ℓ}^{∗}-grading inherited from O. So, for the bimodule R we have

s(sM ⊗_{C} R) = M

for all M ∈ C.

10.3. Let C^{ℤ} denote the category of ℤ-graded objects of C. For an X_{ℓ}^{∗} × X_{ℓ}^{∗}-graded 𝔲-bimodule O we define a functor F_{O; \frac{∞}{2} +•} : C ⟶ C^{ℤ},

F_{O; \frac{∞}{2} +•}(M) := sTor_{\frac{∞}{2} +•}^{C}(sM, O).

Here Tor_{\frac{∞}{2} +•}^{C}(sM, O) has a right 𝔲-module structure and X_{ℓ}^{∗}-grading inherited from O.

10.4. Lemma. (i) Suppose O is 𝔲^{+}-induced as a left 𝔲-module. Then F_{O; \frac{∞}{2} +n} = 0 for n < 0, and F_{O; \frac{∞}{2} +0} = F_{O}.

(ii) Suppose O is projective as a left 𝔲-module. Then F_{O; \frac{∞}{2} +n} = 0 for n ≠ 0, and F_{O; \frac{∞}{2} +0} = F_{O}.

Proof follows immediately from the next

10.4.1. Lemma. Let V ∈ C_{r}, and assume that N ∈ C is 𝔲^{+}-induced. Then

Tor_{\frac{∞}{2} +•}^{C}(V, N) = Tor_{•}^{C}(V, N).

In particular, if N is projective then

Tor_{\frac{∞}{2} +•}^{C}(V, N) = \{ \begin{matrix}0 & if n ≠ 0 \\ V ⊗_{C} N & if n = 0. \end{matrix}

Proof. According to Theorem IV.4.9 we may choose any 𝔲^{+}-induced right convex resolution of N for the computation of Tor_{\frac{∞}{2} +•}^{C}(V, N). So, let us choose N itself. Then we have

H^{•}(R_{↙}^{•}(V) ⊗_{C} N) = H^{•}((R_{↙}^{•}(V) ⊗ sN) ⊗_{C} B).

But R_{↙}^{•}(V) ⊗ sN is a left 𝔲-projective resolution of V ⊗ sN, so

H^{−•}((R_{↙}^{•}(V) ⊗ sN) ⊗_{C} B) = Tor_{•}^{C}((V ⊗ sN), B) = Tor_{•}^{C}(V, N). □

10.5. Corollary.

F_{R; \frac{∞}{2} +n} = \{ \begin{matrix}0 & if n ≠ 0 \\ Id & if n = 0. \end{matrix}

Proof. The bimodule R is projective as a left 𝔲-module. □

10.6. Lemma. Let M_{1}, …, M_{n}, N ∈ C. There is a canonical perfect pairing between Ext_{C}^{\frac{∞}{2} +k}(M_{1} ⊗ … ⊗ M_{n}, N) and Tor_{\frac{∞}{2} +k}^{C}(B, M_{1} ⊗ … ⊗ M_{n} ⊗ N^{∗}) for every k ∈ ℤ.

Proof. This is just IV.4.8. □

10.7. Converse Theorem. Suppose Q ∈ Ind C ⊗ Ind C, and an isomorphism of functors ϕ : F_{R, \frac{∞}{2} +•} ⟶^{∼} F_{Q, \frac{∞}{2} +•} is given. Then ϕ is induced by the isomorphism ϕ̄: R ⟶^{∼} Q.

Proof. We start with the following Lemma:

10.7.1. Lemma. Let M ∈ Ind C. Suppose Ext_{C}^{\frac{∞}{2} +n}(M, N) = 0 for n ≠ 0 and arbitrary N ∈ C. Suppose further that Ext_{C}^{\frac{∞}{2} +0}(M, N) is exact in N. Then M is projective, and Ext_{C}^{\frac{∞}{2} +0}(M, N) = Hom_{C}(M, N).

Proof. By the Theorem IV.4.9, Ext_{C}^{\frac{∞}{2} +•}(M, N) = H^{•}(Hom_{C}C(P^{•}(M), R^{•}(N))) where P^{•}(M) is some 𝔲^{+}-induced convex right resolution of M, and R^{•}(N) is some 𝔲^{−}-induced concave right resolution of N.

Since Ext_{C}^{\frac{∞}{2} +n}(M,?) vanishes for n ≠ 0, and is exact for n = 0, we deduce that Ext_{C}^{\frac{∞}{2} +•}(M, N) = H^{•}(Hom_{C}(P^{•}(M), N)). In particular,

Ext_{C}^{\frac{∞}{2} +0}(M, N) = Coker(Hom_{C}(P^{1}(M), N) ⟶^{d_{0}^{∗}} Hom_{C}(P^{0}(M), N))

Let us denote this latter functor in N by h(N) for brevity. We denote by ξ^{∗} : h(N) ⟶ Hom_{C}(M, N) the evident morphism of functors. By the assumption, h(N) is exact and hence representable by some projective P^{0} ∈ Ind C. We denote by ξ : M ⟶ P^{0} the map inducing ξ^{∗} on the functors they represent.

We denote by θ : P^{0} ⟶ P^{0}(M) the map inducing the evident projection Hom_{C}(P^{0}(M), N) ⟶ h(N) on the functors they represent. Then θ ∘ ξ = ε : M ⟶ P^{0}(M) where ε stands for the augmentation.

Now ε is injective, hence ξ : M ⟶ P^{0} is injective. We may extend ξ : M ⟶ P^{0} to a 𝔲^{+}-induced convex right resolution P^{0} ⟶ P^{1} ⟶ … of M, and use this resolution instead of P^{•}(M) for the calculation of Ext_{C}^{\frac{∞}{2} +•}(M, N). Repeating the above argument for P^{•} we conclude that

Coker(Hom_{C}(P^{1}, N) ⟶^{d_{0}^{∗}} Hom_{C}(P^{0}, N)) = Hom_{C}(P^{0}, N),

i.e. d_{0}^{∗} = 0, whence 0 = d_{0} : P^{0} ⟶ P^{1}, whence M = P^{0}. □

10.7.2. Proof of the Theorem. Applying the above Lemma together with the Lemma IV.4.8, we see that Q is projective as a left 𝔲-module, and sTor_{\frac{∞}{2} +n}^{C}(sN, Q) = sTor_{n}^{C}(sN, Q) = s(sN ⊗_{C} Q) if n = 0, and zero otherwise for any N ∈ C.

So ϕ boils down to the isomorphism of functors in N:

s(sN ⊗_{C} R) ⟶^{∼} s(sN ⊗_{C} Q)

Applying the Lemma IV.4.8 again, we obtain the isomorphism of functors

C ⟶ C : s(Hom_{C}(R,?)) ⟶^{∼} s(Hom_{C}(Q,?)),

whence the isomorphism of representing objects. □

11. The adjoint representation

11.1. For μ ∈ Y_{ℓ}^{∗} ⊂ X_{ℓ}^{∗} (see IV.9.1.2) we denote by T_{μ} an autoequivalence C ⟶ C given by twisting by L(μ) : T_{μ}(N) := N ⊗ L(μ).

For μ, ν ∈ Y_{ℓ}^{∗} ⊂ X_{ℓ}^{∗} we will consider an autoequivalence T_{μ} ⊗ T_{ν} of IndC ⊗ IndC.

The objects R and T_{μ} ⊗ T_{−μ}R give rise to the same functor FR = F_{T_{μ}⊗T_{−μ}R} = Id : IndC ⟶ IndC (notations of 10.2). The equality of functors is induced by the isomorphism of objects t_{μ} : R ⟶^{∼} T_{μ} ⊗ T_{−μ}R.

Consider the tensor product functor ⊗ : IndC ⊗ IndC ⟶ IndC.

Let us denote by âd the object ⊗(R) ∈ IndC. For μ ∈ Y_{ℓ}^{∗} ⊂ X_{ℓ}^{∗} we have ⊗(T_{μ} ⊗T_{−μ}R) = âd. Thus the isomorphism t_{μ} induces the same named automorphism of âd. This way we obtain the action of the group Y_{ℓ}^{∗} on âd.

We define the adjoint representation ad as the quotient âd/Y_{ℓ}^{∗}.

It is easy to see that ad ∈ C ⊂ IndC, and in fact âd = ad ⊗ B[Y_{ℓ}^{∗}] with the trivial action of 𝔲 on the second multiple, and the trivial action of Y_{ℓ}^{∗} on the first multiple.

In concrete terms, ad coincides with 𝔲 considered as a 𝔲-module via the adjoint action. The X_{ℓ}^{∗}-grading on ad is compatible with multiplication on 𝔲, and the generators have the following weights:

θ_{i} ∈ (ad)_{−i′}; ϵ_{i} ∈ (ad)_{i′}; K_{i} ∈ (ad)_{0}.

Note that the highest weight of ad is equal to 2ρ_{ℓ} (notations of IV.9.1.2).

11.2. Recall the notations of IV.9.1, 9.2. Recall that Õ_{ζ} is a semisimple subcategory of C consisting of all the direct sums of simples L(λ), λ ∈ Δ = Δ_{ℓ}.

The natural embedding Õ_{ζ} ↪ C admits the right adjoint functor N ↦ N^{Δ} (the maximal Õ_{ζ}-subobject), as well as the left adjoint functor N ↦ N_{Δ} (the maximal Õ_{ζ}-quotient). Finally,

N ↦ ⟨N⟩_{Δ}

denotes the image functor of the canonical morphism from the right adjoint to the left adjoint.

One checks easily that if M ⊂ N is a direct summand belonging to Õ_{ζ} and not properly contained in any of the kind (i.e. maximal), then we have a canonical isomorphism

⟨N⟩_{Δ} ⟶^{∼} M.

For this reason we call ⟨N⟩_{Δ} the maximal Õ_{ζ}-direct summand of N.

H.Andersen and J.Paradowski introduced the tensor structure ˜⊗ on the category Õ_{ζ} (see [AP]). Arguing like in the proof of Lemma IV.9.3, one can show easily that for M, N ∈ Õ_{ζ} ⊂ C we have

M ˜⊗N = ⟨M ⊗ N⟩_{Δ}

11.2.1. Definition. The semisimple adjoint representation is the following object of Õ_{ζ}:

ãd := ⨁_{λ∈Δ} L(λ) ˜⊗L(λ)^{∗}

Here ∗ is the rigidity (see e.g. IV.4.6.2).

11.3. Theorem. ãd = ⟨ad⟩_{Δ}.

Proof. According to [LM], ad ∈ C represents the functor V ↦ Nat(Id, Id ⊗ V) sending V ∈ C to the vector space of natural transformations between endofunctors C ⟶ C.

Hence ⟨ad⟩_{Δ} represents the functor sending V ∈ Õ_{ζ} to the vector space of natural transformations between endofunctors Id and ⟨Id ⊗ V ⟩_{Δ} : Õ_{ζ} ⟶ Õ_{ζ}.

Similarly, ãd ∈ Õ_{ζ} represents the functor W ↦ Nat(Id, Id ˜⊗W).

It remains to recall that for W ∈ Õ_{ζ} ⊂ C we have Id ˜⊗W = ⟨Id ⊗ W ⟩_{Δ}. □

Chapter 4. Quadratic degeneration in genus zero

This chapter belongs really to the part IV. Here we construct a sheaf R ∈ IndFS ⊗IndFS corresponding to the regular bimodule R ∈ IndC ⊗ IndC under the equivalence Φ.

12. I-sheaves

Since R is an object of IndFS ⊗ IndFS we start with developing a little machinery to describe certain Ind-sheaves.

12.1. We consider the root datum Y^{♢} := Y × Y; X^{♢} = X × X; I^{♢} := I ⊔ I; ⟨(y_{1}, y_{2}), (x_{1}, x_{2})⟩ := ⟨y_{1}, x_{1}⟩ + ⟨y_{2}, x_{2}⟩; I^{♢} ∋ (i_{1}, i_{2}) ↦ (i_{1}, i_{2}) ∈ Y^{♢}; I^{♢} ∋ (i_{1}, i_{2}) ↦ (i′_{1}, i′_{2}) ∈ X^{♢}. The corresponding category FS^{♢} is equivalent to FS ⊗ FS. Recall that an object X of FS^{♢} is the following collection of data (see III.4.2):

(a) a weight λ(X) = (λ_{1}, λ_{2}) ∈ X_{ℓ}^{∗} × X_{ℓ}^{∗};

(b) for each (α_{1}, α_{2}) ∈ ℕ[I] × ℕ[I] = ℕ[I^{♢}], a sheaf X^{α_{1},α_{2}} ∈ M(A_{λ_{1},λ_{2}}^{α_{1},α_{2}}; S); Note that (A_{λ_{1},λ_{2}}^{α_{1},α_{2}}, S) = (A_{λ_{1}}^{α_{1}}, S_{1}) × (A_{λ_{2}}^{α_{2}}, S_{2}); we will denote by X^{α⃗_{1},α⃗_{2}}(d⃗) perverse sheaves over A_{λ_{1},λ_{2}}^{α⃗,α⃗_{2}}(d⃗) obtained by taking the restrictions with respect to the embeddings A_{λ_{1},λ_{2}}^{α⃗_{1},α⃗_{2}}(d⃗) ↪ A_{λ_{1},λ_{2}}^{α_{1},α_{2}};

(c) for each (α_{1}, α_{2}), (β_{1}, β_{2}) ∈ ℕ[I] × ℕ[I], d > 0, a factorization isomorphism

ψ^{α_{1},α_{2};β_{1},β_{2}}(d) : X^{(α_{1},α_{2};β_{1},β_{2})}(d) ⟶^{∼} I^{•}_{λ_{1}−β′_{1},λ_{2}−β′_{2}}^{(α_{1},α_{2};0,0)}(d) ⊠ X^{(0,0;β_{1},β_{2})}(d)  (315)

satisfying the usual associativity conditions. Note that A^{α_{1},α_{2};0,0}(d) = A^{α_{1};0}(d) × A^{α_{2};0}(d), and

I^{•}_{λ_{1}−β′_{1},λ_{2}−β′_{2}}^{(α_{1},α_{2};0,0)}(d) = I^{•}_{λ_{1}−β′_{1}}^{(α_{1};0)}(d) ⊠ I^{•}_{λ_{2}−β′_{2}}^{(α_{2};0)}(d).

12.2. Definition. An I-sheaf is the following collection of data:

(a) a weight χ ∈ X_{ℓ}^{∗}, to be denoted by χ(X_{ℓ}^{∗});

(b) for each (λ_{1}, λ_{2}) ∈ X_{ℓ}^{∗} × X_{ℓ}^{∗} such that λ_{1} + λ_{2} − χ = γ ∈ ℕ[I], and α_{1}, α_{2} ∈ ℕ[I], such that α_{1}, α_{2} ≤ γ, a sheaf X_{λ_{1},λ_{2}}^{α_{1},α_{2}} ∈ M(A_{λ_{1},λ_{2}}^{α_{1},α_{2}}; S);

we will denote by X^{α⃗_{1},α⃗_{2}}(d⃗) perverse sheaves over A_{λ_{1},λ_{2}}^{α⃗,α⃗_{2}}(d⃗) obtained by taking the restrictions with respect to the embeddings A_{λ_{1},λ_{2}}^{α⃗_{1},α⃗_{2}}(d⃗) ↪ A_{λ_{1},λ_{2}}^{α_{1},α_{2}};

(c) for each λ_{1}, λ_{2} ∈ X_{ℓ}^{∗} with λ_{1} + λ_{2} − χ = γ ∈ ℕ[I], and α_{1}, α_{2}, β_{1}, β_{2} ∈ ℕ[I] such that α_{1} + β_{1}, α_{2} + β_{2} ≤ γ, d > 0, a factorization isomorphism

ψ^{α_{1},α_{2};β_{1},β_{2}}(d) : X^{(α_{1},α_{2};β_{1},β_{2})}(d) ⟶^{∼} I^{•}_{λ_{1}−β′_{1},λ_{2}−β′_{2}}^{(α_{1},α_{2};0,0)}(d) ⊠ X^{(0,0;β_{1},β_{2})}(d)  (316)

satisfying the usual associativity conditions;

(d) for each (λ_{1}, λ_{2}) ∈ X_{ℓ}^{∗} × X_{ℓ}^{∗} such that λ_{1} + λ_{2} − χ = γ ∈ ℕ[I], and α_{1}, α_{2} ∈ ℕ[I] with α_{1}, α_{2} ≤ γ, and β_{1}, β_{2} ∈ ℕ[I], an isomorphism

X_{λ_{1}+β′_{1},λ_{2}+β′_{2}}^{α_{1}+β_{1},α_{2}+β_{2}} ⟶^{∼} σ_{∗}X_{λ_{1},λ_{2}}^{α_{1},α_{2}}

satisfying the usual compatibilities.

Here σ stands for the closed embedding A_{λ_{1},λ_{2}}^{α_{1},α_{2}} ↪ A_{λ_{1}+β′_{1},λ_{2}+β′_{2}}^{α_{1}+β_{1},α_{2}+β_{2}}.

12.2.1. Remark. Note that an I-sheaf X can be uniquely reconstructed from the partial set of data (b),(c),(d), namely the data (b),(c),(d) given only for α_{1} = α_{2}, β_{1} = β_{2}. In what follows we will describe I-sheaves by these partial data.

13. Degeneration of quadrics

13.1. We recall the construction of [KL]II, 15.2, 15.3. Consider ℙ^{1} × ℙ^{1} × 𝔸^{1} with coordinates p, q, t. Consider the subvariety Q ⊂ ℙ^{1} × ℙ^{1} × 𝔸^{1} given by the equation pq = t. The fiber Q_{t} of Q over t ∈ 𝔸^{1} is a projective line if t ≠ 0, and the union of two projective lines if t = 0.

Each fiber Q_{t} has two marked points: (p = 0, q = ∞), and (p = ∞, q = 0). For t = 0 each irreducible component of Q_{0} has exactly one marked point, and the marked points lie away from the singularity of Q_{0}. The irreducible components of Q_{0} are denoted by Q_{0v} (for vertical), and Q_{0h} (for horizontal).

There are two maps f_{1}, f_{2} from the standard projective line ℙ_{st}^{1} to Q_{t}, taking 0 ∈ ℙ_{st}^{1} to the first (resp. second) marked point on Q_{t}, and mapping ℙ_{st}^{1} isomorphically onto the irreducible component of Q_{t} containing this marked point. If z is the standard coordinate on ℙ_{st}^{1}, then

f_{1}(z) = (\frac{tz}{z − 1}, \frac{z − 1}{z}); f_{2}(z) = (\frac{z − 1}{z}, \frac{tz}{z − 1}).

Thus, restricting to the open subset ′Q ⊂ Q given by the inequality t ≠ 0, we obtain a map ℂ^{∗} ⟶ P̃^{2}.

The limit for t ⟶ 0 is a boundary point of P̃^{2} (notations of IV.2.3). Composing with the “1-jet at 0” projection P̃^{2} ⟶ T P^{∘2} (notations of IV.2.7) we obtain a map ℂ^{∗} ⟶ T P^{∘2}; and the limit for t ⟶ 0 is a boundary point of T P^{∘2}.

13.2. For α ∈ ℕ[I] let Q^{α} ⟶^{π^{α}} 𝔸^{1} denote the space of relative configurations: (π^{α})^{−1}(t) = Q_{t}^{α}.

13.2.1. Q^{α} contains the open subset ′Q^{α} := (π^{α})^{−1}(ℂ^{∗}).

Q^{•α} ⊂ Q^{α} is the open subset of configurations where none of the points equals the marked ones.

Q^{∘α} ⊂ Q^{•α} is the open subset of configurations where all the points are distinct.

′Q^{•α} := ′Q^{α} ∩ Q^{•α}; ′Q^{∘α} := ′Q^{α} ∩ Q^{∘α}.

The open inclusion ′Q^{∘α} ↪ ′Q^{•α} is denoted by j.

The projection Q^{•α} ⟶ 𝔸^{1} is denoted by π^{•α}.

13.2.2. Let μ_{1}, μ_{2} ∈ X_{ℓ}^{∗} be a pair of weights such that the triple (μ_{1}, μ_{2}, α) is admissible (see IV.3.1).

The above construction defines the map θ^{α} : ′Q^{∘α} ⟶ P_{μ_{1}μ_{2}}^{∘α} (notations of IV.3.2), and we define the local system J_{μ_{1}μ_{2}}^{α} on ′Q^{∘α} as θ^{α∗}I_{μ_{1}μ_{2}}^{α} (notations of IV.3.4).

We denote by J_{μ_{1}μ_{2}}^{•α} the perverse sheaf j_{!∗}J_{μ_{1}μ_{2}}^{α}[dim Q^{α}] on ′Q^{•α}.

13.3. The aim of this Chapter is to compute the nearby cycles Ψ_{π^{•α}}J_{μ_{1}μ_{2}}^{•α} as a perverse sheaf on Q_{0}^{•α}. The problem is not quite trivial, but it appears that one can combine the sheaves Ψ_{π^{•α}}J_{μ_{1}μ_{2}}^{•α} for different α, μ_{1}, μ_{2} into a single I-sheaf; and this I-sheaf is already easy to compute.

14. The I-sheaf R

14.1. We denote by Q_{0}^{•} the open subset of the special fiber Q_{0}^{•} := Q_{0} − {(0, ∞), (∞, 0)} obtained by throwing away the marked points. Evidently, Q_{0}^{•} is the union of two irreducible components 𝔸_{v}^{1} and 𝔸_{h}^{1} given by the equations p = 0 (vertical component), and q = 0 (horizontal component) respectively. They intersect at the point (0, 0) ∈ Q_{0}^{•}.

The irreducible components of Q_{0}^{•α}, α ∈ ℕ[I], are numbered by decompositions α = α_{1} + α_{2}; α_{1}, α_{2} ∈ ℕ[I]. Namely,

Q_{0}^{•α} = ⋃_{α_{1}+α_{2}=α} A_{v}^{α_{1}} × A_{h}^{α_{2}}

Each irreducible component embeds into A_{v}^{α} × A_{h}^{α}.

Namely, (q_{1}, …, q_{|α_{1}|}; p_{1}, …, p_{|α_{2}|}) goes to (q_{1}, …, q_{|α_{1}|}, \overbrace{0, …, 0}^{|α_{2}|}; p_{1}, …, p_{|α_{2}|}, \overbrace{0, …, 0}^{|α_{1}|}).

We see readily that these embeddings agree on the intersections of irreducible components, whence they combine together to the closed embedding

ς^{α} : Q_{0}^{•α} ↪ A_{v}^{α} × A_{h}^{α}  (317)

14.2. We define an I-sheaf R as follows.

(a) We set χ(R) = (l − 1)2ρ;

(b) For μ_{1}, μ_{2} ∈ X_{ℓ}^{∗} such that μ_{1} + μ_{2} − (l − 1)2ρ = α ∈ ℕ[I], we define

R_{μ_{1}μ_{2}}^{α,α} := ς_{∗}^{α}Ψ_{π^{•α}}J_{μ_{2}μ_{1}}^{•α}  (318)

(Note the reverse order of μ_{1}, μ_{2}!) R_{μ_{1}μ_{2}}^{α,α} is a perverse sheaf on A_{μ_{1}}^{α} × A_{μ_{2}}^{α} = A_{μ_{1}μ_{2}}^{α,α}. For β ∈ ℕ[I] we define R_{μ_{1}+β,μ_{2}+β}^{α+β,α+β} := σ_{∗}R_{μ_{1},μ_{2}}^{α,α} where σ stands for the closed embedding A_{μ_{1}μ_{2}}^{α,α} ↪ A_{μ_{1}+β,μ_{2}+β}^{α+β,α+β}. Thus the isomorphisms 12.2(d) are simultaneously defined.

(c) We will construct the factorization isomorphisms 12.2(c) for the case α_{1} = α_{2} = α; β_{1} = β_{2} = β (cf. Remark 12.2.1).

14.2.1. For d > 0 we introduce the following analytic open subsets Q_{<d,<d}, Q_{>d,>d}, Q_{<d,>d}, Q_{>d,<d}: Q_{<d,<d} := {(p, q, t)| |p| < d > |q|}; Q_{>d,>d} := {(p, q, t)| |p| > d < |q|}; Q_{<d,>d} := {(p, q, t)| |p| < d < |q|}; Q_{>d,<d} := {(p, q, t)| |p| > d > |q|}. For α, β, γ ∈ ℕ[I] we introduce the following open subset Q^{α,β,γ}(d) ⊂ Q^{α+β+γ}: it is formed by the configurations such that exactly α points of configuration lie in Q_{<d,<d}; exactly β points lie in Q_{<d,>d}, and exactly γ points lie in Q_{<d,>d}. We have evident decomposition

Q^{α,β,γ}(d) = Q^{0,β,0}(d) × Q^{0,0,γ}(d) × Q^{α,0,0}(d).

Intersecting the above subsets with ′Q^{α+β+γ}, ′Q^{•α+β+γ}, ′Q^{∘α+β+γ}, Q^{•α+β+γ}, etc. we obtain the opens ′Q^{α,β,γ}(d), etc. with similar decompositions. We denote by J_{μ_{1}μ_{2}}^{•α,β,γ}(d) the restriction of J_{μ_{1}μ_{2}}^{•α+β+γ} to ′Q^{•α,β,γ}(d).

14.2.2. We have the canonical isomorphisms

J_{μ_{1}μ_{2}}^{•α,β,γ}(d) ⟶^{∼} I^{•}_{μ_{1}}^{(0,β)}(d) ⊠ I^{•}_{μ_{2}}^{(0,γ)} ⊠ J_{μ_{1}−β′,μ_{2}−γ′}^{•α,0,0}(d)  (319)

satisfying the standard associativity constraints. Here I^{•}_{μ_{1}}^{(0,β)}(d), I^{•}_{μ_{2}}^{(0,γ)} have the following meaning.

The set Q_{<d,>d} (resp. Q_{>d,<d}) projects to the vertical component Q_{0v} (resp. horizontal component Q_{0h}) of the special fiber Q_{0} : (p, q, t) ↦ (0, q, 0) (resp. (p, q, t) ↦ (p, 0, 0)).

This induces the projections

v^{β} : Q^{0,β,0}(d) ⟶ Q_{0v}^{β};
h^{γ} : Q^{0,0,γ}(d) ⟶ Q_{0h}^{γ}.

Recall that Q_{0v} (resp. Q_{0h}) has the marked point (0, ∞) (resp. (∞, 0)) lying on it, and v^{β}(Q^{0,β,0}(d)) (resp. h^{γ}(Q^{0,0,γ}(d))) is a standard open subset of Q_{0v}^{β} (resp. Q_{0h}^{γ}).

Restricting our maps to ′Q^{•} we get

v^{β} : ′Q^{•0,β,0}(d) ⟶ Q_{0v}^{•β};
h^{γ} : ′Q^{•0,0,γ}(d) ⟶ Q_{0h}^{•γ}.

Note that v^{β}( ′Q^{•0,β,0}(d)) = Q_{0v}^{•0,β}(d) — the standard open formed by the configurations with all β points running in the annular neighbourhood of the marked point on Q_{0v}.

Similarly, h^{γ}( ′Q^{•0,0,γ}(d)) = Q_{0h}^{•0,γ}(d) — the standard open formed by the configurations with all γ points running in the annular neighbourhood of the marked point on Q_{0h}.

The sheaf I^{•}_{μ_{1}}^{0,β}(d) (resp. I^{•}_{μ_{2}}^{0,γ}(d)) is defined in III.3.5. We keep the same notation for its inverse image to ′Q^{•0,β,0}(d) (resp. ′Q^{•0,0,γ}(d)) with respect to v^{β} (resp. h^{γ}).

14.2.3. One final remark is in order. Note that Q_{0v}^{•} = 𝔸_{v}^{1}; Q^{•}0h = 𝔸_{h}^{1} (notations of 14.1). Hence Q^{•}β_{0v} = A_{v}^{β}; Q^{•}γ_{0h} = A_{h}^{γ}.

Under this identification the open Q_{0v}^{•0,β}(d) corresponds to the standard open A_{v}^{β,0}(d), and the open Q_{0h}^{•0,γ}(d) corresponds to the standard open A_{h}^{γ,0}(d) (notations of III.2.2).

Warning. Note the opposite roles played by the marked points on Q_{0v} and 𝔸_{v}^{1}: the open Q_{0v}^{•0,β}(d) = A_{v}^{β,0}(d) is formed by configurations where all points run near the marked point of Q_{0v}, or equivalently, where all points run far from the marked point 0 ∈ 𝔸^{1}.

The sheaf I^{•}_{μ_{1}}^{0,β}(d) on Q_{0v}^{•0,β}(d) corresponds to the sheaf I^{•}_{(l−1)2ρ+β−μ_{1}}^{β,0}(d) on A_{v}^{β,0}(d) (notations of III.3.5).

The sheaf I^{•}_{μ_{2}}^{0,γ}(d) on Q_{0h}^{•0,γ}(d) corresponds to the sheaf I^{•}_{(l−1)2ρ+γ−μ_{2}}^{γ,0}(d) on A_{h}^{γ,0}(d).

14.2.4. Putting all the above together, the desired factorization isomorphisms 12.2(c) for the sheaf R are induced by the factorization isomorphisms (319) for the sheaf J^{•}.

15. Convolution

15.1. Let us throw away the line p = ∞, q = 0 from Q. What remains is the degenerating family Q_{a} of affine lines with the marked point (0, ∞). The notations ′Q_{a}, Q_{a}^{•}, etc. speak for themselves.

Given a finite factorizable sheaf X ∈ FS we can glue it into the marked point (0, ∞) and obtain the factorizable sheaf X̃ over the family ′Q_{a} of affine lines. For each t ∈ ℂ^{∗} the restriction of X̃ to the fiber (Q_{a})_{t} = A is the FFC X we started with.

More precisely, let λ = λ(X), α ∈ ℕ[I]. We have the sheaf X̃_{λ}^{α} over ′Q_{a}^{α} such that for any t ∈ ℂ^{∗} its restriction to each fiber ( ′Q_{a}^{α})_{t} = A^{α} coincides with X_{λ}^{α}.

We have the projection

pr : Q_{a} ⟶ 𝔸_{h}^{1} × ℂ, pr(p, q, t) = (p, t).

The restriction of pr to ′Q_{a} is one-to-one onto 𝔸_{h}^{1} × ℂ^{∗}, so pr_{∗}X̃ is the constant family of FFC on A_{h} × ℂ^{∗}.

Recall that π denotes the projection to the t-coordinate. The triangle

Q_{a}  ⟶^{π} ℂ
pr↓  ^{id}↑
𝔸^{1} × ℂ ⟶^{π} ℂ

commutes, and the map pr is proper.

By the proper base change for nearby cycles we have

pr_{∗}^{α}Ψ_{π^{α}} X̃_{λ}^{α} = Ψ_{π^{α}}(pr_{∗}^{α}X̃_{λ}^{α}) = X_{λ}^{α}

Here π^{α}, pr^{α} form the evident commutative diagram

Q_{a}^{α}  ⟶^{π^{α}} ℂ
pr^{α}↓  ^{id}↑
A_{h}^{α} × ℂ ⟶^{π^{α}} ℂ

15.2. Let us compute pr_{∗}^{α}Ψ_{π^{α}} X̃_{λ}^{α} in another way. First we describe Ψ_{π^{α}} X̃_{λ}^{α}. It is a sheaf on (Q_{a}^{α})_{0} which will be denoted by Q_{0a}^{α} for short.

Note that Q_{0a} = ℙ_{v}^{1} ∪ 𝔸_{h}^{1} is the union of two irreducible components intersecting at the point (0, 0). Hence Q_{0a}^{α} is the union of irreducible components numbered by the decompositions α = α_{1} + α_{2}, α_{1}, α_{2} ∈ ℕ[I]. Namely,

Q_{0a}^{α} = ⋃_{α_{1}+α_{2}=α} P_{v}^{α_{1}} × A_{h}^{α_{2}}.

Here P_{v}^{α_{1}} stands for the space of configurations on ℙ_{v}^{1}.

Each irreducible component embeds into P_{v}^{α} × A_{h}^{α} as in 14.1, and one checks readily that these embeddings agree on intersections, whence they combine to the closed embedding

ς^{α} : Q_{0a}^{α} ↪ P_{v}^{α} × A_{h}^{α}.

We will describe ς_{∗}^{α}Ψ_{π^{α}} X̃_{λ}^{α}.

15.2.1. Note that ℙ_{v}^{1} is equipped with two marked points with tangent vectors. The marked points are ∞, 0. The tangent vector at 0 is ∂_{q}, and the tangent vector at ∞ was defined in 13.1.

Recall that for μ ∈ X_{ℓ}^{∗} such that the triple (λ, μ, α) is admissible, the sheaf ς_{∗}^{α}Ψ_{π^{•α}}J_{λμ}^{•α} on A_{v}^{α} × A_{h}^{α} was introduced in 14.2(b).

According to the Theorem IV.3.6 we can glue the sheaves ς_{∗}^{α}Ψ_{π^{•α}}J_{λμ}^{•α} and X_{λ}^{α} to obtain the sheaf X̄_{λμ}^{α} on the space P_{v}^{α} × A_{h}^{α}.

Here X_{λ}^{α} is viewed as a sheaf on Q_{0v}^{0,α}(d) × A_{h}^{α} (notations of 13.1 and 14.2.2) constant along A_{h}^{α}. So it is glued into ∞ × A_{h}^{α}, while ς_{∗}^{α}Ψ_{π^{•α}}J_{λμ}^{•α} is glued into 0 × A_{h}^{α}.

15.2.2. Claim.

ς_{∗}^{α}Ψ_{π^{•α}} X̃_{λ}^{α} = X̄_{λμ}^{α}

Proof. Evident. □

15.3. We proceed with the computation of pr_{∗}^{α}Ψ_{π^{α}} X̃_{λ}^{α}.

Recall the sheaf R introduced in 14.2.

15.3.1. We define the convolution X ⋆ R on P_{v} × A_{h} as the following collection of data:

For μ ∈ X_{ℓ}^{∗} such that λ + μ − (l − 1)2ρ = α ∈ ℕ[I], the sheaf

(X ⋆ R)_{λμλ}^{α} := X̄_{λμ}^{α}

on the space P_{λμ}^{α} × A_{λ}^{α}.

The notation P_{λμ}^{α} suggests that the monodromy around ∞ is λ, and the monodromy around 0 is μ.

We leave the formulation of factorization isomorphisms to the interested reader. 15.3.2. Denote by pr^{α} the projection P^{α} × A^{α} ⟶ A^{α}. Then, evidently,

pr_{∗}^{α}(X ⋆ R)_{λμλ}^{α} = pr_{∗}^{α}X̄_{λμ}^{α} = pr_{∗}^{α}ς_{∗}^{α}Ψ_{π^{α}} X̃_{λ}^{α} = X_{λ}^{α}

Here the second equality holds by the Claim 15.2.2, and the last equality was explained in 15.1. 15.4. Thus, for each X ∈ FS, we have the natural isomorphism pr_{∗}(X ⋆ R) = X. That is, for λ = λ(X), α ∈ ℕ[I], μ = α + (l − 1)2ρ − λ, we have pr_{∗}(X ⋆ R)_{λμλ}^{α} = X_{λ}^{α}. 15.4.1. Lemma. There is a natural isomorphism of functors in X:

Φpr_{∗}(X ⋆ R) ⟶^{∼} F_{Φ(R); \frac{∞}{2} +•}(Φ(X))

(notations of 10.3). Proof. This is just the Theorem IV.8.4. □ 15.5. Theorem. There is a natural isomorphism in IndC ⊗ IndC:

Φ(R) ⟶^{∼} R.

Proof. This is just the above Lemma combined with the Converse Theorem 10.7. □

Chapter 5. Modular functor

16. Gluing over C

This section is quite parallel to the Chapter 1 of IV.

We will change our notations slightly to make them closer to those of loc. cit.

16.1. Let again C ⟶ S be a smooth proper relative curve of genus g.

Let us assume that g > 1 for a moment. It is well known that the isomorphism classes of complex structures on a surface of genus g correspond bijectively to the conformal equivalence classes of Riemann metrics on this surface.

Each conformal equivalence class contains a unique metric of constant curvature −1. Thus we may assume that each fiber C_{s} is equipped with such a metric.

Given 0 < ε ≤ 1 and a finite set K, let C̃_{ε}^{K} denote the space of K-tuples (u_{k})_{k∈K} of analytic morphisms over S : S ×D_{ε} ⟶ C inducing isometry on each fiber, such that the images u_{k}(S × D_{ε}) do not intersect. Here we equip the disk D_{ε} with the Poincaré metric of constant curvature −1.

Given a K-tuple α⃗ = (α_{k}) ∈ ℕ[X_{ℓ}^{∗}]^{K} such that α = ∑_{k} α_{k}, define a space

Cα⃗ ; ε := C̃_{ε}^{K} × ∏_{k∈K} T^{•} D_{ε}^{α_{k}}

and an open subspace

C_{ε}^{oα⃗} := C̃_{ε}^{K} × ∏_{k∈K} T D_{ε}^{oα_{k}}.

We have an evident ”substitution” map

qα⃗ : Cα⃗ ; ε ⟶ T^{•} C^{α}

which restricts to qα⃗ : C_{ε}^{oα⃗} ⟶ T C^{oα}.

16.2. Let us define an element ρ_{ℓ} ∈ X_{ℓ}^{∗} by the condition ⟨i, ρ_{ℓ}⟩ = ℓ_{i} − 1 for all i ∈ I (see IV.9.1.2). ¿From now on we choose a balance function n in the form

n(μ) = \frac{1}{2}μ · μ − μ · ρ_{ℓ}.

In other words, we set

ν_{0} = −ρ_{ℓ}.

We pick a corresponding Heisenberg system H.

16.2.1. Remark. The monodromy of H around the zero section of determinant line bundle is equal to (−1)^{rkX_{ℓ}^{∗}}ζ^{12ρ_{ℓ}·ρ_{ℓ}}. According to the strange formula of Freudenthal–deVries (see e.g. [FZ]) we have 12ρ · ρ = dȟ dim 𝔤. Here 𝔤 is the Lie algebra with the root datum X, Y, …, and ȟ is the dual Coxeter number of 𝔤, while d := max_{i∈I} d_{i}.

Let ζ = exp(\frac{π\sqrt{−1}}{dκ}) for some positive integer κ. Then the above monodromy equals exp(π\sqrt{−1}(rk𝔤 + \frac{ȟ dim 𝔤}{κ})) which coincides with the multiplicative central charge of the conformal field theory associated with the affine Lie algebra 𝔤̂ at level κ (see e.g. [BFM]).

We will offer an explanation of this coincidence in Chapter 6.

16.2.2. Given α = ∑ a_{i}i ∈ ℕ[I] and μ⃗ = (μ_{k}) ∈ X_{ℓ}^{∗}^{K}, we define an element

αμ⃗ = ∑ a_{i} · (−i′) + ∑_{k} μ_{k} ∈ ℕ[X_{ℓ}^{∗}]

where the sum in the right hand side is a formal one. We say that a pair (μ⃗, α) is g-admissible if

∑_{k} μ_{k} − α ≡ (2 − 2g)ρ_{ℓ} mod Y_{ℓ}^{∗}.

Note that given μ⃗, there exists α ∈ ℕ[I] such that (μ⃗, α) is admissible if and only if ∑_{k} μ_{k} ∈ Y; if this holds true, such elements α form an obvious countable set.

We will denote by

e : ℕ[I] ⟶ ℕ[X_{ℓ}^{∗}]

a unique homomorphism sending i ∈ I to −i′ ∈ X_{ℓ}^{∗}.

16.3. Let us consider the space T^{•} C^{K} × T^{•} C^{e(α)}; its points are quadruples ((z_{k}), (τ_{k}), (x_{j}), (ω_{j})) where (z_{k}) ∈ C^{K}, τ_{k} — a non-zero (relative) tangent vector to C at z_{k}, (x_{j}) ∈ C^{e(α)}, ω_{j} — a non-zero tangent vector at x_{j}. To a point z_{k} is assigned a weight μ_{k}, and to x_{j} — a weight −π(j)′. Here π : J ⟶ I is an unfolding of α (implicit in the notation (x_{j}) = (x_{j})_{j∈J}).

We will be interested in some open subspaces:

T^{•} Cα ; μ⃗ := T C^{oK} × T^{•} C^{e(α)} ⊂ T^{•} C^{αμ⃗}

and

T Coα ; μ⃗ ⊂ T^{•} Cα ; μ⃗

whose points are quadruples ((z_{k}), (τ_{k}), (x_{j}), (ω_{j})) ∈ T^{•} Cα ; μ⃗ with all z_{k} ≠ x_{j}. We have an obvious symmetrization projection

pα ; μ⃗ : T Coα ; μ⃗ ⟶ T C^{oαμ⃗}.

Define a space

Cα ; μ⃗ = T C^{oK} × C^{e(α)};

its points are triples ((z_{k}), (τ_{k}), (x_{j})) where (z_{k}), (τ_{k}) and (x_{j}) are as above; and to z_{k} and x_{j} the weights as above are assigned. We have the canonical projection

T^{•} Cα ; μ⃗ ⟶ Cα ; μ⃗.

We define the open subspaces

Coα ; μ⃗ ⊂ C•α ; μ⃗ ⊂ Cα ; μ⃗.

Here the •-subspace (resp., o-subspace) consists of all ((z_{k}), (τ_{k}), (x_{j})) with z_{k} ≠ x_{j} for all k, j (resp., with all z_{k} and x_{j} distinct).

We define the principal stratification S of Cα ; μ⃗ as the stratification generated by subspaces z_{k} = x_{j} and x_{j} = x_{j′} with π(j) ≠ π(j′). Thus, Coα ; μ⃗ is the open stratum of S. As usually, we will denote by the same letter the induced stratifications on subspaces.

The above projection restricts to

T Coα ; μ⃗ ⟶ Coα ; μ⃗.

16.4. Factorization structure.

16.4.1. Suppose we are given α⃗ ∈ ℕ[I]^{K}, β ∈ ℕ[I]; set α := ∑_{k} α_{k}. Define a space

C_{μ⃗,ε}^{α⃗,β} ⊂ C̃_{ε}^{K} × ∏_{k} D_{ε}^{α_{k}} × C^{e(β)}

consisting of all collections ((u_{k}), ((x_{j}^{(k)})_{k}), (y_{j})) where (u_{k}) ∈ C̃_{ε}^{K}, (x_{j}^{(k)})_{k} ∈ D^{α_{k}}, (y_{j}) ∈ C^{e(β)}, such that

y_{j} ∈ C − ⋃_{k∈K} \overline{u_{k}(S × D_{ε})}

for all j (the bar means closure).

We have canonical maps

q_{α⃗,β} : C_{μ⃗,ε}^{α⃗,β} ⟶ Cα+β ; μ⃗,

assigning to ((u_{k}), ((x_{j}^{(k)})_{k}), (y_{j})) a configuration (u_{k}(0)), (u^{•}_{k}(τ)), (u_{k}(x_{j}^{(k)})), (y_{j})), where τ is the unit tangent vector to D_{ε} at 0, and

p_{α⃗,β} : C_{μ⃗,ε}^{α⃗,β} ⟶ ∏_{k} D_{ε}^{α_{k}} × C_{μ⃗−α⃗}^{•β}

sending ((u_{k}), ((x_{j}^{(k)})_{k}), (y_{j})) to ((u_{k}(0)), (u^{•}_{k}(τ)), (y_{j})).

16.5. From now on we change the base to δ̇ ⟶ S in all the above spaces and morphisms. We preserve the old notations.

Given a g-admissible pair (μ⃗, α) we can find another K-tuple of weights μ′ ∈ X_{ℓ}^{∗}^{K} such that μ′ is congruent to μ⃗ modulo Y_{ℓ}^{∗} (see IV.9.1.2), and α_{}_{μ′} is g-admissible in the sense of 2.1.8.

Let us consider a local system p_{μ′}^{α∗}H^{α_{μ′}} over T C_{μ′}^{oα}.

Note that all the spaces T C_{μ′}^{oα} (for different choices of μ′) are identified with T Coα ; μ⃗, and the local system p_{μ′}^{α∗}H^{α_{μ′}} does not depend on the choice of μ′ by the virtue of the Lemma 5.11.1. Thus we will identify all these local systems and call the result pα∗ ; μ⃗H^{αμ⃗}.

By our choice of the balance function n, its monodromies with respect to the rotating of tangent vectors ω_{j} at points x_{j} corresponding to negative simple roots, are trivial.

Therefore it descends to a unique local system over Coα ; μ⃗, to be denoted by H_{μ}^{α}.

We define a perverse sheaf

H•α ; μ⃗ := j_{!∗}Hα ; μ⃗[dim Cα ; μ⃗] ∈ M(C•α ; μ⃗; S).

16.6. Factorizable sheaves over C. Suppose we are given a K-tuple of FFS’s {X_{k}}, X_{k} ∈ FS_{c_{k}}, k ∈ K, c_{k} ∈ X_{ℓ}^{∗}/Y, where ∑_{k} c_{k} = 0. Let us pick μ⃗ = (μ⃗_{k}) ≥ (λ(X_{k})).

Let us call a factorizable sheaf over C obtained by gluing the sheaves X_{k} the following collection of data which we will denote by g({X_{k}}).

(i) For each α ∈ ℕ[I] such that (μ⃗, α) is g-admissible, a sheaf X α ; μ⃗ ∈ M(Cα ; μ⃗; S).

(ii) For each 0 < ε ≤ 1, α⃗ = (α_{k}) ∈ ℕ[I]^{K}, β ∈ ℕ[I] such that (μ⃗, α + β) is g-admissible (where α = ∑ α_{k}), a factorization isomorphism

ϕ_{α⃗,β} : q_{α⃗,β}^{∗}X α+β ; μ⃗ ⟶^{∼} p_{α⃗,β}^{∗}(( × _{k∈K} X_{μ_{k}}^{α_{k}}) ⊠ H_{μ⃗−α⃗}^{•β}).

These isomorphisms should satisfy the standard associativity property.

16.7. Theorem. There exists a unique up to a canonical isomorphism factorizable sheaf over C obtained by gluing the sheaves {X_{k}}.

Proof is similar to III.10.3. □

16.8. We will need the following version of the above Theorem.

Suppose we are given a K-tuple {X_{k}} of objects of FS, and a J-tuple {Y_{j}} of I-sheaves (see 12.2).

Let us pick λ⃗ = (λ⃗_{k}) ≥ (λ(X_{k})), and (μ⃗, ν⃗) = (μ⃗_{j}, ν⃗_{j}) such that (μ⃗_{j} + ν⃗_{j}) ≥ (χ(Y_{j})).

We will denote the concatenation of λ⃗, μ⃗, and ν⃗ by λ⃗μ⃗ν⃗.

Let us call a factorizable sheaf over C obtained by gluing the sheaves X_{k} and I-sheaves Y_{j} the following collection of data which we will denote by g({X_{k}}, {Y_{j}}).

(i) For each α ∈ ℕ[I] such that (λ⃗μ⃗ν⃗, α) is g-admissible, a sheaf X_{λ⃗μ⃗ν⃗}^{α} = g({X_{k}}, {Y_{j}})_{λ⃗μ⃗ν⃗}^{α} ∈ M(C_{λ⃗μ⃗ν⃗}^{α}; S).

(ii) For each 0 < ε ≤ 1, α⃗ = (α_{k}) ∈ ℕ[I]^{K}, (β⃗, γ⃗) = (β_{j}, γ_{j}) ∈ ℕ[I]^{J⊔J}, ξ ∈ ℕ[I] such that β_{j}, γ_{j} ≤ μ⃗_{j} + ν⃗_{j} − χ(Y_{j}) for any j ∈ J, and (λ⃗μ⃗ν⃗, α + β + γ + ξ) is g-admissible (where α = ∑ α_{k}, β = ∑ β_{j}, γ = ∑ γ_{j}), a factorization isomorphism

ϕ_{α⃗β⃗γ⃗,ξ} : q_{α⃗β⃗γ⃗,ξ}^{∗}X_{λ⃗μ⃗ν⃗}^{α+β+γ+ξ} ⟶^{∼} p_{α⃗β⃗γ⃗,ξ}^{∗}(( × _{k∈K} X_{λ_{k}}^{α_{k}}) ⊠ ( × _{j∈J} Y_{μ_{j},ν_{j}}^{β_{j},γ_{j}}) ⊠ H_{λ⃗μ⃗ν⃗−α⃗β⃗γ⃗}^{•ξ}).

These isomorphisms should satisfy the standard associativity property.

Theorem. There exists a unique up to a canonical isomorphism factorizable sheaf over C obtained by gluing the sheaves {X_{k}} and I-sheaves {Y_{j}}.

Proof is similar to III.10.3. □

16.9. We will apply the above Theorem exclusively to the case Y_{j} = R (see 14.2).

16.9.1. Recall (see 16.3) that C_{λ⃗μ⃗ν⃗}^{α} = T C^{0K⊔J⊔J} × C^{e(α)} is the space of 7-tuples

((z_{k}), (τ_{k}); (x_{j}), (ω_{j}); (y_{j}), (η_{j}); (t_{m})).

Here (t_{m}) ∈ C^{e(α)}; (z_{k}) ∈ C^{K}; (x_{j}), (y_{j}) ∈ C^{J}. All the points z_{k}, x_{j}, y_{j} are distinct, and τ_{k} is a nonzero tangent vector at z_{k}; ω_{j} is a nonzero tangent vector at x_{j}; η_{j} is a nonzero tangent vector at y_{j}.

We define

\underline{C}_{λ⃗μ⃗ν⃗}^{α} := T C^{0K⊔J⊔J} × (C/[(x_{j}) = (y_{j})])^{e(α)}

Note that the natural projection 𝔭 : C_{λ⃗μ⃗ν⃗}^{α} ⟶ \underline{C}_{λ⃗μ⃗ν⃗}^{α} is the normalization map.

16.9.2. Suppose we are in the situation of 16.8, and {Y_{j}} is just a set of J copies of the I-sheaf R (see 14.2). Thus, {Y_{j}} = {R_{j}}; R_{j} = R for any j ∈ J.

It is immediate from the definition of R that the sheaf g({X_{k}}, {R_{j}})_{λ⃗μ⃗ν⃗}^{α} on C_{λ⃗μ⃗ν⃗}^{α} descends to a sheaf on \underline{C}_{λ⃗μ⃗ν⃗}^{α} to be denoted by \underline{g}(\{X_{k}\}, \{R_{j}\})_{λ⃗μ⃗ν⃗}^{α}.

16.9.3. Remark. All the above constructions generalize immediately to the case where C is a union of smooth connected components C_{n} of genera g_{n}, ∑ g_{n} = g.

Then C^{α} is a union of connected components numbered by the partitions α = ∑ α_{n}. Each connected component is the product ∏ C_{n}^{α_{n}}. The Heisenberg local system is just the product of H_{g_{n}}. A factorizable sheaf g({X_{k}}) on each connected component also decomposes into external product of the corresponding sheaves on factors. Note that each connected component gives rise to its own admissibility condition.

On the other hand, a sheaf \underline{g}(\{X_{k}\}, \{R_{j}\}) does not necessarily decompose into external product. In effect, if for some j the sections x_{j} and y_{j} lie on the different connected components of C, then some connected components of C_{λ⃗μ⃗ν⃗}^{α} will be glued together in \underline{C}_{λ⃗μ⃗ν⃗}^{α}.

17. Degeneration of factorizable sheaves

17.1. Definition. An m-tuple of weights μ⃗ ∈ X_{ℓ}^{∗}^{m} is called g-positive if

∑_{j=1}^{m} μ_{j} + 2(g − 1)ρ_{ℓ} ∈ ℕ[I] ⊂ X_{ℓ}^{∗}

(notations of IV.9.1.2). If this is so, we will denote

α_{g}(μ⃗) := ∑_{j=1}^{m} μ_{j} + 2(g − 1)ρ_{ℓ}

17.2. We will use freely all the notations of section 7. In particular, C is the universal curve of genus g.

Given a K-tuple of FFS’s {X_{k}} and a g-positive K-tuple of weights μ⃗ = (μ⃗_{k}) ≥ (λ(X_{k})) we define α := α_{g}(μ⃗).

We consider a sheaf X α ; μ⃗ = gα ; μ⃗({X_{k}}) on Cα ; μ⃗ obtained by gluing the sheaves X_{k} (see 16.6). We will study its specialization along the boundary component 𝔇.

Let C denote the following object:

in case 7.1(a) C is the universal curve over M_{g_{1}} × M_{g_{2}}. It consists of two connected components C_{1} and C_{2}.

in case 7.1(b) C is the universal curve C_{0} over M_{g_{0}}.

The maps (302) and (303) give rise to the following morphisms also denoted by ℘_{α}:

(C_{1})_{μ⃗_{1},μ_{1}}^{α_{1}} × (C_{2})_{μ⃗_{2},μ_{2}}^{α_{2}} ⟶ T^{•}_{(\overline{C}|_{𝔇})α ; μ⃗} \overline{C}α ; μ⃗;  (320)

(C_{0})_{μ⃗,μ_{1},μ_{2}}^{α} ⟶ T^{•}_{(\overline{C}|_{𝔇})α ; μ⃗} \overline{C}α ; μ⃗  (321)

(recall that by default all the bases are changed to the punctured determinant line bundles). In (320) the sets μ⃗_{1} and μ⃗_{2} form an arbitrary partition of the set μ⃗. Finally, we can choose μ_{1}, μ_{2} arbitrarily: the spaces we consider do not depend on a choice of μ_{r}.

Recall that C_{μ⃗,μ_{1},μ_{2}}^{α} is the set of tuples ((z_{k}), (τ_{k}); x, ω; y, η; (t_{m})). Here (t_{m}) ∈ C^{e(α)}; (z_{k}) ∈ C^{K}; (x_{j}), (y_{j}) ∈ C^{J}. All the points z_{k}, x, y are distinct, and τ_{k} is a nonzero tangent vector at z_{k}; ω is a nonzero tangent vector at x; η is a nonzero tangent vector at y (see 16.9.1).

We defined the nodal curve \underline{C} in loc. cit.. One can see that the morphisms (320) for different connected components (resp. the morphism (321)) can be glued together into (resp. factor through) the same named morphism ℘_{α}:

\underline{C}_{μ⃗,μ_{1},μ_{2}}^{α} ⟶ T^{•}_{(\overline{C}|_{𝔇})α ; μ⃗} \overline{C}α ; μ⃗  (322)

17.3. Theorem. There is a canonical isomorphism

℘_{α}^{∗}Spgα ; μ⃗(\{X_{k}\}) = \underline{g}_{μ⃗,μ_{1},μ_{2}}^{α}(\{X_{k}\}, R)

(notations of 16.9.2 and 16.9.3).

We have to specify μ_{1}, μ_{2} in the formulation of the Theorem.

17.3.1. In case (320) the choice of μ_{1}, μ_{2} depends on the connected component (see 16.9.3). Namely, we set μ_{r} = α_{r} − α_{g_{r}}(μ⃗_{r}), r = 1, 2.

In case (321) we sum up over all the choices of (μ_{1}, μ_{2}) ∈ (X_{ℓ}^{∗} × X_{ℓ}^{∗})/Y_{ℓ}^{∗} (antidiagonal action) such that μ_{1} + μ_{2} = 2ρ_{ℓ}. Thus, if we identify all the spaces \underline{C}_{μ⃗,μ_{1},μ_{2}}^{α} with one abstract space \underline{C}_{μ⃗,?,?}^{α}, then ℘_{α}^{∗}Spgα ; μ⃗(\{X_{k}\}) is a direct sum of d_{ℓ} summands. Or else we can view ℘_{α}^{∗}Spgα ; μ⃗(\{X_{k}\}) as a collection of sheaves on different spaces \underline{C}_{μ⃗,μ_{1},μ_{2}}^{α}.

17.3.2. Proof. By the definition of gluing ( 16.8) it suffices to construct the desired isomorphism “near the points x, y” and, separately, “away from the points x, y”.

The situation “near the points x, y” is formally equivalent to (a direct sum of a few copies of) the situation of 14.2. Thus we obtain a copy of the I-sheaf R glued into the node x = y.

The situation “away from the points x, y” was the subject of the Theorem 7.6.

We leave it to the reader to work out the values of μ_{1}, μ_{2}. We just note that these are exactly the values making the tuple (α, μ⃗, μ_{1}, μ_{2}) g-admissible. □

18. Global sections over C

18.1. Given a K-tuple of weights μ⃗ and α = α_{g}(μ⃗), consider the projection

η : Cα ; μ⃗ ⟶ T C^{∘K} = T^{∘} Cμ⃗

(see 16.3).

Given a K-tuple of FFS’s {X_{k}} with (μ⃗_{k}) ≥ (λ(X_{k})) we consider the sheaf g({X_{k}})α ; μ⃗ on Cα ; μ⃗ obtained by gluing the sheaves X_{k}.

In this section we will study the complex of sheaves with smooth cohomology Rη_{∗}g({X_{k}})α ; μ⃗

on T^{∘} Cμ⃗. Note that it does not depend on a choice of (μ⃗_{k}) ≥ (λ(X_{k})) (if we identify the

spaces T^{∘} Cμ⃗ for different choices of μ⃗).

Namely, we want to compute its specialization along the boundary component 𝔇. Recall the notations of 7.

In case 7.1(a) for any partition μ⃗ = μ⃗_{1} ⊔ μ⃗_{2} we have the map (cf. (302)) ℘μ⃗:

T^{∘} C_{1}^{μ⃗_{1},μ_{1}} × T^{∘} C_{2}^{μ⃗_{2},μ_{2}} ⟶ T^{•}_{T^{∘} \overline{C}μ⃗|_{𝔇}} T^{∘} \overline{C}μ⃗;  (323)

in case 7.1(b) we have the map (cf. (303)) ℘μ⃗:

T^{∘} C_{0}^{μ⃗,μ_{1},μ_{2}} ⟶ T^{•}_{T^{∘} \overline{C}μ⃗|_{𝔇}} T^{∘} \overline{C}μ⃗.  (324)

On the other hand, consider the space \underline{C}_{μ⃗,μ_{1},μ_{2}}^{α} (see 17.2). In the case 7.1(b) it projects to

T^{∘} C_{0}^{μ⃗,μ_{1},μ_{2}}. We denote the projection by \underline{η}.

In the case 7.1(a) \underline{C}_{μ⃗,μ_{1},μ_{2}}^{α} is disconnected. Its connected components are numbered by the

partitions μ⃗ = μ⃗_{1} ⊔ μ⃗_{2}. Each component projects to the corresponding space T^{∘} C_{1}^{μ⃗_{1},μ_{1}} × T^{∘} C_{2}^{μ⃗_{2},μ_{2}}. Each one of those projections will be denoted by \underline{η}, as well as their totality.

18.2. Theorem. In the notations of 17.3 and 17.3.1 we have a canonical isomorphism

℘∗ ; μ⃗SpRη_{∗}g(\{X_{k}\})α ; μ⃗ = R\underline{η}_{∗}\underline{g}_{μ⃗,μ_{1},μ_{2}}^{α}(\{X_{k}\}, R)

Proof. This is just the Theorem 17.3 plus the proper base change for nearby cycles. □

One can apply Theorems 17.3 and 18.2 iteratively — degenerating the curve C more and more, and inserting more and more copies of R into new nodes.

In the terminology of [BFM] 4.5 (see especially 4.5.6) the Theorem 18.2 means that the category FS is equipped with the structure of fusion category. The correspondence associating to any K-tuple of FFS’s {X_{k}} the cohomology local systems Rη_{∗}g({X_{k}})α ; μ⃗ on the space T C^{∘K}, satisfying the compatibility conditions 18.2, is called fusion functor, or modular functor.

Chapter 6. Integral representations of conformal blocks

19. Conformal blocks in arbitrary genus

19.1. Recall the notations of IV.9. We fix a positive integer κ and consider the category Õ_{κ} of integrable 𝔤̂-modules of central charge κ−ȟ. We set ζ = exp(\frac{π\sqrt{−1}}{dκ}), so that l = 2dκ, and ℓ = dκ.

We have an equivalence of bbrt categories ϕ : Õ_{κ} ⟶^{∼} Õ_{ζ} where Õ_{ζ} is a semisimple subcategory of C defined in loc. cit. It is well known that Õ_{κ} has a structure of fusion category (see e.g. [BFM] or [TUY]). The category Õ_{ζ} is also equipped with the structure of fusion category via the equivalence ϕ. From now on we will not distinguish between the categories Õ_{κ} and Õ_{ζ}.

19.2. Here is a more elementary way to think of fusion functors. Recall that a fusion functor is a correspondence associating to any K-tuple of objects {L_{k}} of our category the local systems (or, more generally, the complexes of sheaves with smooth cohomology) ⟨{L_{k}}⟩_{C}^{K} on the spaces T C^{∘K}, satisfying a certain compatibility conditions.

We can view those local systems as representations of Modular Teichmüller groups T_{g,K} in the stalks at some base points.

We will choose the base points “at infinity” (see [D4]). Namely, we degenerate the curve C into the projective line ℙ^{1} with g nodal points. More precisely, we choose the real numbers x_{1} < y_{1} < x_{2} < … < x_{g} < y_{g} and degenerate C into ℙ^{1}/(x_{j} = y_{j}, 1 ≤ j ≤ g). The marked points degenerate into the real numbers z_{1} < … < z_{m} (m is the cardinality of K) such that z_{m} < x_{1}. The tangent vectors at all the points are real and directed to the right. We will denote this homotopy base point at infinity by ∞_{g,K}.

19.3. For L_{1}, …, L_{m} ∈ Õ_{ζ} the stalk (⟨{L_{k}}⟩_{C}^{K})_{∞_{g,K}} is canonically isomorphic to

⟨L_{1} ˜⊗ … ˜⊗L_{m} ˜⊗ãd^{˜⊗g}⟩ (see [TUY]). Here we use the notation ⟨?⟩ introduced in IV.5.9 and IV.9 (maximal trivial summand), and the notation ãd introduced in 11.2.1 (semisimple adjoint representation).

The action of T_{g,K} on ⟨L_{1} ˜⊗ … ˜⊗L_{m} ˜⊗ãd^{˜⊗g}⟩ is induced by its action on

L_{1} ˜⊗ … ˜⊗L_{m} ˜⊗ãd^{˜⊗g}: generators act by braiding, balance, and Fourier transforms (see [LM]) of the adjoint representation.

19.4. The Theorem 18.2 equips the category FS (and, via the equivalence Φ, the category C) with the fusion structure. To unburden notations and to distinguish from the case of Õ_{ζ} we will denote the corresponding local systems (or, rather, complexes of sheaves with smooth cohomology) by [L_{1}, …, L_{m}]_{C}^{K}.

Theorem. The stalk ([L_{1}, …, L_{m}]_{C}^{K})_{∞_{g,K}} is canonically isomorphic to

Tor_{\frac{∞}{2} +•}^{C}(B, L_{1} ⊗ … ⊗ L_{m} ⊗ ad^{⊗g})

(notations of IV.4.7 and 10.1).

Proof. In effect, the stalk in question is the global sections of some sheaf on the space C^{α} where C = ℙ^{1}/(x_{j} = y_{j}, 1 ≤ j ≤ g). The global sections will not change if we merge the points y_{j} into x_{j} for any j along the real line. The specialization of our sheaf will live on the space (ℙ^{1})^{α} (no nodal points anymore).

It will be the sheaf obtained by gluing {L_{k}}, T_{1}, …, T_{g} into the marked points z_{1}, …, z_{m}, x_{1}, …, x_{g}. Here L_{k} = Φ^{−1}L_{k}, and the sheaves T_{1}, …, T_{g} are just g copies of the same sheaf T = Φ^{−1}ad. This follows immediately from the Theorem 17.3, the Definition of ad in 11.1, and the Theorem 15.5.

It remains to apply the Theorem IV.8.4. □

19.5. The action of T_{g,K} on Tor_{\frac{∞}{2} +•}^{C}(B, L_{1} ⊗ … ⊗ L_{m} ⊗ ad^{⊗g}) is induced by its action on L_{1} ⊗ … ⊗ L_{m} ⊗ ad^{⊗g}: generators act by braiding, balance, and Fourier transforms of the adjoint representation.

Now suppose L_{k} ∈ Õ_{ζ} for all k ∈ K. Then L_{1} ˜⊗ … ˜⊗L_{m} ˜⊗ãd^{˜⊗g} = ⟨L_{1} ⊗…⊗L_{m} ⊗ad^{⊗g}⟩_{Δ} (notations of 11.2) by the Theorem 11.3. The RHS is a natural subquotient of L_{1} ⊗ … ⊗ L_{m} ⊗ ad^{⊗g}, hence the action of T_{g,K} on the latter module induces some action of T_{g,K} on the RHS. This action coincides with the one of 19.3 by the uniqueness result of [MS] which claims that any two extensions of bbrt structure on a category to a fusion structure coincide.

19.6. By the virtue of Corollary IV.5.11, ⟨L_{1} ˜⊗ … ˜⊗L_{m} ˜⊗ãd^{˜⊗g}⟩ is canonically a subquotient of Tor_{\frac{∞}{2} +•}^{C}(B, (L_{1} ⊗ … ⊗ L_{m} ⊗ ad^{⊗g}) ⊗ L(2ρ_{ℓ})).

The above discussion implies that the action 19.3 of T_{g,K} on the former space is induced by its action 19.5 on the latter space. More precisely, T_{g,K} acts on Tor_{\frac{∞}{2} +•}^{C}(B, (L_{1} ⊗ … ⊗ L_{m} ⊗ ad^{⊗g}) ⊗ L(2ρ_{ℓ})) in the evident way, leaving L(2ρ_{ℓ}) alone.

19.7. Let us translate the above discussion back into geometric language. Let us consider the set J := K ⊔ j. We denote by ξ the natural projection T C^{∘J} ⟶ T C^{∘K}.

Summarizing the above discussion, we obtain Theorem. Let L_{k} ∈ Õ_{ζ}, k ∈ K. Then the local system ξ^{∗}⟨L_{1}, …, L_{m}⟩_{C}^{K} on T C^{∘J} is a natural subquotient of the local system [L_{1}, …, L_{m}, L(2ρ_{ℓ})]_{C}^{J}.

19.8. Let us reformulate the above Theorem in more concrete terms.

Recall that ∇_{κ} = Δ_{ℓ} is the first alcove parametrizing the irreducibles in Õ_{κ}. Let λ_{1}, …, λ_{m} ∈ ∇_{κ} be a K-tuple of weights. We consider the corresponding irreducibles L(λ_{1}), …, L(λ_{m}) ∈ Õ_{κ}, and the corresponding local system of conformal blocks ⟨L(λ_{1}), …, L(λ_{m})⟩_{C}^{K} on the space T C^{∘K}.

On the other hand, consider α := λ_{1} + … + λ_{m} + 2gρ_{ℓ}. We define J := K ⊔ j, and λ_{j} := 2ρ_{ℓ}. Let λ⃗ denote λ_{1}, …, λ_{m}, λ_{j}.

Denote by X α ; λ⃗ the sheaf on Cα ; λ⃗ obtained by gluing L(λ_{1}), …, L(λ_{m}), L(λ_{j}).

Note that X α ; λ⃗ = u_{!∗}Hα ; λ⃗ where u stands for the open embedding C∘α ; λ⃗ ↪ Cα ; λ⃗.

Recall that η stands for the projection Cα ; λ⃗ ⟶ T C^{∘J}.

Theorem. The local system of conformal blocks ξ^{∗}⟨L(λ_{1}), …, L(λ_{m})⟩_{C}^{K} is isomorphic to a canonical subquotient of a “geometric” local system Rη_{∗}u_{!∗}Hα ; λ⃗. □

19.9. Corollary. The above local system of conformal blocks is semisimple. It is a direct summand of the geometric local system above.

Proof. The geometric system is semisimple by Decomposition theorem, [BBD], Théorème 6.2.5. □

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R.B.: School of Mathematical Sciences, Tel-Aviv University, Ramat Aviv, 69978, Israel

E-mail address: roman@math.tau.ac.il, roma@ias.edu

M.F.: Independent Moscow University, Bolshoj Vlasjevskij Pereulok, dom 11, Moscow 121002 Russia

E-mail address: fnklberg@main.mccme.rssi.ru

V.S.: Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, 53225 Bonn Germany

E-mail address: vadik@mpim-bonn.mpg.de