STRONGLY TYPICAL REPRESENTATIONS OF THE BASIC CLASSICAL LIE SUPERALGEBRAS.

MARIA GORELIK

Abstract. We describe the category of representations with a strongly typical central character of a basic classical Lie superalgebra in terms of representations of its even part.

1. Introduction

1.1. In [PS1], I. Penkov and V. Serganova show that the category of representations of a basic classical Lie superalgebra 𝔤 of type I with a fixed typical central character is equivalent to the category of representations of the even part 𝔤_{0} with a suitable central character. A similar result for type II was proven by I. Penkov in [P] for “generic” central characters. The aim of this paper is to understand for which central characters such an equivalence holds. We also study a simplest example when such an equivalence fails to exist, but the corresponding category of representations of a Lie superalgebra still has a good description.

1.2. The basic classical Lie superalgebras were described by V. Kac in [K1]. These Lie superalgebras are the closest to the ordinary simple Lie algebras: for instance, they can be described by their Cartan matrices. The even part of a basic classical Lie superalgebra 𝔤 is a reductive Lie algebra. The centre Z(𝔤) is described in [K2], [S1], [BZV]; it is isomorphic to a subalgebra of S(𝔥)^{W} and their fields of fractions coincide (here 𝔥 is a Cartan subalgebra of 𝔤_{0} and W is the Weyl group of 𝔤_{0}).

1.3. Let 𝔤 = 𝔤_{0} ⊕ 𝔤_{1} be a basic classical Lie superalgebra, Ũ be its universal enveloping superalgebra and Z(𝔤) be the centre of Ũ. Let T be a special ghost element constructed in [G1]— see 2.3. We call a maximal ideal χ̃ of Z(𝔤) strongly typical if it does not contain T^{2}.

For a maximal ideal χ̃ of Z(𝔤) and a 𝔤-module Ñ set Ñ_{χ} := {v ∈ Ñ| χ̃^{r}v = 0, ∀r >> 0}. For a fixed χ̃ ∈ Max Z(𝔤), denote by gr C̃_{r} the category of graded 𝔤-modules Ñ satisfying χ̃^{r}Ñ = 0 and by gr C̃_{∞} the category of graded 𝔤-modules Ñ satisfying Ñ_{χ} = Ñ.

Consider 𝔤_{0} as a purely even Lie superalgebra. Denote by U its enveloping algebra and by Z(𝔤_{0}) the centre of U. For a maximal ideal χ of Z(𝔤_{0}) and a 𝔤_{0}-module N define N_{χ} as above. For a fixed χ ∈ Z(𝔤_{0}), denote by gr C_{r} the category of graded 𝔤_{0}-modules N satisfying χ^{r}N = 0 and by gr C_{∞} the category of graded 𝔤_{0}-modules N satisfying N_{χ} = Ñ. In Sect. 3 we prove the following

The author was partially supported by TMR Grant No. FMRX-CT97-0100. Research at MSRI is supported in part by NSF grant DMS-9701755.

1.3.1. Theorem. For a strongly typical χ̃ ∈ Max Z(𝔤) the category gr C̃_{∞} is equivalent to the category gr C_{∞} where χ is a suitable maximal ideal of Z(𝔤_{0}). The equivalence is given by the functors

Ñ ↦ Ñ_{χ}, N ↦ (Ind_{𝔤_{0}}^{𝔤} N)_{χ}.

The restriction of these functors provide also the equivalence of gr C̃_{r} with gr C_{r}.

A maximal ideal χ ∈ Max Z(𝔤_{0}) is suitable in the sense of the above theorem iff for some projective Verma 𝔤-module M̃ satisfying χ̃M̃ = 0, the 𝔤_{0}-module M̃_{χ} is Verma and, moreover, one has M̃ = ŨM̃_{χ}. Note that if χ is a perfect mate for χ̃ in the sense of [G2], then it fulfills these conditions. In [G2], Sect.8 we prove the existence of a perfect mate χ for each strongly typical χ̃ ∈ Max Z(𝔤). This allows one to describe the category gr C̃_{∞} corresponding to any strongly typical χ̃ in terms of Theorem 1.3.1.

The proof is based on the theorem stating that for a strongly typical χ̃ the annihilator of a Verma 𝔤-module M̃ satisfying χ̃M̃ = 0 is equal to Ũχ̃— see [G2].

An analogue of Theorem 1.3.1 for the basic classical Lie superalgebras of type I is proven in [PS1]; for “generic” χ̃ in type II case, it is proven in [P]. Both proofs are based on a realization of the categories gr C̃_{1}, gr C_{1} as categories of D-modules on the corresponding flag varieties.

1.4. Theorem 1.3.1 suggests a study of categories of representations of 𝔤 corresponding to a non strongly typical character χ̃. Call such χ̃ weakly atypical. If χ̃ is not typical, the category C̃_{∞} has a very complicated structure— for instance, it contains non-trivial extensions of non-isomorphic finite dimensional modules. An “intermediate” case, when χ̃ is typical but not strongly typical (it is possible only for the types B(m, n), G(3)), seems to be less complicated. We consider the case when 𝔤 = 𝔬𝔰𝔭(1, 2l) (B(0, 2l)) and the maximal ideal is a “generic” weakly atypical (see the condition (10)). This means that the highest weight λ of a highest weight module annihilated by 𝔪 belongs to exactly one of the hyperplanes S_{β}, β ∈ Δ_{1}^{+} where S_{β} := {μ ∈ 𝔥^{∗}| (μ + ρ, β) = 0}.

It is convenient to substitute the maximal ideal 𝔪 of Z(𝔤) by the maximal ideal χ̃ of the “ghost centre” Z̃(𝔤) (see [G1] for definition) which contains 𝔪. For 𝔤 = 𝔬𝔰𝔭(1, 2l) one has Z̃(𝔤) = Z(𝔤) ⊕ T Z(𝔤) and the ideal χ̃ := 𝔪 ⊕ T Z(𝔤) is a maximal ideal of Z̃(𝔤). Define the categories gr C̃_{r} and gr C̃_{∞} for such χ̃ ∈ Max Z̃(𝔤) in the same way as it was done above for χ̃ ∈ Max Z(𝔤). For fixed χ, denote by C_{r} the category of non-graded 𝔤_{0}-modules N satisfying χ^{r}N = 0 and by C_{∞} the full subcategory of the category of non-graded 𝔤_{0}-modules consisting of the modules N satisfying N_{χ} = Ñ.

Under the above assumption on 𝔪, we prove in Sect. 4 the following 1.4.1. Theorem. The category gr C̃_{∞} is equivalent to the category C_{∞} where χ is a suitable maximal ideal of Z(𝔤_{0}). The equivalence is given by the functors

Ñ ↦ Ñ_{χ} ∩ Ñ_{0}, N ↦ (Ind_{𝔤_{0}}^{𝔤} N)_{χ}

where the grading on Ind_{𝔤_{0}}^{𝔤} N is determined by the assuming N to be even.

The restriction of these functors provide also the equivalence of gr C̃_{r} with C_{r}.

A maximal ideal χ ∈ Max Z(𝔤_{0}) is suitable in the sense of the above theorem iff for some projective Verma 𝔤-module M̃ satisfying χ̃M̃ = 0, the 𝔤_{0}-module M := M̃_{χ} ∩ M̃_{0} is Verma and M̃ = ŨM. In 4.5.3 we construct such a suitable ideal χ for each 𝔪 ∈ Max Z(𝔤) satisfying the above assumption.

The proof is based on the theorem stating that the annihilator of a Verma 𝔤-module M̃ satisfying 𝔪M̃ = 0 is equal to Ũχ̃ (see [GL1], 6.2) and on Proposition 4.7 which describes the locally finite part F (M̃, M̃) of endomorphisms of a Verma module M̃ through the image of Ũ in it.

1.5. Acknowledgments. The results of this paper were obtained when the author was a visitor at MSRI and at Max-Planck Institut für Mathematik at Bonn. I express my gratitude to these institutions for the hospitality and excellent working conditions. I wish to thank V. Serganova and I. Penkov for helpful discussions.

2. Preliminaries

Everywhere in the paper 𝔤 = 𝔤_{0}⊕𝔤_{1} denotes one (unless otherwise specified, an arbitrary one) of the basic classical complex Lie superalgebras 𝔤𝔩(m, n), 𝔰𝔩(m, n), 𝔬𝔰𝔭(m, n), 𝔭𝔰𝔩(n, n). Each of these Lie superalgebras possesses the following properties: it admits a 𝔤-invariant bilinear form which is non-degenerate on [𝔤, 𝔤] and the even part 𝔤_{0} is a reductive Lie algebra.

2.1. Conventions. In this paper the ground field is ℂ. We denote by ℕ^{+} the set of positive integers. If A is an algebra, N is an A-module and X, Y are subsets of A and N respectively, we denote by XY the submodule spanned by the products xy where x ∈ X, y ∈ Y.

For a ℤ_{2}-homogeneous element u of a superalgebra denote by d(u) its ℤ_{2}-degree. In all formulae where this notation is used, u is assumed to be ℤ_{2}-homogeneous.

For a Lie superalgebra 𝔪 denote by U(𝔪) its universal enveloping algebra and by S(𝔪) its symmetric algebra. All modules in the text are assumed to be left modules unless otherwise specified. An 𝔪-module N is called locally finite if dim U(𝔪)v < ∞ for all v ∈ N. Set Ũ := U(𝔤) and U := U(𝔤_{0}).

The symbol Ṽ (resp., V) is always used for a simple 𝔤 (resp., 𝔤_{0}) module and the symbol M̃ (resp., M) for a Verma 𝔤 (resp., 𝔤_{0}) module.

2.2. Fix a triangular decomposition 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+}— see [PS2] for definition. Denote by Δ the set of all non-zero roots of 𝔤 and by Δ_{0}^{+} (resp., Δ_{1}^{+}) the set of non-zero positive even (resp., odd) roots of 𝔤. Set \overline{Δ}_{1}^{+} := \{β ∈ Δ_{1}^{+}| 2β ∉ Δ_{0}^{+}\}. and

ρ := \frac{1}{2}( ∑_{α∈Δ_{0}^{+}} α − ∑_{β∈Δ_{1}^{+}} β).

Let W be the Weyl group of 𝔤_{0}. For w ∈ W, μ ∈ 𝔥^{∗} set

w.μ = w(μ + ρ) − ρ.

Denote by (−, −) a 𝔤-invariant bilinear form on 𝔤 which is non-degenerate on [𝔤, 𝔤] and also the induced W -invariant bilinear form on 𝔥^{∗}. One has \overline{Δ}_{1}^{+} = \{β ∈ Δ_{1}^{+}| (β, β) = 0\}.

2.3. Define the adjoint action of 𝔤 on Ũ by setting

(ad g)u = gu − (−1)^{d(g)d(u)}ug, ∀g ∈ 𝔤, u ∈ Ũ

By default, the action of 𝔤 on Ũ is assumed to be the adjoint action. The centre Z(𝔤) of Ũ is equal to Ũ^{ad 𝔤}.

Define the twisted adjoint action of 𝔤 on Ũ by setting

(ad′ g)u = gu − (−1)^{d(g)(d(u)+1)}ug, ∀g ∈ 𝔤, u ∈ Ũ.

The anticentre A(𝔤) := Ũ^{ad′ 𝔤} contains an element T defined in [G1]. This is a unique element of A(𝔤) satisfying P(T)(λ) = ∏_{β∈Δ_{1}^{+}}(β, λ + ρ) for any λ ∈ 𝔥^{∗}; here P : Ũ → S(𝔥) is the Harish-Chandra projection. The element T is even. Since T belongs to A(𝔤), it commutes with the even elements of Ũ and anticommutes with the odd ones; in particular, T^{2} ∈ Z(𝔤).

The Harish-Chandra projection P provides a monomorphism ι : Z(𝔤) → S(𝔥)^{W.}. The image of ι is described in [K2], [S1], [BZV]; ι is bijective iff 𝔤 = 𝔬𝔰𝔭(1, 2l). The centre Z(𝔤) contains an element Q such that

P(Q)(λ) = ∏_{β∈\overline{Δ}_{1}^{+}}(β, λ + ρ).

The localized algebra Z(𝔤)[T^{−2}] is isomorphic to a localization of a polynomial algebra S(𝔥)^{W} by P(Q)—see [K3]. The localized algebra Z(𝔤)[T^{−2}] is isomorphic to a localization of a polynomial algebra S(𝔥)^{W}.

2.4. The category Õ and Verma modules. Denote by O the full subcategory of the category of 𝔤_{0}-modules consisting of finitely generated 𝔥-diagonalizable 𝔤_{0}-modules which are 𝔫_{0}^{+}-locally finite. Denote by Õ the similarly defined category of 𝔤-modules. Since U(𝔫^{+}) is finite over U(𝔫_{0}^{+}), a 𝔤-module N belongs Õ iff as a 𝔤_{0}-module N belongs to O. In particular, any module of category Õ has a finite length.

2.4.1. For λ ∈ 𝔥^{∗} denote by ℂ_{λ} a one-dimensional 𝔟-module such that 𝔫^{+}v = 0 and hv = λ(h)v for any h ∈ 𝔥, v ∈ ℂ_{λ}. Define a Verma module M̃(λ) by setting

M̃(λ) := Ũ ⊗_{U(𝔟)} ℂ_{λ}.

A Verma 𝔤_{0}-module M(λ) is defined similarly.

2.4.2. A maximal ideal χ̃ ∈ Max Z(𝔤) is called typical if it does not contain Q defined in 2.3. If χ̃ ∈ Max Z(𝔤) is typical then the set

W (χ̃) := {λ ∈ 𝔥^{∗}| χ̃M̃(λ) = 0}

forms a single W.-orbit. For 𝔤 = 𝔬𝔰𝔭(1, 2l) all χ̃ ∈ Max Z(𝔤) are typical.

Call a 𝔤-module N typical if Ann_{Z(𝔤)} N is a typical maximal ideal of Z(𝔤). Define the partial order on 𝔥^{∗} by setting μ ≥ ν ⟺ (μ − ν) ∈ ∑_{α∈Δ^{+}} ℕα. A typical Verma module M̃(λ) is projective (resp., simple) in Õ if λ is maximal (resp., minimal) in W.λ—see, for instance, [G2], 2.5.3. In particular, for a typical χ̃ ∈ Max Z(𝔤) there exists a projective Verma module M̃ satisfying χ̃M̃ = 0.

A maximal ideal χ̃ ∈ Max Z(𝔤) is called strongly typical if T^{2} ∉ χ̃. A strongly typical central character is typical. Call a 𝔤-module N strongly typical if Ann_{Z(𝔤)} N is a strongly typical maximal ideal of Z(𝔤).

If 𝔤 is not of types B(m, n), G(3) then Δ_{1}^{+} = \overline{Δ}_{1}^{+} and so the notion of typical and strongly typical module coincide.

2.5. Throughout the paper we shall write “ad 𝔤-module” instead “𝔤-module with respect to the adjoint action”. For any 𝔤-modules N_{1}, N_{2} view Hom(N_{1}, N_{2}) := Hom_{ℂ}(N_{1}, N_{2}) as a 𝔤-module with respect to the adjoint action and denote by F (N_{1}, N_{2}) the locally finite part of the ad 𝔤-module Hom(N_{1}, N_{2}). Similarly for any 𝔤_{0}-modules N_{1}, N_{2} denote by F (N_{1}, N_{2}) the locally finite part of the ad 𝔤_{0}-module Hom(N_{1}, N_{2}). Notice that for any 𝔤-module N its locally finite part coincides with its ad 𝔤_{0}-locally finite part, since Ũ is a finite extension of U.

2.5.1. Let N be a 𝔤_{0}-module and E be a finite dimensional 𝔤_{0}-module. It is easy to check that A := F (N, N), B := F (N ⊗ E, N ⊗ E) are subalgebras of Hom(N, N) and Hom(N ⊗ E, N ⊗ E) respectively. The algebra A acts on F (N, N ⊗ E) from the right and the algebra B acts on F (N, N ⊗ E) from the left; these actions commute. We claim that F (N, N ⊗ E) is a free right A-module whose rank is equal to the dimension of E and, moreover, End_{A}(F (N, N ⊗ E)) = B.

Indeed, consider the map ι : F (N, N) ⊗ E → Hom(N, N ⊗ E) given by ι(f ⊗ v)(n) = f(n) ⊗ v for any f ∈ F (N, N), v ∈ E and the map ι′ : F (N, N ⊗ E) → Hom(N, N) ⊗ E given by ψ ↦ ∑_{i} p_{i} ∘ ψ ⊗ e_{i} where {e_{i}} is a basis of E and p_{i} : N ⊗ E → N are given by p_{i}(n⊗e_{j}) := δ_{i,j}n⊗e_{j}. One can easily sees that Im ι ⊆ F (N, N ⊗E), Im ι′ ⊆ F (N, N)⊗E and that ι′ ∘ι = id, ι∘ι′ = id. Thus F (N, N ⊗E) ≅ F (N, N)⊗E is a free right A-module whose rank is equal to the dimension of E. Similarly, the map ι′′ : A⊗End_{ℂ}(E) → B given by ι′′(f ⊗ ϕ)(n ⊗ v) := f(n) ⊗ ϕ(v) for any f ∈ A = F (N, N), n ∈ N, v ∈ E, ϕ ∈ End_{ℂ}(E) is bijective. For any f, f′ ∈ A, v ∈ E, ϕ ∈ End_{ℂ}(E) one has ι′′(f ⊗ ϕ)(ι(f′ ⊗ v)) = ι(ff′ ⊗ ϕ(v)). This implies End_{A}(F (N, N ⊗ E)) = B since the A-module F (N, N ⊗ E) is freely generated by the elements ι(1 ⊗ e_{i}) (here 1 is the unit of A).

2.6. Let M be a Verma 𝔤-module. By Duflo’s theorem Ann M = U Ann_{Z(𝔤_{0})} M—see [D]. By [J1], 6.4 the natural map U/(Ann M) → F (M, M) is bijective.

Let M̃ be a strongly typical Verma 𝔤-module. Then Ann M̃ = Ũ Ann_{Z(𝔤)} M̃ and the natural map Ũ/(Ann M̃) → F (M̃, M̃) is bijective (see [G2], 9.4,9.5).

2.7. For a 𝔤-module Ñ and a maximal ideal χ̃ ∈ Z(𝔤) set

Ñ_{χ} := {v ∈ Ñ| χ̃^{r}v = 0, r >> 0}.

We say that a Ũ-module Ñ has a finite support supp_{Z(𝔤)} Ñ = {χ̃_{1}, …, χ̃_{k}} if for any v ∈ Ñ there exist r_{1}, …, r_{k} ∈ ℕ^{+} such that ∏_{i} χ̃_{i}^{r_{i}}v = 0. In this case,

Ñ = ⊕_{i}Ñ_{χ_{i}}

and each Ñ_{χ_{i}} is canonically isomorphic to the localization of the module Ñ at χ̃_{i}. If Ñ has a finite support and 0 → Ñ′ → Ñ → Ñ′′ → 0 is an exact sequence then, for any χ̃′ ∈ Max Z(𝔤), the sequence 0 → Ñ′_{χ′} → Ñ_{χ′} → Ñ\begin{matrix}′ & ′ \\ & χ′ \end{matrix} → 0 is also exact.

We adopt the similar notation for Z(𝔤_{0}) and U-modules. For a graded 𝔤-module Ñ and a maximal ideal χ of Z(𝔤_{0}) we set

Ñ_{χ;i} := Ñ_{χ} ∩ Ñ_{i}

for i = 0, 1.

2.7.1. Let χ̃ be a strongly typical central character, M̃ be such that χ̃M̃ = 0. Then

Ũχ̃ ∩ Z(𝔤_{0}) = Ann M̃ ∩ Z(𝔤_{0}) = ∏_{χ∈supp_{Z(𝔤_{0})} _{M}} χ^{r(χ)}

where r(χ) is the minimal r such that χ^{r}M̃_{χ} = 0. In particular, if χ ∈ Max Z(𝔤_{0}) is such that M̃_{χ} is a Verma 𝔤_{0}-module then r(χ) = 1. Moreover, any 𝔤-module Ñ satisfying χ̃Ñ = 0 has a finite support in Z(𝔤) (which is a subset of supp_{Z(𝔤_{0})} M̃) and one has χÑ_{χ} = 0 if M̃_{χ} is a Verma 𝔤_{0}-module.

2.7.2. For a strongly typical χ̃ ∈ Max Z(𝔤) call χ ∈ Max Z(𝔤_{0}) a perfect mate if

(i) For any Verma 𝔤-module M̃ annihilated by χ̃, the 𝔤_{0}-module M̃_{χ} is Verma.

(ii) For any non-trivial 𝔤-module Ñ annihilated by χ̃, the 𝔤_{0}-module Ñ_{χ} is non-trivial. In [G2], Sect.8 we describe a perfect mate for each strongly typical χ̃ ∈ Max Z(𝔤).

2.8. For a graded 𝔤_{0}-module L denote by Ind_{𝔤_{0}}^{𝔤} L the vector space Ũ ⊗_{U} L (here Ũ is considered as a right U-module and a left Ũ-module through the multiplication) equipped with the natural structure of a left graded Ũ-module. Denote by Coind_{𝔤_{0}}^{𝔤} L the vector space Hom_{U}(Ũ, L) (here Ũ is considered as a left U-module) equipped with the following structure of a left graded Ũ-module: (uf)(u′) := f(u′u) for any f ∈ Hom_{U}(Ũ, L), u, u′ ∈ Ũ. For a graded 𝔤-module Ñ and a graded 𝔤_{0}-module L one has the canonical bijections

Hom_{𝔤_{0}}(Ñ, L) ⟶^{∼} Hom_{𝔤}(Ñ, Coind_{𝔤_{0}}^{𝔤} L),

Hom_{𝔤_{0}}(L, Ñ) ⟶^{∼} Hom_{𝔤}(Ind_{𝔤_{0}}^{𝔤} L, Ñ).

(1)

By [BF], Ind_{𝔤_{0}}^{𝔤} L ≅ Coind_{𝔤_{0}}^{𝔤} L for any graded 𝔤_{0}-module L.

The same formulae define non-graded versions of Ind_{𝔤_{0}}^{𝔤} L and Coind_{𝔤_{0}}^{𝔤} L. The same canonical bijections (1) take place.

3. Equivalence of Categories for a strongly typical central character

In this section we prove that the category of Ũ-modules Ñ satisfying Ñ = Ñ_{χ} is equivalent to the category of U-modules N satisfying N = N_{χ} provided χ̃ ∈ Max Z(𝔤) is strongly typical and χ ∈ Max Z(𝔤_{0}) is its perfect mate—see 2.7.2. The corresponding quasi-inverse functors are given by Ñ ↦ Ñ_{χ} and by N ↦ (Ind_{𝔤_{0}}^{𝔤} N)_{χ}. There are two versions of this result: graded and non-graded. We consider first a non-graded version. A graded version is easily deduced from the non-graded one.

3.1. Notation. Take a strongly typical χ̃ ∈ Max Z(𝔤) and let χ ∈ Max Z(𝔤_{0}). Denote by C̃_{r} (r ∈ ℕ^{+}) the category of non-graded Ũ-modules Ñ satisfying χ̃^{r}Ñ = 0 and by C̃_{∞} the category of non-graded Ũ-modules Ñ satisfying Ñ = Ñ_{χ}. Similarly, let C_{r} be the category of U-modules N satisfying χ^{r}N = 0 and C_{∞} be the category of U-modules N satisfying N = N_{χ}. Evidently C̃_{r} is a full subcategory of C̃_{r+1} and any module in C̃_{∞} is a direct limit of modules belonging to C̃_{r} for r → ∞.

3.1.1. Let M be a Verma 𝔤_{0}-module such that χM = 0. We shall use the following equality

Ann_{U}(Ind_{𝔤_{0}}^{𝔤} M) = Ũχ

which follows from the fact that Ũ is free over U.

Let us show that

supp_{Z(𝔤)} Ind_{𝔤_{0}}^{𝔤} N ⊆ supp_{Z(𝔤)} Ind_{𝔤_{0}}^{𝔤} M  (2)

for any N ∈ C_{∞}. Indeed, for any N ∈ C_{1}

Ann_{Z(𝔤)}(Ind_{𝔤_{0}}^{𝔤} N) ⊇ Ũχ ∩ Z(𝔤) = Ann_{Z(𝔤)}(Ind_{𝔤_{0}}^{𝔤} M)

that implies the inclusion (2) (for N ∈ C_{1}). Any N ∈ C_{r} admits a finite filtration with the factors belonging to C_{1} and so the inclusion (2) holds for such N. To deduce (2) for any N ∈ C_{∞}, observe that for any v ∈ Ind_{𝔤_{0}}^{𝔤} N there exists a finitely generated submodule N′ of N such that Ind_{𝔤_{0}}^{𝔤} N′ contains v. Since N′ is finitely generated, it lies in C_{r} for a suitable r ∈ ℕ^{+}. This implies the inclusion (2).

Hence for any N ∈ C_{∞}

Ind_{𝔤_{0}}^{𝔤} N = ⊕_{χ′∈supp_{Z(𝔤)} Ind_{𝔤_{0}}^{𝔤} M}(Ind_{𝔤_{0}}^{𝔤} N)_{χ′}  (3)

Similarly, choose M̃ such that Ũχ̃ = Ann M̃. Then for any Ñ ∈ C̃_{∞}

Ñ = ⊕_{χ′∈supp_{Z(𝔤_{0})} _{M}} Ñ_{χ′}.  (4)

3.1.2. Consider the map f : Ũ/(Ũχ) → F (M, Ind_{𝔤_{0}}^{𝔤} M) induced by the natural map Ũ → F (M, Ind_{𝔤_{0}}^{𝔤} M) given by u ↦ (m ↦ u ⊗ m). We claim f is an isomorphism of left Ũ and right U-modules. Since Ũ is free over U and Ann_{U} M = Uχ, the map f is injective. Kostant’s Separation Theorem (see [Ko]) states the existence of an ad 𝔤_{0}-submodule H′ of S(𝔤_{0}) such that the multiplication map provides an isomorphism H′ ⊗ S(𝔤_{0})^{𝔤_{0}} ⟶^{∼} S(𝔤_{0}). Then

S(𝔤) = Λ𝔤_{1} ⊗ S(𝔤_{0}) = (Λ𝔤_{1} ⊗ H′) ⊗ S(𝔤_{0})^{𝔤_{0}}.

Using [BL], 5.4, it is easy to deduce the existence of an ad 𝔤_{0}-submodule H of Ũ such that the multiplication map provides the isomorphism H ⊗ Z(𝔤_{0}) → Ũ. Moreover, H ≅ Λ𝔤_{1} ⊗ H′ as an ad 𝔤_{0}-module. By 2.6, H′ ≅ F (M, M) as ad 𝔤_{0}-modules. The following chain of ad 𝔤_{0}-isomorphisms

Ũ/(Ũχ) ≅ Λ𝔤_{1} ⊗ H′ ≅ Λ𝔤_{1} ⊗ F (M, M) ≅ F (M, Ind_{𝔤_{0}}^{𝔤} M)

implies the surjectivity of f.

The bijectivity of f gives the following useful formula

Ind_{𝔤_{0}}^{𝔤} N = F (M, Ind_{𝔤_{0}}^{𝔤} M) ⊗_{U} N.  (5)

for any N ∈ C_{1}.

3.2. Fix any χ ∈ Max Z(𝔤_{0}), χ̃ ∈ Max Z(𝔤). Define the functors Ψ : C̃_{∞} → C_{∞} and Φ : C_{∞} → C̃_{∞} by the formulae

Ψ(Ñ) = Ñ_{χ}, Φ(N) = (Ind_{𝔤_{0}}^{𝔤} N)_{χ}.

3.2.1. Lemma. The functor Ψ is left and right adjoint to Φ.

Proof. Using (3), one obtains for any N ∈ C_{∞}, Ñ ∈ C̃_{∞}

Hom_{𝔤}(Φ(N), Ñ) = Hom_{𝔤}((Ind_{𝔤_{0}}^{𝔤} N)_{χ}, Ñ) = Hom_{𝔤}(Ind_{𝔤_{0}}^{𝔤} N, Ñ) = Hom_{𝔤_{0}}(N, Ñ)
= Hom_{𝔤_{0}}(N, Ñ_{χ}) = Hom_{𝔤_{0}}(N, Ψ(Ñ))

and also, using (4),

Hom_{𝔤}(Ñ, Φ(N)) = Hom_{𝔤}(Ñ, (Ind_{𝔤_{0}}^{𝔤} N)_{χ}) = Hom_{𝔤}(Ñ, Ind_{𝔤_{0}}^{𝔤} N) ⟶^{∼} Hom_{𝔤}(Ñ, Coind_{𝔤_{0}}^{𝔤} N)
= Hom_{𝔤_{0}}(Ñ, N) = Hom_{𝔤_{0}}(Ñ_{χ}, N) = Hom_{𝔤_{0}}(Ψ(Ñ), N)

where = stands for the natural isomorphisms and ⟶^{∼} is induced by an isomorphism Ind_{𝔤_{0}}^{𝔤} N ≅ Coind_{𝔤_{0}}^{𝔤} N (see 2.8).

3.3. Conventions. Fix a strongly typical χ̃ ∈ Max Z(𝔤) and a Verma 𝔤-module M̃ which is projective in Õ and such that χ̃M̃ = 0. Take χ ∈ Max Z(𝔤_{0}) such that

(a) M := M̃_{χ} is a Verma 𝔤_{0}-module
(b) M̃ = ŨM. (6)

For instance, one can choose χ to be a perfect mate for χ̃ (see 2.7.2).

Till the end of this section χ̃, M̃ and χ chosen as above are assumed to be fixed.

3.3.1. Theorem. The functors

\begin{matrix}Ψ : C̃_{∞} → C_{∞} & Ñ ↦ Ñ_{χ}, \\ Φ : C_{∞} → C̃_{∞} & N ↦ (Ind_{𝔤_{0}}^{𝔤} N)_{χ} \end{matrix}

are mutually quasi-inverse. Moreover, their restrictions provide the equivalence of the categories C̃_{r} and C_{r} for any r ∈ ℕ^{+}.

Outline of the proof. We know by 3.2.1 that the functors Ψ, Φ are adjoint. In 3.4.1 we reduce the required assertion to the “case r = 1” that is to the statement that the restriction Ψ_{1}, Φ_{1} of the functors Ψ, Φ to the categories C̃_{1}, C_{1} provide an equivalence of the categories. Not that 3.4.1 does not use the condition (6).

Observe that the inclusion Ψ_{1}(C̃_{1}) ⊆ C_{1} immediately follows from 2.7.1, but it is not clear apriori that Φ_{1}(C_{1}) ⊆ C̃_{1}. To prove that Ψ_{1}, Φ_{1} provide an equivalence of the categories we show, using Proposition 3.4.3, that the functor Φ′ : N ↦ F (M, M̃) ⊗_{U} N provides an equivalence of the categories C_{1} → C̃_{1}. It is easy to show that Ψ_{1} is the left quasi-inverse to Φ′ and so it provides an equivalence of the categories C̃_{1} → C_{1}. Using this fact, we show that Φ(M) ≅ M̃ and deduce that Φ_{1} is isomorphic to Φ′. This will complete the proof.

3.4. Proof of Theorem 3.3.1.

3.4.1. Reduction to the case r = 1. The formula (3) implies that Φ(N) is isomorphic to the localization of Ind_{𝔤_{0}}^{𝔤} N, considered as a Z(𝔤)-module, at the maximal ideal χ̃ (that is by the set Z(𝔤) \ χ̃).

Similarly, the formula (4) implies that Ψ(Ñ) is isomorphic to the localization of Ñ, considered as a Z(𝔤_{0})-module, at the maximal ideal χ.

Taking into account that the induction and the localization functors are exact and commute with direct limits, one concludes that the assertion of Theorem 3.3.1 is equivalent to the statement that the restrictions Ψ_{1}, Φ_{1} of the functors Ψ, Φ to the categories C̃_{1}, C_{1} provide an equivalence of the categories.

3.4.2. Case r = 1. It remains to show that the restriction Ψ_{1} of Ψ to the subcategory C̃_{1} and the restriction Φ_{1} of Φ to the subcategory C_{1} provide an equivalence of the categories C̃_{1} and C_{1}.

We start with the following technical proposition.

3.4.3. Lemma. Assume that Ñ is a 𝔤-module, N is a 𝔤_{0}-direct summand of Ñ and Ñ is a 𝔤-direct summand of Ind_{𝔤_{0}}^{𝔤} N. Then the F (Ñ, Ñ)-F (N, N) bimodule F (N, Ñ) provides a Morita equivalence between the algebras F (N, N) and F (Ñ, Ñ).

Proof. Denote the algebra F (N, N) by A. For any 𝔤-module X endow the vector space F (N, X) with the natural right A-module structure. Since N is a 𝔤_{0}-direct summand of Ñ, A as a right module over itself is a direct summand of F (N, Ñ). To obtain the statement one has to check that F (N, Ñ) is a finitely generated projective right A-module and that End_{A}(F (N, Ñ)) = F (Ñ, Ñ).

Denote Ind_{𝔤_{0}}^{𝔤} N by I. Recall that as a 𝔤_{0}-module I = Λ𝔤_{1} ⊗ N. By 2.5.1, F (N, I) is a free A-module whose rank is equal to the dimension of Λ𝔤_{1}. Since Ñ is a direct summand of I, the A-module F (N, Ñ) is a direct summand of F (N, I). Hence F (N, Ñ) is a finitely generated projective right A-module. One has End_{A}(F (N, Ñ)) = p End_{A}(F (N, I))p where p ∈ End_{A}(F (N, I)) is the idempotent with the image F (N, Ñ) corresponding to the decomposition F (N, I) = F (N, Ñ)⊕F (N, G). By 2.5.1, the left action of F (I, I) on F (N, I) induces an isomorphism F (I, I) ⟶^{∼} End_{A}(F (N, I)). Thus the left action of F (Ñ, Ñ) on F (N, Ñ) induces an isomorphism F (Ñ, Ñ) ⟶^{∼} End_{A}(F (N, Ñ)). The assertion follows.

3.4.4. Retain the notation of 3.3. Consider a canonical map Ind_{𝔤_{0}}^{𝔤} M → M̃. By the property (b) of (6), this map is surjective. Recall M̃ is projective in Õ. Consequently, M̃ is a 𝔤-direct summand of Ind_{𝔤_{0}}^{𝔤} M and so the pair (M̃, M) satisfies the assumptions of Lemma 3.4.3. From 2.6 it follows that F (M̃, M̃) ≅ Ũ_{χ} and F (M, M) ≅ U_{χ} as algebras. Taking into account Lemma 3.4.3 one concludes that the functor Φ′ : C_{1} → C̃_{1} defined by

Φ′(N) := F (M, M̃) ⊗_{U_{χ}} N = F (M, M̃) ⊗_{U} N

provides an equivalence of the categories.

3.4.5. One has

Ψ_{1} ∘ Φ′(N) = F (M, M̃_{χ}) ⊗_{U} N = F (M, M) ⊗_{U} N = N

since F (M, M) ≅ U_{χ}. Hence Ψ_{1} is quasi-inverse to Φ′ and it provides an equivalence of the categories C̃_{1} → C_{1}.

Let us verify that the functors Φ_{1} and Φ′ are isomorphic. Formula (5) implies that

Φ_{1}(N) = F (M, (Ind_{𝔤_{0}}^{𝔤} M)_{χ}) ⊗_{U} N = F (M, Φ_{1}(M)) ⊗_{U} N  (7)

for any N ∈ C_{1}. Thus to show that Φ_{1}(N) ≅ Φ′(N) for all N ∈ C_{1} it is enough to verify only that

(Ind_{𝔤_{0}}^{𝔤} M)_{χ} ≅ M̃.  (8)

As we have shown above, M̃ is a 𝔤-direct summand of Ind_{𝔤_{0}}^{𝔤} M. Thus it is enough to check that

Hom_{𝔤}(Ñ, Ind_{𝔤_{0}}^{𝔤} M) = 0

for all simple Ñ ∈ C̃_{1} such that Ñ ≅ S̸oc M̃ and that

dim Hom_{𝔤}(Soc M̃, Ind_{𝔤_{0}}^{𝔤} M) = 1.

Recall that Ind_{𝔤_{0}}^{𝔤} M ≅ Coind_{𝔤_{0}}^{𝔤} M (see 2.8) and so

Hom_{𝔤}(Ñ, Ind_{𝔤_{0}}^{𝔤} M) = Hom_{𝔤_{0}}(Ñ, M) = Hom_{𝔤_{0}}(Ñ_{χ}, M).

If Ñ ∈ C̃_{1} is simple, Ñ_{χ} = Ψ_{1}(Ñ) ∈ C_{1} is also simple. Taking into account that the 𝔤_{0}-socle of M is a simple Verma 𝔤_{0}-module, one concludes that Hom_{𝔤_{0}}(Ñ_{χ}, M) = 0 if Ñ_{χ} ≅ S̸oc M and dim Hom_{𝔤_{0}}(Ñ_{χ}, M) = 1 otherwise. If Ψ_{1}(Ñ) = Ñ_{χ} ≅ Soc M then Ñ ≅ Soc M̃ since Ψ_{1} : C̃_{1} → C_{1} provides an equivalence of the categories and Ψ_{1}(M̃) = M. This proves (8).

Hence the functors Φ_{1} and Φ′ are isomorphic. This completes the proof of Theorem 3.3.1.

3.5. Remarks.

3.5.1. Recall that any Ñ ∈ C̃_{∞} is generated by Ñ_{χ} and so is a homomorphic image of Ind_{𝔤_{0}}^{𝔤} Ñ_{χ}. Consequently, the restrictions of the functors Ψ, Φ provide an equivalence between C̃_{∞} ∩ Õ and C_{∞} ∩ O. The same is true for the categories of weight modules. Also, one can easily see that the Verma modules of C̃_{∞} ∩ Õ correspond to the Verma modules of C_{∞} ∩ O.

3.5.2. Formula (8) which says that Φ(M) ≅ M̃ is crucial for the whole proof. If one had an independent proof of this formula, one could deduce from it a straightforward proof of Theorem 3.3.1.

3.6. Graded case. Denote by gr C̃_{∞} (resp., gr C̃_{r}) the full category of ℤ_{2}-graded Ũmodules Ñ satisfying Ñ_{χ} = Ñ (resp., χ̃^{r}Ñ = 0). Consider U as a purely even superalgebra and define similarly gr C_{∞}, gr C_{r}. Denote by # the forgetful functors gr C̃_{∞} → C̃_{∞} and gr C_{∞} → C_{∞}.

3.6.1. Evidently any N′ ∈ C_{∞} is isomorphic to N^{#} for some N ∈ gr C_{∞}— for instance, one can consider N′ as a purely even (odd) module. It turns out that the similar assertion holds for C̃_{∞}. Indeed, since χ̃ is strongly typical, it contains an element (T^{2} −c^{2}) for some non-zero c ∈ ℂ. Denote by Ũ_{0} the even part of Ũ; recall that T lies in the centre of Ũ_{0}. Any Ñ ∈ C̃_{∞} is a direct sum of the Ũ_{0}-modules N_{+}, N_{−} where

N_{+} := {v ∈ M̃| (T − c)^{r}v = 0, ∀r >> 0}, N_{−} := {v ∈ M̃| (T + c)^{r}v = 0, ∀r >> 0}.

One can define a ℤ_{2}-grading on Ñ by putting Ñ_{0} := N_{+}, Ñ_{1} := N_{−}. In such a way, one obtains a functor C̃_{∞} → gr C̃_{∞} which is left quasi-inverse to the functor #. This implies, in particular, that for any irreducible graded Ũ_{χ}-module Ñ, its image Ñ^{#} remains irreducible (non-graded) module.

3.6.2. Theorem. The functors

Ψ^{gr} : gr C̃_{∞} → gr C_{∞}, Ñ ↦ Ñ_{χ},
Φ^{gr} : gr C_{∞} → gr C̃_{∞}, N ↦ (Ind_{𝔤_{0}}^{𝔤} N)_{χ}

are mutually quasi-inverse. Their restriction provides an equivalence of the categories gr C̃_{r} and gr C_{r} for any r ∈ ℕ^{+}.

Proof. Repeating the arguments of Lemma 3.2.1, one shows that the functor Ψ^{gr} is left and right adjoint to the functor Φ^{gr}.

The functors Ψ, Φ are quasi-inverse and so the canonical homomorphism α_{N} : N → Ψ ∘ Φ(N) is an isomorphism for any N ∈ C_{∞}. For any N ∈ gr C_{∞} denote by α_{N} the canonical homomorphism N → Ψ^{gr}∘Φ^{gr}(N). One has #∘Ψ^{gr} = Ψ∘# and #∘Φ^{gr} = Φ∘#. Therefore #(α_{N}) = α_{N^{#}}. Since α_{N^{#}} is an isomorphism, α_{N} is also an isomorphism.

Similarly, the canonical homomorphism β_{N} : Φ^{gr} ∘Ψ^{gr}(Ñ) → Ñ is an isomorphism for any Ñ ∈ gr C̃_{∞}. Hence gr Ψ and gr Φ are quasi-inverse.

The equality Ψ^{gr}(gr C̃_{r}) = gr C_{r} follows from the equality Ψ(gr C̃_{r}) = C_{r}. This implies the second claim.

4. An Example

Say that a Ũ-central character is weakly atypical if it is not strongly typical. In this section we consider a “generic” weakly atypical central characters for 𝔤 := 𝔬𝔰𝔭(1, 2l). We show that, for such a character 𝔪 ∈ Max Z(𝔤), the category gr C̃_{∞} of graded 𝔤-modules Ñ satisfying Ñ_{𝔪} = Ñ is equivalent to the category C_{∞} of 𝔤_{0}-modules N satisfying N_{χ} = N for an appropriate χ. We see that in this case the category gr C̃_{∞} is “twice smaller” than one could expect in the strongly typical case.

Throughout this section all 𝔤-modules are assumed to be graded.

4.1. The superalgebra 𝔬𝔰𝔭(1, 2l) does not have isotropic roots: (β, β) ≠ 0 for any β ∈ Δ_{1}. As a result, 𝔬𝔰𝔭(1, 2l) has many features of the simple Lie algebras. For instance, Ũ is a domain, its centre Z(𝔤) is a polynomial algebra and all finite dimensional representations are completely reducible. In [M1], Musson proved the existence of a harmonic space H in Ũ: this is an ad 𝔤-submodule of Ũ such that the multiplication map provides the isomorphism H ⊗ Z(𝔤) ⟶^{∼} Ũ.

If M̃ is strongly typical its annihilator is a centrally generated ideal.

Suppose that M̃ is not strongly typical; this means that 𝔪 := Ann_{Z(𝔤)} M̃ contains T^{2}. Then, by [GL1], 6.2,

Ann M̃ = Ũ(T Z(𝔤) + 𝔪)

and χ̃ := (T Z(𝔤) + 𝔪) is the maximal ideal of the “ghost centre” Z̃(𝔤) = Z(𝔤) + T Z(𝔤). The ideal Ũχ̃ is primitive since a Verma module M̃(μ) is simple if μ ∈ W (𝔪) is minimal.

Moreover, by [GL1], 5.3, for any simple 𝔤_{0}-module V one has

dim Hom_{𝔤_{0}}(V, Ũ/(Ann M̃)) = \frac{1}{2} dim Hom_{𝔤_{0}}(V, H)  (9)

where H is the harmonic space mentioned above.

4.2. Retain the notation of 2.4.2. Recall that 𝔪 ∈ Max Z(𝔤) is strongly typical iff the elements λ ∈ W (𝔪) do not belong to the hyperplanes

S_{β} := {μ ∈ 𝔥^{∗}| (μ + ρ, β) = 0}

for β ∈ Δ_{1}^{+}. In this section we consider 𝔪 ∈ Max Z(𝔤) such that any λ ∈ W (𝔪) belongs to exactly one hyperplane S_{β} that is

∃!β ∈ Δ_{1}^{+} : (λ + ρ, β) = 0.  (10)

In particular, 𝔪 is not strongly typical and so T^{2} ∈ 𝔪. Set

χ̃ := T Z(𝔤) + 𝔪.

For any 𝔤-module Ñ the vector space

Ñ_{χ} := {v ∈ Ñ| χ̃^{r}v = 0, r >> 0}

is a 𝔤-submodule of Ñ. Let gr C̃_{r} (resp., gr C̃_{∞}) be the category of graded 𝔤-modules Ñ satisfying χ̃^{r}Ñ = 0 (resp., Ñ_{χ} = Ñ). Observe that C̃_{1} ⊆ C̃′_{1} ⊆ C̃_{2} where C̃′_{1} is the category of graded 𝔤-modules Ñ satisfying 𝔪Ñ = 0.

Fix a projective Verma module M̃ annihilated by χ̃. Fix χ ∈ Max Z(𝔤_{0}) such that

(a) M := M̃_{χ;0} is a Verma 𝔤_{0}-module
(b) M̃ = ŨM. (11)

(the existence of χ satisfying (11) will be shown in 4.5). Let C_{∞} (resp., C_{r}) be the category of 𝔤_{0}-modules N satisfying N_{χ} = N (resp., χ^{r}N = 0).

4.3. Theorem. The functors

Ψ : gr C̃_{∞} → C_{∞}  Ñ ↦ (Ñ_{0})_{χ},
Φ : C_{∞} → gr C̃_{∞} N ↦ (Ind_{𝔤_{0}}^{𝔤} N)_{χ}

where the grading on Ind_{𝔤_{0}}^{𝔤} N = Ũ⊗_{U} N is determined by assuming N to be even, are quasiinverse. Moreover their restrictions to gr C̃_{r} and C_{r} respectively provide an equivalence of these categories.

Proof. First steps of the proof are the same as those in the proof of Theorem 3.3.1. Considering Ind_{𝔤_{0}}^{𝔤} N as the graded module with respect to the grading defined above one obtains

Hom_{𝔤}(Ind_{𝔤_{0}}^{𝔤} N, Ñ) = Hom_{𝔤_{0}}(N, Ñ_{0})

for any graded 𝔤-module Ñ. Repeating 3.2.1, one concludes that the functors Ψ, Φ are adjoint. Repeating 3.4.1, one reduces the required assertion to “the case r = 1” that is to the statement that the restrictions Ψ_{1}, Φ_{1} of the functors Ψ, Φ to the categories gr C̃_{1}, C_{1} provide an equivalence of the categories.

Evidently, M is a 𝔤_{0}-direct summand of M̃. The embedding M → M̃ gives rise to a non-zero 𝔤-map Ind_{𝔤_{0}}^{𝔤} M → M̃. Using M̃ = ŨM and the projectivity of M̃, one concludes that M̃ is a 𝔤-direct summand of Ind_{𝔤_{0}}^{𝔤} M. Denote by B the algebra F (M̃, M̃) and by B-Mod the category of B-modules. Taking into account Lemma 3.4.3, one concludes that the functor Φ′′ : C_{1} → B-Mod defined by

Φ′′(N) := F (M, M̃) ⊗_{U} N

provides an equivalence of the categories.

The category gr C̃_{1} is the category of graded modules over the superalgebra A := Ũ/(Ũχ̃). In Proposition 4.7 below we prove that B = A ⊕ Aθ where θ ∈ F (M̃, M̃) is given by

θ(v) = (−1)^{d(v)}v.

In particular, θ^{2} = 1 and θa = (−1)^{d(a)}aθ for any a ∈ A. Define the functor Gr : B-Mod → gr C̃_{1} as follows: for any X ∈ B-Mod the A-module structure on Gr(X) is given by “the restriction of scalars” and the grading is given by

Gr(X)_{0} := {v ∈ X| θv = v}, Gr(X)_{1} := {v ∈ X| θv = −v}.

It is easy to see that Gr provides an equivalence of the categories B-Mod and gr C̃_{1}. Hence Φ′ := Gr ∘Φ′′ provides an equivalence of the categories C_{1} and gr C̃_{1}.

The functor Φ′ : C_{1} → gr C̃_{1} is given by the formula N ↦ F (M, M̃) ⊗_{U} N where the grading on F (M, M̃) is defined by F (M, M̃)_{i} := F (M, M̃_{i}) (i = 0, 1).

The last step of the proof repeats 3.4.5. Indeed,

Ψ_{1} ∘ Φ′(N) = F (M, (M̃_{χ;0}) ⊗_{U} N = F (M, M) ⊗_{U} N = N

since F (M, M) ≅ U_{χ}. Hence Ψ_{1} is quasi-inverse to Φ′ and it provides an equivalence of the categories C̃_{1} → C_{1}. Now, in order to verify that the functors Φ_{1} and Φ′ are isomorphic, one can simply repeat 3.4.5 for the graded modules.

4.4. The restrictions of the functors Ψ, Φ provide an equivalence between C̃_{∞} ∩ Õ and C_{∞} ∩ O and between the corresponding categories of weight modules. The Verma modules of C̃_{∞} ∩ Õ correspond to the Verma modules of C_{∞} ∩ O.

4.4.1. The category gr C̃_{∞} has a canonical involution Π given by

(ΠÑ)_{0} = Ñ_{1}, (ΠÑ)_{1} = Ñ_{0}.

This leads to an interesting involution Π′ = Ψ ∘ Π ∘ Φ on the category C_{∞}. One has

\begin{aligned}N & = ((Ind_{𝔤_{0}}^{𝔤} N)_{χ})_{χ;0}, \\ Π′(N) & = ((Ind_{𝔤_{0}}^{𝔤} N)_{χ})_{χ;1} \end{aligned}

Theorem 4.3 implies that M′ := M̃_{χ;1} = Ψ(Π(M̃)) is a Verma 𝔤_{0}-module. For N ∈ C_{1} one has Φ(N) ≅ Φ′(N) and so

Π′(N) = F (M, M′) ⊗_{U} N.

4.5. It remains to prove Proposition 4.7 and to find χ ∈ Max Z(𝔤_{0}) satisfying (11). To achieve these goals we proceed in a manner similar to [G2], Sect.8.

Call χ ∈ Max Z(𝔤_{0}) a mate for χ̃ if for a Verma module M̃ annihilated by χ̃ the 𝔤_{0}- modules M̃_{χ;0}, M̃_{χ;1} are Verma modules. Call χ ∈ Max Z(𝔤_{0}) a perfect mate for χ̃ if it is a mate and for a simple highest weight module Ṽ (λ) annihilated by χ̃ the 𝔤_{0}-module Ṽ (λ)_{χ;0} is non-zero.

In this subsection we construct a perfect mate χ for χ̃ satisfying (10).

4.5.1. The root system of 𝔤 takes form

Δ_{1}^{+} = {σ_{i}}_{1≤i≤l}, Δ_{0}^{+} = {σ_{i} ± σ_{j}; 2σ_{i}}_{1≤i<j≤l}

and (σ_{i}, σ_{j}) = δ_{i,j}. The Weyl group W acts on {σ_{i}}_{1}^{l} by the signed permutations.

Set

Γ : = {∑_{1}^{l} r_{i}σ_{i}| r_{i} ∈ {0, 1}} ,
Γ_{0} : = {∑_{1}^{l} r_{i}σ_{i}| r_{i} ∈ {0, 1}, ∑_{1}^{l} r_{i} is even} ,
Γ_{1} : = {∑_{1}^{l} r_{i}σ_{i}| r_{i} ∈ {0, 1}, ∑_{1}^{l} r_{i} is odd} .

Define the action of the Weyl group W on Γ by setting

w_{∗}γ = w(γ − ρ_{1}) + ρ_{1}.

Take an arbitrary λ ∈ 𝔥^{∗} and fix a ℤ_{2}-grading on a Verma module M̃(λ) in such a way that a highest weight vector becomes even. As a 𝔤_{0}-module, M̃ = M̃_{0} ⊕ M̃_{1}; the module M̃(λ)_{i} has a filtration such that the set of factors coincides with the set {M(λ − γ) : γ ∈ Γ_{i}}— see [M1], 3.2.

It is easy to check that for any w ∈ W, γ ∈ Γ

w.λ − w_{∗}γ + ρ_{0} = w(λ − γ + ρ_{0}).  (12)

Therefore the 𝔤_{0}-central characters of M(w.λ − w_{∗}γ) and M(λ − γ) coincide. Thus the multiset of 𝔤_{0}-central characters of {M(w.λ − γ) : γ ∈ Γ} does not depend on the choice of w ∈ W. Recall that the set

W (χ̃) := {μ ∈ 𝔥^{∗}|χ̃M̃(μ) = 0}

forms a single W.-orbit.

4.5.2. Say that μ, μ′ ∈ ∑_{i} ℤσ_{i} have the same parity if (μ − μ′) ∈ ℤΔ_{0}^{+}. Suppose that γ, γ′ ∈ Γ both lie either in Γ_{0} or in Γ_{1}. This is equivalent to the condition that the elements γ, γ′ have the same parity. For any w ∈ W one has w_{∗}γ − w_{∗}γ′ = w(γ − γ′) and so the elements w_{∗}γ, w_{∗}γ′ also have the same parity. Hence both w_{∗}γ, w_{∗}γ′ lie either in Γ_{0} or in Γ_{1}. This has the following consequence: the multiset of 𝔤_{0}-central characters of {M(w.λ − γ) : γ ∈ Γ_{0}} coincides either with the multiset of 𝔤_{0}-central characters of {M(λ − γ) : γ ∈ Γ_{0}} or with the multiset of 𝔤_{0}-central characters of {M(λ − γ) : γ ∈ Γ_{1}}. In particular, if χ ∈ Max Z(𝔤_{0}) is such that for some λ ∈ W (χ̃) one has M̃(λ)_{χ} = M(λ − γ_{0}) ⊕ M(λ − γ_{1}) for certain γ_{i} ∈ Γ_{i}(i = 0, 1) then for any λ′ ∈ W (χ̃) the 𝔤_{0}-modules M̃(λ′)_{χ;i} are Verma modules.

4.5.3. Define a lexicographic order on ℂ by setting c_{1} > c_{2} if Re c_{1} > Re c_{2} or Re c_{1} = Re c_{2} and Im c_{1} > Im c_{2}.

The condition (10) implies the existence of λ ∈ W (χ̃) such that

λ + ρ = ∑_{1}^{l} k_{i}σ_{i}, k_{1}, …, k_{l−1} > k_{l} = 0.

Set M̃ := M̃(λ) and

χ = Ann_{Z(𝔤_{0})} M(λ).

One has ρ_{1} = \frac{1}{2} ∑_{1}^{l} σ_{i} and so

λ + ρ_{0} = ∑_{1}^{l}(k_{i} + \frac{1}{2})σ_{i}.

One easily sees that

λ + ρ_{0} − γ ∈ W (λ + ρ_{0})

only for γ = σ_{l}, 0. Therefore χM(λ − γ) = 0 iff γ = σ_{l}, 0. This implies M̃_{χ} = M(λ) ⊕ M(λ − σ_{l}). By 4.5.2, χ is a mate for χ̃.

4.5.4. Let us show that χ is a perfect mate. Suppose this is not true. Then Ṽ (w.λ)_{χ;i} = 0 for some w ∈ W and i ∈ {0, 1}. The equality (12) implies that for any y ∈ W

M̃(y.λ)_{χ} = M(y.λ − y_{∗}0) ⊕ M(λ − y_{∗}σ_{l})

and so Ṽ (y.λ)_{χ} is a homomorphic image of M(y.λ − y_{∗}0) ⊕ M(λ − y_{∗}σ_{l}). The module Ṽ (w.λ) is a homomorphic image of the module M̃(w.λ); denote the kernel of this homomorphism by Ñ. The module Ñ has finite length and the factors of its Jordan-Gölder series have form Ṽ (μ) for some μ ∈ W.λ satisfying μ < w.λ. Since 0 = Ṽ (w.λ)_{χ;i} = (M̃(w.λ)/Ñ)_{χ;i}, one concludes that the 𝔤_{0}-module Ñ_{χ;i} = M̃(w.λ)_{χ;i} has a finite filtration whose factors are quotients of either M(y.λ − y_{∗}0) or M(y.λ − y_{∗}σ_{l}) for some y ∈ W satisfying y.λ < w.λ. Hence the highest weight of M̃(w.λ)_{χ;i} belongs to the set

X := {y.λ − y_{∗}0, y.λ − y_{∗}σ_{l}}_{y∈W s.t. y.λ<w.λ}.

The module M̃(w.λ)_{χ;_{i}} is isomorphic either to M(w.λ − w_{∗}0) or to M(w.λ − w_{∗}σ_{l}). Therefore either w.λ − w_{∗}0 or w.λ − w_{∗}σ_{l} belongs to the set X.

If w.λ − w_{∗}0 = y.λ − y_{∗}0 then, by (12), w(λ + ρ_{0}) = y(λ + ρ_{0}), that is w^{−1}y ∈ Stab_{W} (λ + ρ_{0}). One has

Stab_{W} (λ + ρ_{0}) = Stab_{W} (∑_{1}^{l}(k_{j} + \frac{1}{2})σ_{j}) ⊆ Stab_{W} (∑_{1}^{l} k_{j}σ_{j}) = Stab_{W} (λ + ρ)

since k_{1}, …, k_{l−1} > k_{l} = 0. Thus w.λ − w_{∗}0 = y.λ − y_{∗}0 implies w.λ = y.λ. Hence (w.λ − w_{∗}0) ∉ {y.λ − y_{∗}0}_{y∈W,y.λ<w.λ}.

If w.λ − w_{∗}0 = y.λ − y_{∗}σ_{l} then, by (12), w(λ + ρ_{0}) = y(λ + ρ_{0} − σ_{l}) or, equivalently, w(λ + ρ_{0}) = ys_{β_{l}}(λ + ρ_{0}) where s_{β_{l}} ∈ W is the reflection with respect to the root β_{l}. Therefore w^{−1}ys_{β_{l}} ∈ Stab_{W} (λ+ρ_{0}). As we already saw Stab_{W} (λ+ρ_{0}) ⊆ Stab_{W} (λ+ρ) and so w^{−1}ys_{β_{l}} ∈ Stab_{W} (λ + ρ). Then w^{−1}y ∈ Stab_{W} (λ + ρ) since s_{β_{l}} ∈ Stab_{W} (λ + ρ). Hence w.λ − w_{∗}0 = y.λ − y_{∗}σ_{l} forces w^{−1}y ∈ Stab_{W} (λ + ρ). We conclude that (w.λ − w_{∗}0) ∉ X.

Similarly, if w.λ − w_{∗}σ_{l} = y.λ − y_{∗}0 then, by (12), w(λ + ρ_{0} − σ_{l}) = y(λ + ρ_{0}). As we have shown above, this implies w^{−1}y ∈ Stab_{W} (λ + ρ). Hence (w.λ − w_{∗}σ_{l}) ∉ {y.λ − y_{∗}0}_{y∈W,y.λ<w.λ}.

Finally, if w.λ − w_{∗}σ_{l} = y.λ − y_{∗}σ_{l} then, by (12), w^{−1}y ∈ Stab_{W} (λ + ρ_{0} − σ_{l}). One can easily deduce from the equality

λ + ρ_{0} − σ_{l} = ∑_{1}^{l−1}(k_{j} + \frac{1}{2})σ_{j} − \frac{1}{2}σ_{l}

that Stab_{W} (λ + ρ_{0} − σ_{l}) ⊆ Stab_{W} (λ + ρ). Thus (w.λ − w_{∗}σ_{l}) ∉ X as required.

Hence {w.λ − w_{∗}0; w.λ − w_{∗}σ_{l}} ∩ X = ∅. This proves that Ṽ (w.λ)_{χ;i} ≠ 0 for i = 0, 1.

4.5.5. Corollary. The ideal χ ∈ Max Z(𝔤_{0}) described in 4.5.3 is a perfect mate for χ̃.

4.6. Suppose that χ ∈ Max Z(𝔤_{0}) is a perfect mate for χ̃.

The ideal Ũχ̃ is equal to the annihilator of a Verma module M̃ which has a finite support in Z(𝔤_{0}). Therefore any 𝔤-module annihilated by Ũχ̃ has a finite support in Z(𝔤_{0}). Arguing as in [G2], 8.3.2, 8.3.3, one can deduce from the definition of perfect mate that for a graded 𝔤-module Ñ annihilated by Ũχ̃ one has Ñ = ŨÑ_{χ;0}. In particular, Ñ_{χ;0} ≠ 0 if Ñ ≠ 0.

For a graded 𝔤-bimodule L set

χL_{χ;0} := {f ∈ L_{0}| χ^{r}f = fχ^{r} = 0, ∀r >> 0}.

Arguing as in [G2], 8.4 one concludes _{χ}L_{χ;0} ≠ 0 provided L ≠ 0 and χ̃L = Lχ̃ = 0.

4.7. Proposition. Let M̃ be a Verma module annihilated by χ̃ and let A be the image of Ũ in F (M̃, M̃) under the natural map. Then

F (M̃, M̃) = A ⊕ Aθ

where θ ∈ F (M̃, M̃) is given by

θ(v) = (−1)^{d(v)}v.

Proof. Suppose that M̃ is simple. By [G2], 11.1.5, for any simple 𝔤-module Ṽ one has dim Hom_{𝔤}(Ṽ , F (M̃, M̃)) = dim Hom_{𝔤}(Ṽ , H) where H is a harmonic space —see 4.1. The complete reducibility of both H and F (M̃, M̃) forces H ≅ F (M̃, M̃). The element θ is ad 𝔤_{0}-invariant and so A ≅ Aθ as ad 𝔤_{0}-modules. In the light of (9), as ad 𝔤_{0}-modules H ≅ A ⊕ A. Since the multiplicity of any finite dimensional 𝔤_{0}-module in H is finite, in order to prove the equality F (M̃, M̃) = A ⊕ Aθ it is enough to show that F (M̃, M̃) = A + Aθ. Since θu = (−1)^{d(u)}uθ for any u ∈ Ũ, (A + Aθ) is a graded Ũ-subbimodule of F (M̃, M̃). Let χ ∈ Z(𝔤_{0}) be a perfect mate for χ̃. Then both M := M̃_{χ;0} and M′ := M̃_{χ;1} are Verma 𝔤_{0}-modules. The restriction of endomorphisms of M̃ to M ⊕ M′ induces an isomorphism

χF (M̃, M̃)_{χ;0} ⟶^{∼} F (M, M) ⊕ F (M′, M′).

Recall that the natural maps U/(Uχ) → F (M, M) and U/(Uχ) → F (M′, M′) are bijective; identify _{χ}F (M̃, M̃)_{χ;0} with U/(Uχ) ⊕ U/(Uχ) through these maps. Write

Ann_{Z(𝔤_{0})} M̃ = χ ∏_{1}^{s} χ_{j}^{r_{j}}

where supp_{Z(𝔤_{0})} = {χ, χ_{1}, …, χ_{s}} and r_{1}, …, r_{s} ∈ ℕ^{+}. Take a ∈ ∏_{1}^{s} χ_{j}^{r_{j}} such that a = 1 modulo χ. The image of Ua in F (M̃, M̃) lies in _{χ}F (M̃, M̃)_{χ;0}. Since for any u ∈ U the element ua acts on both M and M′ by the multiplication by u, the image J of Ua in χF (M̃, M̃)_{χ;0} is equal to the diagonal copy of U/(Uχ) inside U/(Uχ) ⊕ U/(Uχ):

J = {(u, u)| u ∈ U/(Uχ)}.

Then Jθ = {(u, −u)| u ∈ U/(Uχ)} and thus

J + Jθ = U/(Uχ) ⊕ U/(Uχ) =_{χ} F (M̃, M̃)_{χ;0}.

Hence

χF (M̃, M̃)_{χ;0} =_{χ} (A + Aθ)_{χ;0}.

By 4.6, this forces F (M̃, M̃) = A + Aθ.

We have shown that F (M̃, M̃) = Im Ũ⊕θ Im Ũ provided M̃ is simple. Take an arbitrary Verma module M̃ satisfying χ̃M̃ = 0. The module M̃ contains a simple Verma submodule M̃′ since Ũ is a domain. Let E be an ad 𝔤-submodule of Ũ such that the restriction of the natural map f′ : Ũ → F (M̃′, M̃′) to E provides a bijection E → Im f′. Then F (M̃′, M̃′) = f′(E) ⊕ f′(E)θ. The restriction of the natural map f : Ũ → F (M̃, M̃) to E provides a bijection E → Im f since ker f = ker f′ = Ũχ̃. Suppose that f(u_{1}) = f(u_{2})θ for some u_{1}, u_{2} ∈ E; then (u_{1} − u_{2})M̃_{0} = 0 and so f′(u_{1}) = f′(u_{2})θ that implies u_{1} = u_{2} = 0 since f′(E) ∩ f′(E)θ = 0. Therefore f(E) ∩ f(E)θ = 0. Thus F (M̃, M̃) contains f(E) ⊕ f(E)θ. A Joseph’s reasoning based on the use of GK-dimension, shows that the map F (M̃′, M̃′) → F (M̃′, M̃) is bijective and the map F (M̃, M̃) → F (M̃′, M̃) is injective (both maps are induced by the embedding M̃′ to M̃)— see [J2], 8.3.9 or [G2], 9.2. In particular, F (M̃, M̃) is isomorphic to an ad 𝔤-submodule of F (M̃′, M̃′). On the other hand, F (M̃, M̃) contains f(E) ⊕ f(E)θ which is isomorphic, as an ad 𝔤_{0}-module, to F (M̃′, M̃′). Since the multiplicity of each finite dimensional 𝔤_{0}-module in F (M̃′, M̃′) is finite, one concludes that F (M̃, M̃) = f(E) ⊕ f(E)θ = Im f ⊕ θ Im f as required.

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  email:  gorelik@mpim-bonn.mpg.de