A REDUCTION METHOD FOR ATYPICAL REPRESENTATIONS OF CLASSICAL LIE SUPERALGEBRAS

VERA SERGANOVA

Abstract. We introduce a reduction method for studying representations of classical Lie superalgebras with atypical central character. We show that the atypical quotient of universal enveloping algebra has a non-trivial Jacobson radical. The factor by this radical has a new center, which is calculated for sl (1|n) and psl (2|2). Using this center we obtain new character formulae, generalization of Borel-Weil-Bott and Beilinson-Bernstein localization theorems.

Contents

1. Introduction 1 2. Preliminaries 3 3. The Jacobson radical of U^{χ} 4 4. The case 𝔤 = sl (1|2) 6 5. The case 𝔤 = psl (2|2) 8 6. The center of Ū for 𝔤 = sl (1|n) 10 7. On structure of Ū-modules for 𝔤 = sl (1|n) 14 8. Degenerate case 18 9. Category O and character formulae 19 10. Geometric realization of Ū and localization theorem 21 References 24

1. Introduction

In this paper we suggest a way to treat representations of classical Lie superalgebras with an atypical central character. Let 𝔤 be a classical Lie superalgebra (see [9]), U be its universal enveloping algebra and Z be the center of U. By Schur’s lemma an irreducible representation of 𝔤 admits a central character χ ∈ Spec Z, therefore the structure of the ring Z should play an important role in representation theory of 𝔤. In contrast with classical Lie algebras, where Z is isomorphic to a polynomial ring, for Lie superalgebras Z is not Noetherian. However, for generic points χ ∈ Spec Z, called typical, the situation is completely under control. V. Kac was first to show that all theorems about finite-dimensional Lie algebra representations can be easily generalized to typical representations of classical Lie superalgebras. I. Penkov generalized this in [15] by proving the following theorem. If χ ∈ Spec Z is a generic typical central character then the category of all 𝔤-modules which admit the central character χ is equivalent to the category of all modules over its even part 𝔤_{0} which admit a certain central character χ_{0}. He proves this theorem by extending D-module localization theorem of Beilinson and Bernstein to representations of classical Lie superalgebras with generic typical central characters. Recently, M. Gorelik reproved this theorem by purely algebraic methods in [7]. She managed to get rid off the assumption that χ is generic.

Date: October 30, 2001.

I. Penkov also described a certain class of atypical irreducible representations which he called generic. He was able to describe these representations in terms of its 𝔤_{0}- components. This description generalizes Bernstein-Leites formula introduced in [3].

People still know very little about atypical general representations. To mention few things, finite-dimensional modules are not completely reducible, and the category of finite-dimensional modules is described only for sl (1|n) in [4]. Character formulae for finite-dimensional modules are very complicated. Primitive ideals are classified (see[12]), but the relations between them are unknown. Geometric facts such as Borel-Weil-Bott theorem or Beilinson-Bernstein theorem do hold in atypical case.

Here is what we suggest to do. Let χ ∈ Spec Z be atypical, and U^{χ} denote the quotient U/ (Ker χ). In contrast with a typical χ the algebra U^{χ} has a non-trivial Jacobson radical J^{χ}. Let us study the representations of Ū^{χ} = U^{χ}/J^{χ}. This algebra Ū^{χ} has a new center Z \binom{}{Ū^{χ}} which we are still unable to describe in general. However, this center is sufficiently large and we conjecture that Spec Z \binom{}{Ū^{χ}} separates almost all finite-dimensional representation. Probably it is possible to prove Penkov’s theorem for generic φ ∈ Spec Z \binom{}{Ū^{χ}} and the analog of Beilinson-Bernstein theorem.

In this paper we fulfill this program for 𝔤 = sl (1|n) and psl (2|2). We obtain several new results for sl (1|n) including analogs of Penkov and Beilinson-Bernstein theorem and characters formulae for infinite-dimensional highest weight representations. We also do some initial steps for sl (m|n) for m > 1. We did not discuss the case of osp (m|2n) here, but we believe that all initial steps can be done for this superalgebra as well. The method can be used successfully for representation theory of Lie superalgebra P (n), but this is a subject of another paper.

It would be interesting to give an interpretation of our approach in terms of the orbit method. Indeed, the situation we deal resembles pretty much the solvable case. Typical central characters correspond to generic values of invariants in 𝔤^{∗}, i.e. to generic orbits. Atypical central characters should probably correspond to singular levels of invariants. In the solvable case these submanifolds may contain infinite families of orbits, which correspond to the new center phenomenon in super case.

The author would like to thank I. Penkov, I. Musson and especially M. Gorelik for stimulating discussions and interest.

2. Preliminaries

Let 𝔤 denote one of the following superalgebras gl (m|n), sl (m|n) or psl (n|n), U be the universal enveloping algebra of 𝔤, Z be the center of U. Recall that 𝔤 is Z graded algebra: 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}, and naturally U is also ℤ-graded: U = U_{−mn} ⊕ ⋯ ⊕ U_{mn}.

From now on we fix a Cartan subalgebra 𝔥 ⊂ 𝔤_{0} and denote the set of roots with respect to 𝔥 by Δ. Recall that 𝔤 admits a non-degenerate invariant symmetric form B (x, y) =str xy. It induces a non-degenerate form on 𝔥^{∗} which we denote by (,). If 𝔟_{0} is a Borel subalgebra of 𝔤_{0}, 𝔥 ⊂ 𝔟_{0}, then there are finitely many Borel subalgebras 𝔟 ⊂ 𝔤 with 𝔟_{0} ⊂ 𝔟. All such Borel subalgebras can be obtained from one of them by odd reflections (see [16] ). The Borel subalgebra 𝔟_{d} = 𝔟_{0} + 𝔤_{1} is called distinguished. By 𝔟_{d}^{op} we denote the Borel subalgebra 𝔟_{0} + 𝔤_{−1}. For any subalgebra 𝔩 ⊂ 𝔤 invariant under the adjoint action of 𝔥, we denote by Δ (𝔩) the set of roots of 𝔩.

Put ρ_{0} = \frac{1}{2} ∑_{α∈Δ(𝔟_{0})} α, ρ_{1} = \frac{1}{2} ∑_{α∈Δ(𝔟_{1})} α, ρ_{𝔟} = ρ_{0} − ρ_{1} for a Borel subalgebra 𝔟 ⊂ 𝔤. A choice of 𝔟 determines the triangular decomposition 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+} and the decomposition U = U (𝔫^{−}) ⊗ U (𝔥) ⊗ U (𝔫^{+}). The Harish-Chandra projection h_{𝔟} is defined as the natural map U → U/ (𝔫^{−}U + U𝔫^{+}) = U (𝔥) = S (𝔥). The restriction of h_{𝔟} to Z is an injective homomorphism Z → S (𝔥). The image h_{𝔟} (Z) is described in [10] and [17] as follows. Consider S (𝔥) as the polynomial ring on 𝔥^{∗}. If W is the Weyl group of 𝔤_{0}, define an action of W on 𝔥^{∗} by μ^{w} = w (μ + ρ_{𝔟})−ρ_{𝔟}, and let S^{W} (𝔥) denote the subring of W -invariant polynomial in S (𝔥). Then

h_{𝔟} (Z) = {p ∈ S^{W} (𝔥) | p (μ + tα) = p (μ) for any μ ∈ 𝔥^{∗}, α ∈ Δ_{1}, (α, μ + ρ) = 0} (2.1)

Let χ : Z → ℂ be an arbitrary character. There exists μ ∈ 𝔥^{∗} such that χ (z) = χ_{μ} (z) := (h_{𝔟} (z)) (μ). Let m_{χ} = {μ ∈ 𝔥^{∗} | χ = χ_{μ}}. The following proposition is an easy consequence of the description of h_{𝔟} (Z).

Lemma 2.1. There exist a set of odd roots α_{1}, …, α_{k} ∈ Δ (𝔤_{1}) and a weight μ ∈ 𝔥^{∗} such that (α_{i}, α_{j}) = 0 and m_{χ} = ∪_{w∈W} l_{χ}^{w}, where l_{χ} = μ + ℂα_{1} + ⋯ + ℂα_{k}.

If k = 0 the central character χ is called typical. If k > 0 the central character χ is called atypical, and the number k is called the degree of atypicality of χ.

Let U^{χ} = U/ (Ker χ). For a typical χ the structure of U^{χ} is well understood. More precisely, the following theorem is proved in [15] (see also [7] ). Let Z^{0} be the center of U (𝔤_{0}), for a central character φ : Z^{0} → ℂ define U^{φ} (𝔤_{0}) = U (𝔤_{0}) / (Ker φ). If χ: Z → ℂ is typical, one can find φ : Z^{0} → ℂ such that the algebras U^{χ} and U^{φ} (𝔤_{0}) are Morita equivalent. Moreover, φ can be chosen such that U^{χ} is a matrix algebra over U^{φ} (𝔤_{0}). Thus, the study of representation of 𝔤 admitting a typical central character can be reduced to the similar problem for 𝔤_{0}.

The goal of this paper is to generalize the above result for atypical central characters.

3. The Jacobson radical of U^{χ}

In this section we show that for atypical central character χ the algebra U^{χ} has a non-trivial Jacobson radical and give some description of this radical.

We will use the notations of Lemma 2.1. Since the roots α_{1}, …, α_{k} are mutually orthogonal one can find a Borel subalgebra 𝔟 so that α_{1}, …, α_{k} are simple positive roots. Then

𝔭 = 𝔤_{−α_{1}} ⊕ ⋯ ⊕ 𝔤_{−α_{k}} ⊕ 𝔟

is a parabolic subalgebra of 𝔤. We put ρ = ρ_{𝔟}. Since (ρ, α_{i}) = 0, (μ, α_{i}) = 0 for any μ ∈ l_{χ}, and therefore μ defines one dimensional 𝔭-module C_{μ}. Put

F (μ) = U ⊗_{U(𝔭)} C_{μ}.

Here are some properties of a 𝔤-module F (μ).

Lemma 3.1. (a) The center Z acts by scalar operators on F (μ), and z_{|F (μ)} = χ_{μ} (z) Id.

(b) F (μ) is free U (𝔫^{−})-module.

(c) F (μ) admits a 𝔤_{0}-module filtration with subquotients isomorphic to 𝔤_{0}-Verma modules M^{0} (ν), where ν = μ − γ and γ runs the set of sums of non-repeating roots from Δ (𝔫_{1}^{+}) \ {α_{1}, …, α_{k}}.

(d) Let l′_{χ} = {μ ∈ l_{χ} | (μ, α) ∉ ℤ, α ∈ Δ (𝔤_{0})}. If μ ∈ l′_{χ}, then F (μ) splits into direct sum of 𝔤_{0}-Verma modules M^{0} (ν), and F (μ) is an irreducible 𝔤-module.

(e) ∩_{μ∈l} Ann F (μ) = ∩_{μ∈l′} Ann F (μ).

Proof. (a) follows from the fact that F (μ) is a highest weight module with highest weight μ ∈ l_{χ}. To prove (b) and (c) let

𝔪 = ⊕_{α∈Δ(𝔫^{−})\{−α_{1},...,−α_{k}}}𝔤_{α}.

Then 𝔪 is a subalgebra of 𝔤, 𝔤 = 𝔭 ⊕ 𝔪, 𝔪_{0} = 𝔫_{0}^{−} and both statement follow from the following isomorphisms of vector spaces

U ≅ U (𝔪) ⊗ U (𝔭) , U (𝔪) ≅ U (𝔫_{0}^{−}) ⊗ S (𝔪_{1}) , F (μ) ≅ U (𝔫_{0}^{−}) ⊗ S (𝔪_{1}) ⊗ C_{μ}.

To prove (d) note that in case μ ∈ l′_{χ}, all M^{0} (ν) are irreducible and

Ext_{𝔤_{0}}^{1} (M^{0} (ν) , M^{0} (ν′)) = 0

for ν ≠ ν′. Since 𝔥 acts semisimply on F (μ), F (μ) could not contain self-extensions of M. Therefore F (μ) is a semisimple 𝔤_{0}-module.

Now we will show that F (μ) is an irreducible 𝔤-module. For this we will use the odd reflection method. Let L (ν) denote an irreducible highest weight module with 𝔟-highest weight ν. Suppose ν ∈ l′_{χ}. Then the 𝔟_{d}-highest weight of L (ν) is equal to ν + δ, where δ equals the sum of all roots from Δ (𝔤_{1}) \Δ (𝔭). In the same way 𝔟_{d}^{op}-highest weight of L (ν) is ν + δ′, where δ′ is the sum of all odd roots from Δ (𝔤_{−1}) \Δ (𝔭). We call the length of 𝔤-module M the maximal positive p such that S^{p} (𝔤_{1}) M ≠ 0^{1}. The above implies that for any ν ∈ l′_{χ} the length of L (ν) is mn − k. On the other hand the length of F (μ) also equals mn − k. Assume that L (ν) is some irreducible submodule of F (μ). Then χ_{ν} = χ_{μ}, and ν is as in (c). One can check easily that ν ∈ l′_{χ}, and therefore the lengths of L (ν) and F (μ) are equal. Hence M^{0} (μ)∩L (ν) ≠ 0. But M^{0} (μ) is an irreducible 𝔤_{0}-module, therefore M^{0} (μ) ⊂ L (ν). Thus, F (μ) = L (ν), and therefore F (μ) is irreducible.

Finally, (e) follows from Zariski density of l′_{χ} in l_{χ}.

Let J^{χ} = ∩_{μ∈l} Ann_{U^{χ}} F (μ), and Ū^{χ} = U^{χ}/J^{χ}.

Theorem 3.2. The ideal J^{χ} coincides with the Jacobson radical of U^{χ}.

Proof. The inclusion rad U^{χ} ⊆ J^{χ} follows from Lemma 3.1 (d) and (e). To show the inverse inclusion we will use the result of Musson (see [12]) that every primitive ideal of U coincides with the annihilator of some irreducible highest weight module. Let L_{d} (λ) denote the irreducible 𝔟_{d}-highest weight module with highest weight λ. We have to show that J^{χ}L_{d} (λ) = 0 whenever L_{d} (λ) admits central character χ. Let δ be the sum of all roots from Δ (𝔤_{1}) \Δ (𝔭), l_{d} = l_{χ} + δ. One can check easily using Lemma 2.1 that L_{d} (λ) admits the central character χ if and only if w (λ + ρ_{0})−ρ_{0} ∈ l_{d} for some w ∈ W. We have to show that J^{χ}L_{d} (λ) = 0 for all such λ. First, let λ ∈ l_{d}. Then L_{d} (λ) is a subquotient of F (λ − δ), and therefore J^{χ}L_{d} (λ) = 0. Let now ν = w (λ + ρ_{0})−ρ_{0} ∈ l_{d}. Then ν and λ induce the same central character φ : Z^{0} → ℂ, and the Verma modules M^{0} (λ), M^{0} (ν) are both faithful U^{φ} (𝔤_{0})-modules by Duflo’s theorem. Let \binom{}{Ū^{χ}}^{+} be the image of U (𝔤_{0} ⊕ 𝔤_{1}) in Ū^{χ}. Then U^{φ} (𝔤_{0}) is a quotient algebra of \binom{}{Ū^{χ}}^{+} by some ideal I^{+}, and therefore the U^{φ} (𝔤_{0})-module M^{0} (λ) can be made an \binom{}{Ū^{χ}}^{+}-module such that 𝔤_{1} acts trivially on M^{0} (λ). Then the Ū-module M = Ū^{χ} ⊗_{\binom{}{Ū^{χ}}}^{+} M^{0} (λ) is a highest weight module with 𝔟_{d}-highest weight λ and

J^{χ}M = 0. Since L_{d} (λ) is a quotient of M, J^{χ}L_{d} (λ) = 0.

Now we plan to study the structure of the algebra Ū^{χ}. Start with the following simple observations.

Corollary 3.3. The natural map 𝔤 → Ū^{χ} is injective. Ū^{χ} inherits the ℤ-grading of U and Ū^{χ} = Ū_{−s}^{χ} ⊕ ⋯ ⊕ Ū_{s}^{χ}, where s = mn − k. In particular, if x ∈ U_{i} for i > s, then its image in Ū^{χ} is zero.

It is more convenient to study the larger algebra Ū defined in the following way. Let S = {μ ∈ 𝔥^{∗} | (μ, α_{1}) = ⋯ = (μ, α_{k}) = 0}. Put J = ∩_{μ∈S} Ann F (μ). Define Ū as the quotient U/J. One can see immediately that all statements of Corollary 3.3 are true for Ū as well. Every irreducible 𝔤-module with degree of atypicality k is an Ū-module.

1If z_{0} is a central element of 𝔤_{0} and z_{0} acts semi-simply on a cyclic 𝔤-module M, then the length of M is the number of distinct eigenvalues of z_{0} on M.

4. The case 𝔤 = sl (1|2)

Let now 𝔤 = sl (1|2). Choose a basis of 𝔥 as follows:

z_{0} = diag (2,1,1), h = diag (0, 1, −1)

According to this choice of a basis we numerate weights μ ∈ 𝔥^{∗} by pairs (a, b) where a = μ (z_{0}) and b = μ (h). Assume furthermore that 𝔟 = 𝔟_{d}. There exists exactly one atypical central character χ, and l_{χ} is given by the equation a = −b. In this case l_{χ} = S and Ū = Ū^{χ}.

Let Ω_{0} be the Casimir operator of sl (2) ⊂ 𝔤, Ω be the Casimir operator in osp (1|2) ⊂ sl (1|2)^{2}. Finally let L = X_{−α}X_{α} + X_{−β}X_{β} where α and β are two roots of Δ (𝔤_{1}). Note that z_{0}, Ω_{0}, Ω and L lie in the centralizer of 𝔤_{0}. By an abuse of notations we denote the images of these elements in Ū by the same letters.

The following lemma is a very easy calculation.

Lemma 4.1. If μ ∈ l_{χ}, then F (μ) considered as an osp (1|2)-module is isomorphic to Verma module with highest weight b. Furthermore as an sl (2)-module F (μ) is isomorphic to the sum of two Verma sl (2) − modules with highest weights b and b − 1.

Corollary 4.2. Ω acts by the scalar b (b + 1) on F (μ), and therefore belongs to the center Z \binom{}{Ū} of Ū.

Lemma 4.3. The following relations hold in Ū:

z_{0}^{2} + Ω_{0} = 2Ω
(Ω − Ω_{0})^{2} = z_{0}^{2}
z_{0} − 2L = Ω − Ω_{0}

Proof. Since F (μ) for μ ∈ l_{χ} forms a faithful system of modules it is sufficient to check the relations on F (μ). This can be done easily using Lemma 4.1.

In the similar way the following lemma can be proved.

Lemma 4.4. C = −z_{0}^{3} − \frac{3}{2}z_{0} (Ω − Ω_{0}) − \frac{1}{2}z_{0} − \frac{1}{2}Ω belongs to Z \binom{}{Ū}, and acts on F (μ) by b^{2} (b + 1).

Let Ū (b) be the quotient of Ū by the ideal (Ω − b (b + 1) , C − b^{2} (b + 1)), U^{0} (b) denote the quotient of U (gl (2)) by the ideal (Ω_{0} − b (b + 2) , z_{0} + b) and U′ (b) denote the quotient of U (osp (1|2)) by the ideal (Ω − b (b + 1)).

Theorem 4.5. Let b ≠ 0, −1, then Ū (b) is Morita equivalent to U^{0} (b). If b ≠ −\frac{1}{2}, then Ū (b) is isomorphic to U′ (b).

2We normalize Ω and Ω_{0} so that Ω acts as a (a + 1) on the irreducible osp (1|2)-module with highest weight a and Ω_{0} acts as a (a + 2) on the irreducible sl (2)-module with highest weight a.

Proof. Let M be a Ū (b)-module and M^{𝔤_{1}} be the space of 𝔤_{1}-invariant vectors in M. Clearly M^{𝔤_{1}} is a 𝔤_{0}-module, M^{𝔤_{1}} ≠ 0, and LM^{𝔤_{1}} = 0. The relations of Lemma 4.3 after substituting L = 0 imply one of the following

1. z_{0} acts on M^{𝔤_{1}} by the scalar −b, Ω_{0} acts on M^{𝔤_{1}} by the scalar b^{2} + 2b;

2. z_{0} acts on M^{𝔤_{1}} by the scalar b + 1, Ω_{0} acts on M^{𝔤_{1}} by the scalar b^{2} − 1.

However, the condition C = b^{2} (b + 1) excludes the second case for b ≠ 0, −1 and therefore M^{𝔤_{1}} is a U^{0} (b)-module. Furthermore, M^{𝔤_{1}} = Ker (z + b). Define a functor Inv from the category of Ū (b)-modules to the category of U^{0} (b)-modules by putting Inv M = M^{𝔤_{1}} = Ker (z + b). This functor is clearly faithful and exact. It is easy to see that Inv F (μ) = M^{0} (μ). Let Ū^{+} (b) denote the subalgebra in Ū (b) generated by 𝔤_{0} and 𝔤_{1}. Every U^{0} (b)-module N can be equipped with the structure of Ū^{+} (b)-module by letting 𝔤_{1} act by zero. Put

Ind N = Ū (b) ⊗Ū^{+}_{(b)} N.

Then (Ind N)^{𝔤_{1}} = Ker (z_{0} + b) = N. Therefore the functors Ind and Inv establish the equivalence of the categories of U^{0} (b)-modules and Ū (b)-modules.

Finally, to establish the isomorphism between Ū (b) and U′ (b), consider the following sequence

U (osp (1|2)) ⟶^{i} Ū (b) ⟶^{σ} F (F (μ) , F (μ)) ,

where for arbitrary 𝔤-modules M and N, F (M, N) denote the subspace of Hom (M, N) locally finite as ad 𝔤_{0}-module, the first map i is induced by the embedding of osp (1|2) into 𝔤 and the second map σ is the representation of Ū (b). As it follows from [6] the composite map σ ∘ i is an isomorphism for b ≠ −\frac{1}{2}. Therefore σ (Ū (b)) is isomorphic to U′ (b). On the other hand, F (μ) = Ind M^{0} (μ) and M^{0} (μ) is a faithful U^{0} (b)-module. Hence F (μ) is a faithful Ū (b)-module and therefore σ is injective. Thus

Ū (b) ≅ F (F (μ) , F (μ)) ≅ U′ (b) .

Remark 4.6. The isomorphism between U′ (b) and Ū (b) has its origin in the following geometric fact. If B′ is a Borel subgroup of OSP (1|2) and P is the parabolic subgroup of SL (1|2) with the Lie algebra 𝔭, then the algebraic supervarieties OSP (1|2) /B′ and SL (1|2) /P are isomorphic.

Finally let us look what happens in case b = 0, −1. Then Ū (0) is isomorphic to Ū (−1), and z_{0}^{2} = Ω_{0}^{2} = −Ω_{0}. One can prove similarly to the above theorem that Ū (0) / (z_{0} − Ω_{0}) ≅ Ū (0) / (z_{0} + Ω_{0}) ≅ U′ (0). One one can construct the injective homomorphism

f : Ū (0) → F (F (0) , F (0)) ⊕ F (F (−1) , F (−1)) ≅ U′ (0) ⊕ U′ (0) .

One can show that Ū (0) ≅ Im f = {(x, y) ∈ U′ (0) ⊕ U′ (0) | p (x) = p (y)}, where p : U′ (0) → ℂ is the projection to the factor by augmentation ideal.

5. The case 𝔤 = psl (2|2)

The Lie algebra 𝔤_{0} is isomorphic to sl (2) ⊕ sl (2). The Cartan subalgebra is 2dimensional, and we choose a basis {h_{1}, h_{2}} in it so that h_{i} is the standard element of a copy of sl (2). We choose a Borel subalgebra 𝔟 so that all simple roots are odd. In this case ρ_{𝔟} = 0. There is exactly one atypical central character χ, which is induced by the trivial module. In this case S = l_{χ} = {μ ∈ 𝔥^{∗} | μ (h_{1}) = −μ (h_{2})}, and again Ū = Ū^{χ}. By F (a) we denote F (μ) with μ (h_{1}) = −μ (h_{2}) = a. Let Ω_{1} and Ω_{2} be the Casimir elements of two sl (2)-subalgebras of 𝔤_{0} and Ω be the Casimir element of 𝔤. Let X_{1}, X_{2}, X_{3}, X_{4} be a basis of 𝔤_{1} and Y_{1}, Y_{2}, Y_{3}, Y_{4} be the dual basis of 𝔤_{−1}. Then the element L = ∑_{i=1}^{4} Y_{i}X_{i} = ∑_{i=1}^{4} −X_{i}Y_{i} belongs to the centralizer of 𝔤_{0} in U, and we have the relation

Ω_{1} − Ω_{2} + 2L = Ω. (5.1)

Note that the image of Ω in U^{χ} is zero, and therefore 2L = Ω_{2} − Ω_{1}.

Let M^{0} (a, b) denote sl (2) ⊕ sl (2)-Verma module with highest weight (a, b). The following lemma is obtained by direct calculation^{3}.

Lemma 5.1. F (a) ≅ M^{0} (a, −a) ⊕ 2M^{0} (a − 1, −1 − a) ⊕ M^{0} (a − 2, −2 − a) as 𝔤_{0}- module, the element

C_{1} = \frac{(Ω_{1} + Ω_{2})^{2}}{4} + \frac{Ω_{1} + Ω_{2}}{2} − \frac{(Ω_{1} − Ω_{2})^{2}}{8} (5.2)

acts by the scalar a^{2} (a^{2} − 1) on F (a), and the element

(5.3) C_{2} = \frac{(Ω_{1} + Ω_{2})^{3}}{8} + \frac{(Ω_{1} + Ω_{2})^{2}}{4} − 3\frac{(Ω_{1} − Ω_{2})^{4}}{256} + \frac{(Ω_{1} − Ω_{2})^{2}}{16}

acts by the scalar a^{2} (a^{2} − 1)^{2} on F (a).

Thus, the images of C_{1} and C_{2} in Ū belong to the center Z \binom{}{Ū} of Ū. Again we denote their images in Ū by the same letters C_{1} and C_{2}. Now put

Ū (a) = Ū/ (C_{1} − a^{2} (a^{2} − 1) , C_{2} − a^{2} (a^{2} − 1)^{2})

and

U^{0} (a) = U (𝔤_{0}) / (Ω_{1} − a^{2} + 1, Ω_{2} − a^{2} + 1) .

Theorem 5.2. If a ≠ 0, ±1, then Ū (a) and U^{0} (a) are Morita equivalent.

The rest of this section will contain the proof of Theorem 5.2.

For any 𝔤-module M we denote by M^{𝔤_{±1}} the space of 𝔤_{±1}-invariants. Obviously M^{𝔤_{±1}} is 𝔤_{0}-invariant and LM^{𝔤_{±1}} = 0.

3Use that Ω_{1} and Ω_{2} act on M^{0} (a, b) as a (a + 2) and b (b + 2) correspondingly.

Lemma 5.3. Let a ≠ 0, ±1 and M be a Ū (a)-module, then M^{𝔤_{1}} and M^{𝔤_{−1}} are U^{0} (a)-modules.

Proof. Since LM^{𝔤_{±1}} = 0, it follows from (5.1) that Ω_{1} = Ω_{2} on M^{𝔤_{1}}. The relation (5.2) implies that

Ω_{1}^{2} + Ω_{1} = a^{2} (a^{2} − 1) ,

and hence Ω_{1} = Ω_{2} = a^{2} − 1 or Ω_{1} = Ω_{2} = −a^{2}. The relation (5.3) implies

Ω_{1}^{3} + Ω_{1}^{2} = a^{2} (a^{2} − 1)^{2} .

If a^{2} ≠ 0, 1 we have Ω_{1} = Ω_{2} = a^{2} − 1. Therefore M^{𝔤_{±1}} is U^{0} (a)-module.

Denote by Inv the functor from the category of Ū (a)-modules to the category of U^{0} (a)-modules which sends M to M^{𝔤_{1}}. Obviously Inv is faithful and exact on the left.

Lemma 5.4. Let V be the ad 𝔤-submodule of Ū generated by L, and Ū^{2} denote the second term in the filtration of Ū induced by the natural filtration of U. Then Ū^{2} = V ⊕ ℂ ⊕ 𝔤.

Proof. By Poincare-Birkhoff-Witt’s theorem U^{2} ≅ ℂ⊕𝔤⊕S^{2} (𝔤). A direct calculation shows that S^{2} (𝔤) ≅ ℂΩ⊕L (2, 0)⊕L (0, 2). Therefore Ū^{2} ≅ ℂ⊕𝔤⊕L (2, 0)⊕L (0, 2). Furthermore, L ∈ L (2, 0) ⊕ L (0, 2), but L does not belong to L (2, 0) or L (0, 2). Hence L generates V.

Lemma 5.5. Let M be a Ū-module such that M = M^{𝔤_{1}}+M^{𝔤_{−1}}. Then M is a trivial 𝔤-module.

Proof. Since LM = 0, we have S^{2} (𝔤) ⊂ Ann_{U} M by Lemma 5.4. Hence X^{2}M = 0 for any X ∈ 𝔤. This implies that 𝔤M = 0.

Now we will construct a functor Ind form the category of U^{0} (a)-modules to the category of Ū (a)-modules. Let Ū^{+} (a) be the subalgebra of Ū (a) generated by 𝔤_{0} and 𝔤_{1}. Every U^{0} (a)-module N becomes an Ū^{+} (a)-module if we put 𝔤_{1}N = 0. Put

Ind N = Ū (a) ⊗Ū^{+}_{(a)} N.

Lemma 5.6. If a ≠ 0, 1, −1, then for every Ū^{0} (a)-module N, Inv (Ind N) ≅ N, and for every Ū (a)-module M, Ind (Inv M) ≅ M.

Proof. Ind N = N ⊕ 𝔤_{1}N ⊕ S^{2} (𝔤_{1}) N. Let N′ = Inv (Ind N) ∩ (𝔤_{1}N ⊕ S^{2} (𝔤_{1}) N). It suffices to show that N′ = 0. Indeed let M′ be the 𝔤-submodule of Ind N generated by N′. Then M′ ⊂ 𝔤_{−1}N ⊕S^{2} (𝔤_{−1}) N. Then M′ satisfies the condition of Lemma 5.5, and therefore M′ = N′ is trivial. This is possible only for a = 0, ±1.

To prove the second statement consider the natural surjection

π : Ind (Inv M) → M.

Let Ker π = R. Then Inv R ∩ M = {0} and Inv R ⊂ Inv (Ind M) = M. Therefore Inv R = {0}, and R = {0} because Inv is faithful.

Corollary 5.7. If a ≠ 0, ±1, then the functors Ind and Inv establish the equivalence between the categories of U^{0} (a)-modules and Ū (a)-modules.

6. The center of Ū for 𝔤 = sl (1|n)

In this section 𝔤 = sl (1|n), χ is some atypical character, 𝔟 can be always chosen 𝔟_{d}. We denote the simple roots of 𝔟 by α, β_{1}, …, β_{n−1}, where α is odd and β_{1}, …, β_{n−1} are even simple roots of sl (n) ⊂ 𝔤_{0}. Then

l_{χ} = {μ ∈ 𝔥^{∗} | (μ, α) = 0, (μ, β_{2}) = μ_{2}, …, (μ, β_{n−1}) = μ_{n−1}} ,

where μ_{2}, …, μ_{n−1} ∈ ℂ. We put μ_{1} = (μ, β_{1}) and use μ_{1} as the coordinate on the line l_{χ}. Note that S is the hyperplane given by the equation (μ, α) = 0, and we use μ_{1}, …, μ_{n−1} as coordinates on S.

Let Z \binom{}{Ū} denote the center of Ū. By Z̄ we denote the image of Z in Ū under the natural projection. Clearly Z̄ ⊆ Z \binom{}{Ū}. In this section we describe Z \binom{}{Ū}.

To proceed farther we need to recall certain facts from finite-dimensional representation theory of sl (1|n). See [4] for reference. By L (λ) we denote the irreducible 𝔤module with highest weight λ. By K (λ) we denote Kac module U (𝔤)⊗_{U(𝔤_{0}+𝔤_{1})}L^{0} (λ), where L^{0} (λ) is the irreducible 𝔤_{0}-module with trivial action of 𝔤_{1}. By K′ (λ) we denote the induced module U (𝔤) ⊗_{U(𝔤_{0}+𝔤_{−1})} L^{0} (λ). Finally by P (λ) we denote the projective cover^{4} of L (λ), which is an indecomposable submodule in U ⊗_{U(𝔤_{0})} L^{0} (λ). P (λ) is also injective, contragredient and has a unique irreducible submodule and unique irreducible quotient isomorphic to L (λ). We are interested in the structure of P (0). P (0) has four irreducible subquotient which could be calculated from the following exact sequences

\begin{matrix}0 → K′ (−2ρ_{1}) → P (0) → K′ (0) → 0 \\ 0 → L (2ρ_{1}) → K′ (0) → L (0) → 0 \\ 0 → L (0) → K′ (−2ρ_{1}) → L (−α) → 0 \end{matrix} (6.1)

Let X_{β} ∈ 𝔤_{β} such that β′ ([X_{β}, X_{−β}]) = (β′, β) for any β′ ∈ Δ. Let

X = ∏_{β∈Δ(𝔤_{1})}X_{β}, Y = ∏_{β∈Δ(𝔤_{−1})}X_{β},

and

X^{ad} = ∏_{β∈Δ(𝔤_{1})}ad X_{β}, Y^{ad} = ∏_{β∈Δ(𝔤_{−1})}ad X_{β}.

The following three lemmas and corollary hold for sl (m|n) with arbitrary m and n. They can be extracted from [5], but we give independent proofs here.

Lemma 6.1. Y^{ad} (U (𝔤_{0})) = Y U (𝔤_{0}). In particular Y^{ad} (Z^{0}) = Y Z^{0}.

4P (λ) is projective in the category of all finite-dimensional 𝔤-modules, semi-simple over 𝔥.

Proof. It is obvious that Y^{ad} (U (𝔤_{0})) ⊆ Y U (𝔤_{0}). To check the equality note that Y^{ad} : U (𝔤_{0}) → Y U (𝔤_{0}) is a homomorphism of ad sl (n)-modules. A simple calculation shows that Y^{ad} (U (𝔥)) = Y U (𝔥). Since U (𝔥) generates U (𝔤_{0}) as ad sl (n)- module, we have Y^{ad} (U (𝔤_{0})) = Y U (𝔤_{0}).

Recall that h denotes the Harish-Chandra homomorphism. We put h_{u} = h (u) for u ∈ U.

Lemma 6.2. Let Z′ = X^{ad} (Y Z^{0}). Then Z′ ⊂ Z. For any x ∈ Z^{0}

h (X^{ad} (Y x)) = Δ (μ) h_{x},

where Δ (μ) = ∏_{β∈Δ(𝔤_{1})} (μ + ρ, β).

Proof. Let y ∈ Y Z^{0}. Then y is ad 𝔤_{−1}-invariant. If M is the ad 𝔤-submodule generated by y, then obviously M is a quotient of K′ (−2ρ_{1}). From the exact sequence

0 → L (0) → K′ (−2ρ_{1}) → L (−α) → 0

it is clear that X^{ad} (y) = 0 or X^{ad} (y) generates a trivial ad 𝔤-submodule. Thus X^{ad} (y) ∈ Z. To find h (X^{ad} (Y x)) apply X^{ad} (Y x) to the highest vector v of a Verma module M (μ). We have

X^{ad} (Y x) v = XY xv = Δ (μ) h_{x} (μ) v.

Corollary 6.3. h (Z′) = Δ (μ) h (Z^{0}).

Lemma 6.4. Every z ∈ Z′ is contained in some ad 𝔤-submodule N_{z} ⊂ U isomorphic to P (0).

Proof. Let z ∈ Z′. Then by Lemma 6.1 z = X^{ad}Y^{ad} (c) for some c ∈ Z^{0}. Let M_{z} be the ad 𝔤-submodule in U generated by c. Clearly z generates a trivial submodule of M_{z}. Let M be the injective hull of M_{z} with injection i : M_{z} → M. Then M ≅ P (0) ⊕ R for some injective module R, and i (z) ∈ P (0). Let π : M → P (0) denote the projection, then π (c) ≠ 0, and therefore π ∘ i is surjective. Since P (0) is projective, there exist an injective homomorphism j : P (0) → M_{z} such that i∘j = id. Put N_{z} = j (P (0)). Then z ∈ N_{z}.

Staring from this point all results are proven only for 𝔤 = sl (1|n).

Lemma 6.5. Let j : P (0) → U be a non-trivial homomorphism of 𝔤-modules, where 𝔤-action on U is adjoint. Then either j is injective or j (P (0)) is a trivial ad 𝔤submodule in U.

Proof. Let c be an ad 𝔤_{0}-invariant generator of j (P (0)). If Y^{ad} (c) ≠ 0, then Y^{ad} (c) = Y x for some x ∈ Z^{0} and, as follows from Lemma 6.2, X^{ad}Y^{ad} (c) ≠ 0. Therefore j (P (0)) contains a trivial submodule, and by exact sequences (6.1) j is injective. In the same way it can be proven that if X^{ad} (c) ≠ 0, then j is injective as well.

Therefore if j is not injective, then Y^{ad} (c) = X^{ad} (c) = 0. Then looking at the exact sequences (6.1) one can conclude that j (P (0)) = ℂc and therefore j (P (0)) is isomorphic to a trivial module.

Lemma 6.6. Z′ = Z ∩ J, and therefore Z̄ ≅ Z/Z′.

Proof. Let z ∈ Z′. Then by Corollary 6.3 zM (μ) = 0 for all atypical μ ∈ 𝔥^{∗}. Therefore z ∈ ∩_{μ∈S} Ann M (μ) ⊂ ∩_{μ∈S} Ann F (μ) = J. Hence Z′ ⊂ Z ∩ J.

Now suppose z ∈ Z ∩ J. Then z acts trivially on M (μ) for all μ ∈ S. Hence z acts trivially on M (μ) for all atypical μ. Therefore h_{z} (μ) = 0 whenever (μ + ρ, β) = 0 for some β ∈ Δ (𝔤_{1}). Therefore Δ (μ) divides h_{z} (μ). Then by Corollary 6.3 z ∈ Z′, and we have Z ∩ J ⊆ Z′.

Lemma 6.7. Let N̄_{z} be the image of N_{z} in Ū under the natural projection. Then

N̄_{z} ⊂ Z \binom{}{Ū}.

Proof. As follows from the exact sequences (6.1) U_{n} ∩ N_{z} and U_{−n} ∩ N_{z} generate the unique maximal submodule R_{z} of N_{z}. By Corollary 3.3 Ū_{n} = Ū_{−n} = 0. Therefore R̄_{z} = 0. Thus, the image N̄_{z} in Ū is either 0 or a trivial ad 𝔤-module, and hence

N̄_{z} ⊂ Z \binom{}{Ū}.

Lemma 6.8. Z \binom{}{Ū} is spanned by Z̄ and N̄_{z} for all z ∈ Z′.

Proof. Let z ∈ Z \binom{}{Ū}. There exists a homomorphism j : P (0) → U such that the image of j (P (0)) in Ū coincides with ℂz. Let c be an ad 𝔤_{0}-invariant generator of j (P (0)) such that c̄ = z. As follows from Lemma 6.5 either j (P (0)) is a trivial ad 𝔤-submodule in U or j is injective. In the former case c ∈ Z and z = c̄ ∈ Z̄. In the latter case y = X^{ad}Y^{ad} (c) ≠ 0 and y ∈ Z′. Let us choose a generator u ∈ U^{ad 𝔤_{0}} of N_{y} such that X^{ad}Y^{ad} (u) = y. Then X^{ad}Y^{ad} (c − u) = 0, and hence c − u ∈ Z by Lemma 6.5. Then z = c̄ = ū + (c̄ − ū), and therefore z ∈ Z̄ + N̄_{y}.

To describe Z \binom{}{Ū} we define the Harish-Chandra homomorphism h̄ : Z \binom{}{Ū} → S (S^{∗}) ≅ ℂ [μ_{1}, …, μ_{n−1}] by h̄_{z} (μ) = h_{x} (μ) for any x ∈ U with x̄ = z and any μ ∈ 𝔥^{∗}. To see that h̄_{z} does not depend on choice of x note that, since any element z ∈ Z \binom{}{Ū} acts by the scalar c_{z,μ} on F (μ) for any μ ∈ S, we have h̄_{z} (μ) = c_{z,μ}. Since the family F (μ) is faithful over Ū, h̄ is an injective homomorphism. Let A be the image of Z \binom{}{Ū} under the Harish-Chandra homomorphism h̄.

Let W′ be the subgroup of W which preserves α. Clearly W′ is isomorphic to S_{n−1}, and S^{w} = S for any w ∈ W′. Let R denote the ring of W′-invariant polynomials on S, R′ be the subring of polynomials in R constant on all lines l_{χ}. In other words polynomials from R′ do not depend on μ_{1}.

Lemma 6.9. (a) Let r: S (𝔥) → S (S^{∗}) be the restriction homomorphism. Then r ∘ h (Z^{0}) = R.

(b) A ⊂ R.

(c) h̄ \binom{}{Z̄} = R′.

Proof. (a) follows from the fact that h (Z^{0}) = S^{W} (𝔥) and r (S^{W} (𝔥)) = R. (b) follows from the very definition of h̄. Indeed choose ad 𝔤_{0}-invariant x ∈ U such that x̄ = z. Then h̄_{z} = r (h_{x}). Since x is ad 𝔤_{0}-invariant, h_{x} is W -invariant and r (h_{x}) is W′-invariant. (c) follows from the formula (2.1).

Now we will describe h̄ \binom{}{N̄_{z}}. The following polynomial plays a special role

Θ (μ) = ∏_{k=1}^{n−1} (μ + ρ_{0}, β_{1} + ⋯ + β_{k}) = ∏_{i=1}^{n−1} (i + μ_{1} + ⋯ + μ_{i}) = ∏_{γ∈Δ(𝔤_{1})\α} (μ + ρ, γ) .

Lemma 6.10. Let z ∈ Z′, z = X^{ad} (Y u) for some u ∈ Z^{0} and p_{z} ∈ h̄ \binom{}{N̄_{z}}. Then

p_{z} (μ − α) − p_{z} (μ) = aΘ (μ) h_{u} (μ) (6.2)

for some a ∈ ℂ.

Proof. The idea is to look at the zeros of p_{z} (μ) − p_{z} (μ − α). Suppose p_{z} (μ) = p_{z} (μ − α). Let c ∈ U^{ad 𝔤_{0}} be the generator of N_{z} such that Y^{ad} (c) = Y u. Then h̄_{c} is proportional to p_{z}, and c acts by the same constant ν(proportional to p_{z} (μ)) on F (μ) and F (μ − α). We will use the exact sequence

0 → F (μ − α) → M (μ) → F (μ) → 0.

Put c′ = c − ν Id. Then c′M (μ) ⊂ F (μ − α) and (c′)^{2} M (μ) = 0. There are two cases

1. c′ = 0 on M (μ);

2. c′ ≠ 0 on M (μ).

Recall that Y^{ad} (c) = Y^{ad} (c′) = Y u. Therefore in the first case we must have Y u (M (μ)) = 0. Let M^{0} (μ) be the 𝔤_{0} submodule of M (μ) generated by a highest vector. Then u (M^{0} (μ)) = 0, i.e. h_{u} (μ) = 0.

Consider the second case, assuming h_{u} (μ) ≠ 0. In this case c′ is a non-zero element of Hom_{𝔤_{0}} (F (μ) , F (μ − α)). Lemma 3.1 (c) implies that for some distinct γ_{1}, …, γ_{s} ∈ Δ (𝔟_{1}) \α and some distinct γ′_{1}, …, γ′_{s−1} ∈ Δ (𝔟_{1}) \α

Hom_{𝔤_{0}} (M^{0} (μ − γ_{1} − ⋯ − γ_{s}) , M^{0} (μ − α − γ′_{1} − ⋯ − γ′_{s−1})) ≠ 0.

Therefore for some w ∈ W

w (μ − γ_{1} − ⋯ − γ_{s} + ρ_{0}) = μ − α − γ′_{1} − ⋯ − γ′_{s−1} + ρ_{0}.

This is possible if and only if for some γ_{j} and some u ∈ W

u (μ − γ_{j} + ρ_{0}) = μ − α.

Let γ_{j} = α+β_{1}+⋯+β_{k}. Then (μ − α + ρ_{0}, β_{1} + ⋯ + β_{k}) = 1, or using (α, β_{1}) = −1, (α, β_{i}) = 0 for i > 1, (μ + ρ_{0}, β_{1} + ⋯ + β_{k}) = 0.

Thus we obtained that the zeros of the polynomial p_{z} (μ − α)−p_{z} (μ) are the zeros of Θ (μ) h_{u} (μ). The condition Y^{ad} (c) = Y u implies deg h̄_{c} = deg h_{u} + n. Since h̄_{c} and p_{z} are proportional we obtain

deg (p_{z} (μ − α) − p_{z} (μ)) = deg h_{u} + n − 1 = deg Θ (μ) h_{u} (μ) .

Hence the polynomials p_{z} (μ − α) − p_{z} (μ) and Θ (μ) h_{u} (μ) must be proportional.

Remark 6.11. Note that the equation (6.2) in coordinates μ_{1}, …, μ_{n−1} looks as follows

p_{z} (μ_{1}, …, μ_{n−1}) − p_{z} (μ_{1} + 1, …, μ_{n−1}) = aΘ (μ) h_{u} (μ)

Theorem 6.12. Let (Θ) be the ideal in R generated by the polynomial Θ. Then

A = {p ∈ R | p (μ − α) − p (μ) ∈ (Θ)}

Proof. The inclusion

A ⊆ {p ∈ R | p (μ − α) − p (μ) ∈ (Θ)}

follows from Lemma 6.8, Lemma 6.9 (b) and Lemma 6.10.

To show the inverse inclusion let p ∈ R such that p (μ − α) − p (μ) = q (μ) Θ (μ) for some q (μ) ∈ R. By Lemma 6.9 (a) q = h_{u} for some u ∈ Z^{0}. Let z = X^{ad} (Y u). Choose p_{z} ∈ h̄ \binom{}{N̄_{z}} satisfying the equation (6.2) with a = 1. Then

p_{z} (μ − α) − p_{z} (μ) = p (μ − α) − p (μ) .

Therefore p − p_{z} ∈ R′. Since p_{z} ∈ A, p ∈ A.

Let S_{Θ} be the set of zeros of the polynomial Θ (μ).

Corollary 6.13.

A = {p ∈ R | p (μ) = p (μ − α) for all μ ∈ S_{Θ}}

Corollary 6.14. Let S′ = S\ (S_{Θ} ∪ S_{Θ} − α). Then the natural map Spec R → Spec A is an isomorphism when restricted to S′.

Proof. Let μ ∈ S′, ν ∈ S and μ ≠ ν. Then by Corollary 6.13 there exists p ∈ A such that p (μ) = 0 and p (ν) ≠ 0

7. On structure of Ū-modules for 𝔤 = sl (1|n)

We start with the following simple observation.

Theorem 7.1. Let M be a finite-dimensional Ū-module. Then M is completely reducible.

Proof. Let L (λ) and L (μ) be irreducible finite-dimensional Ū-modules with highest weights λ, μ ∈ S. It is sufficient to show that Ext^{1} _{\overline{U}} (L (μ) , L (λ)) = 0. Consider some extension E over U. We have the following exact sequence

0 → L (μ) → E → L (λ) → 0.

Since this sequence does not split λ and μ must be compatible with respect to Bruhat order. Therefore we have three cases: λ = μ, λ > μ and λ < μ. However, as it is shown in [4] , there is no self-extensions of finite-dimensional representations. Thus, the first case is impossible. If λ > μ then E is isomorphic to K (λ). If λ < μ, let us consider the contragredient exact sequence

0 → L′ (λ) → E′ → L′ (μ) → 0.

Note that L′ (λ) ≅ L (λ) and L′ (μ) ≅ L (μ), and therefore E′ is isomorphic to K (μ).

Thus, our task is to show that a Kac module K (λ) is not a Ū-module, i.e. JK (λ) ≠ 0. Indeed, U_{−n}K (λ) ≠ 0 because K (λ) is free over U (𝔤_{−1}). But U_{−n} ⊂ J by Corollary 3.3. Contradiction.

Our next step is to generalize Penkov’s theorem for Ū. For any μ ∈ S let ϕ_{μ} : Z \binom{}{Ū} → ℂ be the character defined by ϕ_{μ} (z) = h̄_{z} (μ) for any z ∈ Z \binom{}{Ū}. The weight μ also defines the character φ_{μ} : Z^{0} → ℂ by φ_{μ} (z) = h_{z} (μ) for any z ∈ Z^{0}. Let Ū^{μ} = Ū/ (Ker ϕ_{μ}) and U^{μ} (𝔤_{0}) = U (𝔤_{0}) / (Ker φ_{μ}). We will prove the following theorem.

Theorem 7.2. If μ ∈ S′, then Ū^{μ} and U^{μ} (𝔤_{0}) are Morita equivalent.

The proof of this theorem is similar to ones given for examples in Sections 4 and 5. We will construct two functors

Inv: Ū^{μ} − mod → U^{μ} (𝔤_{0}) − mod
Ind: U^{μ} (𝔤_{0}) − mod → Ū^{μ} − mod

which establish the equivalence between the categories of Ū^{μ}-modules and U^{μ} (𝔤_{0})- modules.

Let p : U_{0} → U (𝔤_{0}) be the projection along U_{0} ∩ U𝔤_{1}. If κ is the 𝔤_{0}-invariant projection F (μ) → M^{0} (μ) then

p (x) κv = κxv for all x ∈ U_{0}, v ∈ F (μ) .

Therefore p induces p̄: Ū_{0} → Ū (𝔤_{0}), where

Ū (𝔤_{0}) = U (𝔤_{0}) / ∩_{ν∈S} Ann_{U(𝔤_{0})} M^{0} (ν) .

Note that p maps the space of ad 𝔤_{0}-invariants of U to Z^{0}. Therefore p̄ maps ad 𝔤_{0}- invariants of Ū to Z̃ = Z^{0}/ ∩_{ν∈S} Ker φ_{ν}. In particular, p̄(Z \binom{}{Ū}) ⊂ Z̃. Using the identity h = h ∘ p one can easily check that

h̄ (z) = r ∘ h ∘ p̄(z) for any z ∈ Z \binom{}{Ū} . (7.1)

Recall that r is the restriction homomorphism S (𝔥) → S (S^{∗}).

Lemma 7.3. Let μ ∈ S′. Then Z̃p̄(Ker ϕ_{μ}) = Ker φ_{μ}.^{5}

Proof. By Lemma 6.9 (a) the map r ∘ h: Z^{0} → R is surjective. Obviously

(Ker r ∘ h) ∩ Z^{0} = ∩_{μ∈S} Ker φ_{μ}.

Thus r∘h : Z̃ → R is a well-defined isomorphism. By (7.1) r∘h (p̄(Z \binom{}{Ū})) = A. Let I_{μ} ⊂ S (S^{∗}) be the maximal ideal of the point μ ∈ S. Clearly r∘h∘p̄(Ker ϕ_{μ}) = I_{μ}∩A and r ∘ h (Ker φ_{μ}) = I_{μ} ∩ R. It suffices to show that I_{μ} ∩ R = R (I_{μ} ∩ A). But this follows from Corollary 6.14.

Lemma 7.4. Let μ ∈ S′ and M be an Ū^{μ}-module. Let M^{𝔤_{1}} = Ker 𝔤_{1}. Then (a) M^{𝔤_{1}} is an U^{μ} (𝔤_{0})-module.

(b) If z_{0} = diag (n, 1, …, 1), ξ = μ (z_{0}), then M^{𝔤_{1}} = Ker (z_{0} − ξ id).

Proof. (a)For any z ∈ Z \binom{}{Ū}, m ∈ M^{𝔤_{1}} we have zm = p̄(z) m. Since Ker ϕ_{μ} annihilates M^{𝔤_{1}}, then Z̃p̄(Ker ϕ_{μ}) annihilates M^{𝔤_{1}}. By Lemma 7.3 Z̃p̄(Ker ϕ_{μ}) = Ker φ_{μ}. Thus Ker φ_{μ} annihilates M^{𝔤_{1}}, i.e. M^{𝔤_{1}} is U^{μ} (𝔤_{0})-module.

(b) Since φ_{μ} (z_{0}) = ξ, we have M^{𝔤_{1}} ⊆ Ker (z_{0} − ξ id). On the other hand M is a quotient of M′ = U ⊗_{U(𝔤_{0}+𝔤_{1})}M^{𝔤_{1}}, Ker (z_{0} − ξ id) in M′ coincides with M^{𝔤_{1}}. Therefore M^{𝔤_{1}} = Ker (z_{0} − ξ id).

Corollary 7.5. Let Inv: Ū^{μ} − mod → U^{μ} (𝔤_{0}) − mod be the functor defined by Inv (M) = M^{𝔤_{1}}. Then Inv is always faithful and exact for μ ∈ S′.

Proof. Faithfulness is obvious, and exactness follows from Lemma 7.4 (b).

Lemma 7.6. Let s : U (𝔤_{0}) → Ū^{μ} be the composition of embedding U (𝔤_{0}) → U and the natural projection U → Ū^{μ}. Then Ker s ⊂ Ker φ_{μ}.

Proof. Let M be some faithful Ū^{μ}-module. Consider the homomorphism π : Ū^{μ} → End (M). Then

Ker s = Ker π ∘ s = Ann_{U(𝔤_{0})} M ⊂ Ann_{U(𝔤_{0})} M^{𝔤_{1}} ⊂ Ker φ_{μ}.

Let \binom{}{Ū^{μ}}^{+} denote the image of U (𝔤_{0} + 𝔤_{1}) in Ū^{μ} under the natural projection. By the above Lemma every U^{μ} (𝔤_{0})-module N has the unique structure of \binom{}{Ū^{μ}}^{+}-module such that 𝔤_{1}N = 0. Let

Ind N = Ū^{μ} ⊗_{\binom{}{Ū^{μ}}}^{+} N.

Then Ind is a faithful functor from U^{μ} (𝔤_{0}) − mod to Ū^{μ} − mod .

Lemma 7.7. Let μ ∈ S′, N be any U^{μ} (𝔤_{0})-module. Then Inv (Ind N) ≅ N.

5Note that since μ ∈ S, φ_{μ} (∩_{ν∈S} Ker φ_{ν}) = 0. Therefore φ_{μ} induces the character on Z̃.

Proof. Immediately follows from Lemma 7.4 (b).

Corollary 7.8. Functors Inv and Ind establish the equivalence of the category of Ū^{μ}-modules and the category U^{μ} (𝔤_{0})-modules.

Corollary 7.8 implies Theorem 7.2. It also implies

Corollary 7.9. For any μ ∈ S′, F (μ) is a faithful Ū^{μ}-module.

Remark 7.10. Note that in the same way one can prove that the category of right Ū^{μ}-modules and the category of right U^{μ} (𝔤_{0})-modules are equivalent. Indeed, define functors Rinv: mod − Ū^{μ} → mod − U^{μ} (𝔤_{0}) and Rind: mod − U^{μ} (𝔤_{0}) → mod − Ū^{μ} in the following way

Rinv (M) = \{m ∈ M | m𝔤_{−1} = 0\}, Rind (N) = N ⊗_{\binom{}{Ū^{μ}}}^{−} Ū^{μ},

where \binom{}{Ū^{μ}}^{−} denotes the image of U (𝔤_{0} ⊕ 𝔤_{−1}) under the natural projection U → Ū^{μ}. Then the functors Rinv and Rind establish equivalence of mod − Ū^{μ} and mod − U^{μ} (𝔤_{0}).

Let M and N be 𝔤_{0}-modules. Recall that F (M, N) denotes the maximal subspace of Hom (M, N) on which ad 𝔤_{0} acts locally finitely. The classical result of Joseph (see [8]) claims that U^{μ} (𝔤_{0}) is isomorphic to F (M^{0} (μ) , M^{0} (μ)). The next theorem is an analog of this result for Ū^{μ}.

Theorem 7.11. Let μ ∈ S′. Then the natural map π: Ū^{μ} → F (F (μ) , F (μ)) is an isomorphism.

Proof. Theorem 7.2 implies that F (μ) is a faithful Ū^{μ}-modules, and therefore π is injective. To show surjectivity note that F (F (μ) , F (μ)) is a left Ū^{μ}-module. Therefore by Theorem 7.2

F (F (μ) , F (μ)) = Ind (Inv F (F (μ) , F (μ))) = Ind F (F (μ) , M^{0} (μ)) .

Therefore it suffices to show that F (F (μ) , M^{0} (μ)) ⊂ Im π. On the other hand, F (F (μ) , M^{0} (μ)) is a right Ū^{μ}-module, and hence by Remark 7.10

F (F (μ) , M^{0} (μ)) = Rind (Rinv F (F (μ) , M^{0} (μ))) .

Therefore it suffices to show that Rinv F (F (μ) , M^{0} (μ)) ⊂ Im π. Let κ denote the U (𝔤_{0})-invariant projection F (μ) → M^{0} (μ). It is clear that

Rinv F (F (μ) , M^{0} (μ)) = F (M^{0} (μ) , M^{0} (μ)) κ.

Let K ∈ S^{n−1} (𝔤_{1}) ⊗ S^{n−1} (𝔤_{−1}) be ad 𝔤_{0}-invariant element, defined uniquely up to proportionality. One can check easily that for a highest vector v ∈ F (μ), Kv is proportional to v and is not zero for μ ∈ S′. Therefore π (K) is proportional to κ and hence

F (M^{0} (μ) , M^{0} (μ)) κ = π (U (𝔤_{0}) K) .

8. Degenerate case

Theorem 7.2 is not true if μ ∉ S′. Here we give a generalization which works for all μ.

Let μ ∈ S, S_{μ} = {ν ∈ S | ϕ_{ν} = ϕ_{μ}} and T_{μ} = ∪_{w∈W} S_{μ}^{w}. By Corollary 6.13, S_{μ} = {μ} iff μ ∈ S′.

Recall that μ ∈ 𝔥^{∗} is integer if (μ, β_{i}) ∈ ℤ, dominant if (μ + ρ, β_{i}) > 0 and regular if (μ + ρ, β_{i}) ≠ 0 for all i ≤ n − 1. The irreducible highest weight module L (μ) is finite-dimensional iff μ is integer dominant (see [9]).

Lemma 8.1. (a) For any ν ∈ S there exists μ ∈ S\S_{Θ} and k ≥ 0 such that S_{ν} = S_{μ} = {μ + jα | j = 0, …, k}.

(b) T_{μ} contains at most one regular W -orbit.

(c) T_{μ} contains at most one integer dominant weight.

Proof. (a) easily follows from Corollary 6.13.

(b) Note that all weights from S_{Θ} are not regular. If S_{μ} contains more than one element, then as follows from (a) and Corollary 6.13 S_{μ}\ {μ} ⊂ S_{Θ}, and hence all weights from S_{μ}\ {μ} are not regular. Thus S_{μ} contains at most one regular weight, and therefore T_{μ} contains at most one regular orbit.

(c) follows from (b).

Lemma 8.2. Let μ ∈ S\S_{Θ}.

(a) Z̃p̄(Ker ϕ_{μ}) = ∏_{j=0}^{k} Ker φ_{μ+jα}

(b) If M is a Ū^{μ}-module then Inv M = M′_{0} ⊕ ⋯ ⊕ M′_{k}, where M′_{j} is a U^{μ+jα} (𝔤_{0})- module.

Proof. (a) The proof is similar to the proof of Lemma 7.3. Recall that r ∘ h : Z̃ → R is an isomorphism. By Corollary 6.13 and Lemma 8.1 (a)

∏_{j=0}^{k} (I_{μ+jα} ∩ R) = R (I_{μ} ∩ A) = r ∘ h (Z̃p̄(Ker ϕ_{μ})) .

Thus

Z̃p̄(Ker ϕ_{μ}) = (r ∘ h)^{−1} ∏_{j=0}^{k} (I_{μ+jα} ∩ R) = ∏_{j=0}^{k} Ker φ_{μ+jα}.

(b) follows from (a).

Corollary 8.3. The irreducible highest weight 𝔤-module L (λ) is an Ū^{μ}-module iff λ ∈ T_{μ}.

Proof. Follows from Lemma 8.2 (b) and Inv L (λ) = L^{0} (λ).

Corollary 8.4. The category of Ū^{μ}-modules has at most one up to an isomorphism irreducible finite dimensional module.

Let μ ∈ S\S_{Θ}. Put z_{μ} = ∏_{i=0}^{n−1} (z_{0} − μ (z_{0}) − i) and Y^{μ} = Ū^{μ}/ (z_{μ}).

Lemma 8.5. (a)If μ ∈ S′, then Y^{μ} = Ū^{μ}.

(b)If μ ∈ S\S_{Θ} and M is a Y^{μ}-module, then Inv M = Ker (z_{0} − μ (z_{0}) id) and Inv M is a U^{μ} (𝔤_{0})-module.

Proof. (a) If μ ∈ S′, then Inv M is a U^{μ} (𝔤_{0})-module, and M ≅ Ind (Inv M). Since (z_{0} − μ (z_{0})) Inv M = 0, z_{μ}M = 0.

(b) By Lemma 8.2 (b) Inv M = M′_{0} ⊕ ⋯ ⊕ M′_{k}. But z_{μ} (M′_{j}) ≠ 0 for j > 0. Hence Inv M = M′_{0} is a U^{μ} (𝔤_{0})-module. Clearly, M′_{0} = Ker (z_{0} − μ (z_{0}) id).

One can see immediately from Lemma 8.5 (b) that Inv: Y^{μ} − mod → U^{μ} (𝔤_{0}) − mod is an exact faithful functor. One can define a right adjoint functor Ind: U^{μ} (𝔤_{0})− mod → Y^{μ} − mod by putting

Ind N = Y^{μ} ⊗_{(Y^{μ})}^{+} N,

where (Y^{μ})^{+} is the image of U (𝔤_{0} ⊕ 𝔤_{1}) in Y^{μ} and (Y^{μ})^{+}-action on N is induced by 𝔤_{1}N = 0.

Lemma 8.2 (b) implies Inv (Ind N) = N and therefore the following theorem is true.

Theorem 8.6. Let μ ∈ S\S_{Θ}. Then Y^{μ} is Morita equivalent to U^{μ} (𝔤_{0}). Functors Inv and Ind establish the equivalence between the category of Y^{μ}-modules and the category of U^{μ} (𝔤_{0})-modules.

9. Category O and character formulae

In this section we obtain some formulae for characters of irreducible highest weight sl (1|n)-modules L (λ) (possibly infinite-dimensional). Note that if λ is typical then

L (λ) = U (𝔤) ⊗_{U(𝔤_{0}⊕𝔤_{1})} L^{0} (λ) ,

and therefore

ch L (λ) = ch L^{0} (λ) ∏_{α∈Δ(𝔤_{1})}(1 + e^{−α})

We assume now that λ is atypical. Then L (λ) is an Ū-module and therefore an Ū^{μ}-module for some μ ∈ S\S_{Θ}. We are able to obtain a formula for ch L (λ) if λ = μ^{w}. For example this formula works for arbitrary regular λ.

Denote by O (𝔤_{0}) the category of 𝔤_{0}-modules, which are finitely generated, 𝔥semisimple and locally finite over 𝔟_{0} (see [2]). Let O^{μ} (𝔤) denote the category of Y^{μ}-modules which belong to the category O (𝔤_{0}) as 𝔤_{0}-modules. Finally, let O^{μ} (𝔤_{0}) be the subcategory of O (𝔤_{0}) admitting central character φ_{μ}. The following lemma easily follows from Theorem 8.6.

Lemma 9.1. If μ ∈ S\S_{Θ} then O^{μ} (𝔤) and O^{μ} (𝔤_{0}) are equivalent. For any λ = μ^{w}, Ind L^{0} (λ) = L (λ).

If μ ∈ S\S_{Θ}, then Inv F (μ) = M^{0} (μ) and Ind M^{0} (μ) = F (μ). Let now λ = μ^{w} and α′ = w (α)^{6}. Let 𝔟′ be the Borel subalgebra such that 𝔟_{0} ⊂ 𝔟′ and α′ is a simple root. Let 𝔭′ = 𝔟′ ⊕ 𝔤_{−α′} and λ′ = λ − ∑_{β∈Δ(𝔤_{1})\Δ(𝔭′_{1})} β. We put

F (λ) = U (𝔤) ⊗_{U(𝔭′)} C_{λ′}.

Let us start with calculating the character of F (λ). By Lemma 3.1

ch F (λ) = ch M^{0} (λ′) ∏_{β∈Δ(𝔭′_{1})}(1 + e^{−β}) .

Furthermore

ch M^{0} (λ′) = e^{λ′+ρ_{0}} ∏_{β∈Δ(𝔟_{0})}(e^{β/2} − e^{−β/2}) ,

and using

∏_{β∈Δ(𝔭′_{1})}(1 + e^{−β}) = \frac{∏_{β∈Δ(𝔟′_{1})} (1 + e^{−β})}{(1 + e^{−α′})}

and

∏_{β∈Δ(𝔟′_{1})}(1 + e^{−β}) = e^{ρ_{𝔟′}−ρ_{0}} ∏_{β∈Δ(𝔟′_{1})}(e^{β/2} + e^{−β/2})

we obtain

ch F (λ) = \frac{e^{λ′+ρ_{𝔟′}} ∏_{β∈Δ(𝔟′_{1})} (e^{β/2} + e^{−β/2})}{(1 + e^{−α′}) ∏_{β∈Δ(𝔟_{0})} (e^{β/2} − e^{−β/2})}.

Finally using λ′ + ρ_{𝔟′} = λ + ρ, we obtain

ch F (λ) = \frac{e^{λ+ρ} ∏_{β∈Δ(𝔤_{1})} (e^{β/2} + e^{−β/2})}{(1 + e^{−α′}) ∏_{β∈Δ(𝔟_{0})} (e^{β/2} − e^{−β/2})}. (9.1)

Lemma 9.2. If μ ∈ S\S_{Θ} and λ = μ^{w}, then F (λ) ≅ Ind M^{0} (λ).

Proof. Note that F (λ) is a 𝔟′-highest weight module, i.e. it is generated by a 𝔟′- highest vector v of weight λ′. Let

u = ∏_{β∈Δ(𝔤_{1})\Δ(𝔭′_{1})}X_{β}v, v′ = ∏_{β∈Δ(𝔤_{1})\Δ(𝔭′_{1})}X_{−β}u.

6Since μ ∉ S_{Θ}, α′ is defined uniquely.

Then u is 𝔟-highest vector, v′ is proportional to v and the condition (λ + ρ, β) ≠ 0 for β ≠ α′ insures that v′ ≠ 0. Therefore u also generates F (λ), and F (λ) is a 𝔟-highest weight module with highest weight λ. Since λ is atypical, F (λ) is a Ū-module. Let

F′ (λ) = \{m ∈ F (λ) | zm = ϕ_{μ} (z) m, z ∈ Z \binom{}{Ū}\} .

Then F′ (λ) is 𝔤-invariant and u ∈ F′ (λ). Therefore F′ (λ) = F (λ). Hence F (λ) is a Ū^{μ}-module. It follows from (9.1) that z_{μ} (F (λ)) = 0. Thus F (λ) is a Y^{μ}-module. Furthermore,

Inv F (λ) = Ker (z_{0} − μ (z_{0}) id) = Ker (z_{0} − λ (z_{0}) id) = M^{0} (λ) .

Therefore Ind M^{0} (λ) = F (λ) by Theorem 8.6.

Let M be an irreducible module, and N be a module with finite Jordan-Holder series. Then [N : M] denotes the multiplicity of M in N. Lemma 9.2 and Theorem 8.6 imply that for any μ ∈ S\S_{Θ} and λ, ν ∈ μ^{W}

[F (λ) : L (ν)] = [M^{0} (λ) : L^{0} (ν)] .

The last formula implies

Corollary 9.3. If μ ∈ S\S_{Θ}, λ = μ^{w} and

ch L^{0} (λ) = ∑_{ν} a_{λ,ν} ch M^{0} (ν) ,

then

ch L (λ) = ∑_{ν} a_{λ,ν} ch F (ν) .

10. Geometric realization of Ū and localization theorem

First, we assume that 𝔤 = sl (m|n) for arbitrary m and n and keep the notations of Sections 2 and 3. One can see immediately that 𝔭/ [𝔭, 𝔭] is isomorphic to 𝔥̄ = 𝔥/ (𝔥 ∩ [𝔭, 𝔭]). There is the canonical isomorphism identifying 𝔥̄^{∗} with S.

Consider the algebraic flag supervariety X = G/P, where P ⊂ G is a supergroup corresponding to Lie supealgebra 𝔭. For definition and properties of flag supervarieties see [11, 14] and [13]. Let 𝔭_{x} ⊂ 𝔤 be the Lie superalgebra of the stabilizer of the point x ∈ X.

Let O_{X} be the structure sheaf on X, T_{X} be the sheaf of vector fields on X and D_{X} be the sheaf of differential operators. The action of G on X defines the natural homomorphism a : 𝔤 → T_{X}. Define a Lie algebra sheaf 𝔤_{X} and a ring sheaf U_{X} by

𝔤_{X} = O_{X} ⊗ 𝔤, U_{X} = O_{X} ⊗ U, [g, f] = a (g) f for f ∈ O_{X}, g ∈ 𝔤.

Let

N_{X} = {ξ ∈ 𝔤_{X} | ξ_{x} ∈ [𝔭_{x}, 𝔭_{x}] for all x ∈ X} ,

and put D̄_{X} = U_{X}/N_{X}.

Lemma 10.1. The sheaf D̄_{X} is locally isomorphic to S \binom{}{𝔥̄} ⊗ D_{X}.

Proof. It is sufficient to show that D̄_{X} (V) ≅ S \binom{}{𝔥̄} ⊗ D_{X} (V) for a big Schubert cell V ⊂ X. The natural map u: D̄_{X} (V) → S \binom{}{𝔥̄} ⊗ D_{X} (V) is obviously injective. To prove its surjectivity consider the 𝔥-invariant Lie sublgebra 𝔪 such that 𝔤 = 𝔪 ⊕ 𝔭. Since 𝔪 acts transitively on V and the map O_{X} (V) ⊗ U (𝔪) → D_{X} (V) is surjective. Then the map O_{X} (V) ⊗ U (𝔪 + 𝔥) → S \binom{}{𝔥̄} ⊗ D_{X} (V) is also surjective, and that implies the surjectivity of u.

Consider now two natural maps

γ: U → Γ \binom{}{D̄_{X}} and τ : S \binom{}{𝔥̄} → Γ \binom{}{D̄_{X}} .

Lemma 10.2. Ker γ = J.

Proof. Let M be the irreducible D̄_{X} (V)-module with support at x ∈ X such that 𝔭_{x} = 𝔭. The map γ defines a 𝔤-module structure on M and one can check that

M ≅ U ⊗_{U(𝔭)} (S \binom{}{𝔥̄} ⊗ C_{−σ}) ,

where σ = 2ρ + α_{1} + ⋯ + α_{k}. Since M is a faithful D̄_{X} (V)-module

Ker γ = Ann_{U} M = ∩_{μ∈S} Ann_{U} F (μ) = J.

Let γ̄ be the natural injection Ū → Γ \binom{}{D̄_{X}}. One can see that γ̄ (Z \binom{}{Ū}) ⊂

τ (S \binom{}{𝔥̄}) and that

γ̄ (z) = τ \binom{}{h̄_{z}} for any z ∈ Z \binom{}{Ū} , (10.1)

where h̄ : Z \binom{}{Ū} → S \binom{}{𝔥̄} is the Harish-Chandra homomorphism. Hence it is possible to define the homomorphism γ̃ : S \binom{}{𝔥̄} ⊗_{Z\binom{}{Ū}} Ū → Γ \binom{}{D̄_{X}}.

Now take an arbitrary μ ∈ S. Let I_{μ} ⊂ S \binom{}{𝔥̄} be the maximal ideal of μ ∈ S, and D_{X}^{μ} be the quotient sheaf D̄_{X}/τ (I_{μ+σ}) D̄_{X}. It is obvious that D_{X}^{μ} is a twisted differential operators sheaf on X, i.e. D_{X}^{μ} is locally isomorphic to D_{X}.

For the rest of the section we assume that 𝔤 = sl (1|n). It is clear that the homomorphism γ̃ can be pushed down to

γ̃_{μ}: Ū^{μ} → Γ (D_{X}^{μ}) .

Theorem 10.3. Let 𝔤 = sl (1|n) and μ ∈ S′. Then Ū^{μ} is isomorphic to Γ (D_{X}^{μ}).

Proof. We want to show that γ̃_{μ} is an isomorphism. Let M be the irreducible D_{X}^{μ} (V)- module with the support at x ∈ V such that 𝔭_{x} = 𝔭. Denote by θ the homomorphism Γ (D_{X}^{μ}) → End (M). Since M is a faithful Γ (D_{X}^{μ})-module, Ker θ = 0. The composition θ ∘ γ̃_{μ} induces a Ū^{μ}-module structure on M. It is clear that M is isomorphic to F (μ) as Ū^{μ}-module. By Corollary 7.9 F (μ) is a faithful Ū^{μ}-module, and hence γ̃_{μ} is injective. On the other hand, Γ (D_{X}^{μ}) is a G-module, and therefore θ (Γ (D_{X}^{μ})) ⊆ F (M, M). Now using Theorem 7.11 we obtain θ ∘γ̃_{μ} \binom{}{Ū^{μ}} = F (M, M). Thus, both θ and γ̃_{μ} are isomorphisms.

Furthermore, in case of sl (1|n) one can formulate Beilinson-Bernstein localization theorem for atypical modules.

Corollary 10.4. Let 𝔤 = sl (1|n), μ ∈ S′ and (μ, β_{i}) ∉ ℤ^{+}. Then the category of Ū^{μ}-modules is equivalent to the category of D_{X}^{μ}-modules.

Proof. Follows from Beilinson-Bernstein theorem [1] and Theorem 7.2.

Finally note that, in case when μ ∈ S is integer, D_{X}^{μ} is the sheaf of differential operators on some invertible sheaf O_{X}^{μ}. It is interesting that the analog of Borel-Weil-Bott theorem on X holds even if μ ∉ S′.

Let μ ∈ S be some integer weight, and O (μ) be the sheaf of sections of the linear bundle G/P ⊗_{P} C_{−μ} on X. Then the following theorem is true.

Theorem 10.5. (a) If S_{μ} does not contain regular weights, then H^{i} (X, O (μ)) = 0 for all i.

(b) Let λ ∈ T_{μ} be the unique dominant weight and μ = λ^{w} + jα for some j ≥ 0 and w ∈ W, then H^{i} (X, O (μ)) = 0 for for all i ≠ l (w) + j and H^{l(w)+j} (X, O (μ)) ≅ L (λ)^{∗}.

Proof. (a) Note that H^{i} (X, O (μ))^{∗} is a finite dimensional Ū^{μ}-module. If S_{μ} does not contain regular weights, then Ū^{μ} does not have finite-dimensional modules, and therefore the cohomology must vanish.

(b) Since all H^{i} (X, O (μ))^{∗} are finite-dimensional, Corollary 8.4 implies that H^{i} (X, O (μ))^{∗} is completely reducible and must be isomorphic to a direct sum of several copies of L (λ)^{∗}.

Let X_{0} = G_{0}/B_{0} be the underlying flag variety and O_{X_{0}} (υ) denotes the invertible sheaf induced by the infinitesimal character −υ on X_{0}. The structure sheaf O_{X} has a filtration by O_{X_{0}}-modules with irreducible quotients O_{X_{0}} (γ) where γ runs the set of weights of Λ (𝔪_{1}). (We recall that 𝔪 is the 𝔥-invariant subalgebra such that 𝔪 ⊕ 𝔭 = 𝔤). This filtration induces the filtration of O_{X}-module O (μ) with quotients O_{X_{0}} (μ + γ). Thus, by usual Borel-Weil-Bott theorem for X_{0}, all 𝔤_{0}-components which appear in H^{i} (X, O (μ))^{∗} have highest weight (μ + γ)^{u}, where u ∈ W with l (u) = i. Thus, we have to find all u ∈ W and γ such that

μ + γ = λ^{u}.

Substituting μ = λ^{w} + jα we obtain λ^{w} + jα + γ = λ^{u}, or λ + w^{−1} (jα + γ) = λ^{w^{−1}u}. Let s = w^{−1}u. Then λ − λ^{s} = w^{−1} (jα + γ). That implies

w^{−1} (jα + γ) = jβ_{i} + (j − 1) β_{i+1} + ⋯ + β_{i+j−1}, s = (i + j − 1, …, i) .

Then l (s) = j, l (u) = l (w) + j, that implies (b).

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Dept. of Mathematics, University of California at Berkeley, Berkeley, CA 94720

E-mail address: serganov@math.berkeley.edu