ON THE 𝔫-COHOMOLOGY OF LIMITS OF DISCRETE SERIES REPRESENTATIONS

WOLFGANG SOERGEL

Abstract. Using localization we translate the problem of computing the 𝔫-cohomology of limits of discrete series representations into a problem about the combinatorics of Weyl groups.

1. Introduction

In this article I will show how to reduce the problem of computing the 𝔫-cohomology of arbitrary limits of discrete series representations to some explicit combinatorics of Weyl groups, for 𝔫 the nilradical of a Borel containing a compact Cartan.

More precisely, I will write down very explicit complexes of vector spaces over ℂ with bases consisting of certain subsets of the Weyl group and differentials given in these bases by explicit matrices of zeroes, ones and minus ones. The main result then says that the cohomology of these “combinatorial” complexes is the 𝔫-cohomology we were looking for. This recovers old results of Schmid [Sch76] and Williams [Wil88] concerning the discrete series and their nondegenerate limits respectively. In these cases the differentials of our combinatorial complexes all vanish. We also recover the vanishing result of Mirkovic [Mir90], Corollary 4.8 for degenerate limits under the hypothesis that 𝔫 is holomorphic.

Let me explain briefly how these results are obtained. The first step is to use localization to translate our problem into geometry. Since we are interested in singular central characters, this is not entirely straightforward. To describe the outcome I need some notations. Let G ⊃ K ⊃ T be complexifications of our group, of a maximal compact subgroup and of a compact maximal torus. Let X be the flag variety, i.e. the variety of all Borel subgroups of G. Our discrete series becomes under localization the standard module corresponding to some closed K-orbit i : Y ↪ X.

Now 𝔫 is the nilradical of LieB for some Borel B ⊃ T. The singularity of the central character of our limit M of discrete series determines a parabolic subgroup P ⊃ B. Choosing also a possible eigenvalue λ for the action of LieT on the 𝔫-cohomology determines a subvariety j : O ↪ X of the form O = BgP/B ⊂ G/B = X for suitable g ∈ G. Now consider the category P_{B}(O) of perverse sheaves on O which are smooth along B-orbits. Let A ∈ P_{B}(O) be the projective cover in this category of the simple standard object corresponding to the unique closed B-orbit in O. We will show that calculating H^{•}(𝔫, M)_{λ} corresponds to the geometric problem of calculating the hypercohomology groups ℍ^{•}(Y, i^{!}j_{∗}A).

1991 Mathematics Subject Classification. 22E46, 17B20.

Now A is not as untractable an object as the quick (but not very useful) definition given above suggests. It is known that A is selfdual and admits a filtration with each standard object of P_{B}(O) occurring exactly once as a subquotient. This filtration on A leads to some spectral sequence calculating the hypercohomology above. We write down its first stage (with even the differentials given explicitly) and prove that from the second stage on all differentials vanish. This then gives the main result.

This article owes its existence to H. Carayol who explained the problem to me and supplied illuminating examples as well as motivation [Car96].

2. Combinatorics

Let me describe in more detail the combinatorics of the result. I will give two descriptions, one in terms of geometry and one in terms of Coxeter groups. I start with the geometric one.

Given O = BgP/B and a closed K-orbit Y, the combinatorics computing ℍ^{•}(Y, i^{!}j_{∗}A) goes as follows: Call D the set of B-orbits in O and write D = ⋃_{ν} D(ν), where D(ν) ⊂ D denotes the subset of all orbits of dimension ν. Consider the free ℂ-vectorspace ℂD over D and give it the ℤ-grading (ℂD)^{ν} = ℂD(ν).

We make ℂD into a complex and need a differential. Let us say a four-tuple of orbits (x; y, z; w) from D forms a “small square” if and only if {y, z} = {v ∈ D | x ⊂ v̄, x ≠ v, v ⊂ w̄, v ≠ w}, our orbits are all different from one another and dim x = dim w − 2. Now choose for every pair (x, y) of elements of D such that x ⊂ ȳ and dim x = dim y−1 a sign s(x, y) such that for every small square (x; y, z; w) the four signs multiply to −1 = s(x, y)s(x, z)s(y, w)s(z, w). It is always possible to make such choices, see e.g. [BGG75]. We define the differential d : (ℂD)^{ν} → (ℂD)^{ν−1} to be given by the matrix s(x, y), where it is understood that s(x, y) = 0 unless x ⊂ \overline{y}.

𝔫-COHOMOLOGY OF LIMITS OF DISCRETE SERIES 3

Now what we really need are the pieces of this complex “cut by Y in a fixed codimension”. More precisely, for given c ∈ ℕ put

D_{Y}^{c} = {x ∈ D | x meets Y, and c is the codimension of Y ∩ x in x}.

We make ℂD_{Y}^{c} into a complex with ℤ-grading (ℂD_{Y}^{c})^{ν} = ℂ(D_{Y}^{c} ∩D(ν)) and differential d given by the matrix s(x, y). It is not immediately clear but true that d^{2} = 0. We will prove that

ℍ^{n}(Y, i^{!}j_{∗}A) ≅ ⨁_{c} H^{2c−n}(ℂD_{Y}^{c} , d).

Let us reformulate this in combinatorial terms. Suppose given a finite crystallographic Coxeter group W with Bruhat order ≤ and length l : W → ℕ. Suppose V ⊂ W is a subset containing with any two elements x, y the whole interval {z ∈ W | x ≤ z ≤ y}. We will associate to V a complex C^{•}V. As a graded vector space, we simply define C^{ν}V to be the free vector space with basis {x ∈ V | l(x) = ν}. To define the differential, let us call a 4-tuple (x; y, z; w) of elements of W a “small square” if and only if {y, z} = {v | x < v < w} where y ≠ z and l(x) = l(w) − 2. Choose for every pair x < y in W with l(x) = l(y) − 1 a sign s(x, y) in such a way that for every small square (x; y, z; w) the four signs multiply to −1 = s(x, y)s(x, z)s(y, w)s(z, w). It is always possible to make such choices, see e.g. [BGG75]. Now we define the differential d : C^{ν}V → C^{ν−1}V to be given by the matrix s(x, y), where we put s(x, y) = 0 if x and y are not comparable in the Bruhat order. As will be discussed in Lemma 8.1, we have d^{2} = 0 and the complex C^{•}V is independent of the choice of signs s(x, y) up to isomorphism. We denote by H^{•}V its homology.

Let W ⊃ W_{K} be the Weyl group and the compact Weyl group respectively. Let S ⊂ W be the system of simple reflections determined by our Borel B. The singularity of the central character of our limit of discrete series determines in a way to be made more precise later a subset S_{P} ⊂ S. We let W_{P} ⊂ W be the subgroup generated by S_{P}.

The length l(x) of x ∈ W can also be interpreted as the number of walls separating the antidominant Weyl chamber C from xC. We will also need the “K-length” l_{K} : W → ℤ, where l_{K}(x) denotes the number of compact walls separating C from xC.

To any triple (A, D, c) consisting of a left W_{K}-orbit A ⊂ W, a right W_{P} -orbit D ⊂ W and an integer c we associate the set

V(A, D, c) = {x ∈ A ∩ D | l(x) = l_{K}(x) + c}.

We will express the 𝔫-cohomology of limits of discrete series as sums of suitable H^{•}V(A, D, c).

To write down precise formulas we need more precise notations. Let C be the Weyl chamber on which all coroots α^{∨} with α a root of B take negative values. Let 𝔤 ⊃ 𝔟 ⊃ 𝔥 be the Lie algebras of G ⊃ B ⊃ T and let 𝔫 ⊂ 𝔟 be the nilradical. Let U = U(𝔤) ⊃ Z be the universal enveloping algebra and its center. For any integral central character χ ∈ MaxZ let T^{χ} denote the translation functor from the trivial central character to χ. Define the 𝔤-module L(Y, χ) = T^{χ}Γ(G/B, i_{∗}O_{Y}) where i : Y ↪ G/B is the inclusion of a closed K-orbit and i_{∗} denotes the direct image of D-modules. Every limit of discrete series occurs among the L(Y, χ). We want to compute H^{n}(𝔫, L(Y, χ)).

Now this space admits a natural action by 𝔥. Let us normalize the Harish-Chandra homomorphism ξ^{♯} : Z → S(𝔥) by requiring ξ^{♯}(z)−z ∈ U𝔫. Let ξ : 𝔥^{∗} → MaxZ be the corresponding map on points. Its fibres are the orbits for the dot-action of W on 𝔥^{∗}, given by x·λ = x(λ+ρ)−ρ with ρ the half-sum of the roots of 𝔫. For any 𝔤-module L which is annihilated by χ we have the generalized eigenspace decomposition H^{n}(𝔫, L) = ⨁_{λ∈ξ^{−1}(χ)} H^{n}(𝔫, L)_{λ}. We want to establish for our problem the following solution.

Theorem 2.1. The λ-weight space in the 𝔫-cohomology of the limit of discrete series L(Y, χ) is given by the formula

H^{n}(𝔫, L(Y, χ))_{λ} = ⨁_{c} H^{2c+dim Y −n}V(A, D, c)

where A = A(Y) = {x ∈ W | xB ∈ Y } and D = D(λ) = {x ∈ W | x^{−1}(λ + ρ) ∈ \overline{C}\}.

Remark that somewhat implicitly in this is hidden the statement that we have to take W_{P} the isotropy group (for the linear W-action) of the unique weight in \overline{C} ∩ (ξ^{−1}(χ) + ρ). The summands for different c correspond to parts of various “weight” in the 𝔫-cohomology, as will be explained in the proof of the theorem at the very end of this paper.

I don’t know how to further simplify the right hand side in general. However I want to remark on a special case: If W_{K} is generated by W_{K} ∩ S, i.e. for “𝔫 holomorphic”, we see that c is constant on each A, and for this c the complex C^{•}V(A, D, c) will look like the analogous complex for the group generated by all compact reflections with fixed point λ + ρ. It will thus have zero cohomology if there are any such reflections, i.e. in the degenerate case. This recovers Corollary 4.8 of [Mir90].

3. Preliminaries on 𝔫-cohomology

Let 𝔤 = \overline{𝔫}⊕𝔥⊕𝔫 be a complex reductive Lie algebra with a triangular decomposition. Put 𝔟 = 𝔥⊕𝔫. Let U(𝔤) ⊃ Z be the enveloping algebra of 𝔤 and its center. For any ideal χ ⊂ Z put U_{χ} = U(𝔤)/χU(𝔤).

Lemma 3.1. For any U_{χ}-module M we have

H^{i}(𝔫, M) = Ext_{U_{χ}}^{i}(U_{χ} ⊗_{U(𝔫)} ℂ, M).

Proof. To compute the 𝔫-cohomology H^{i}(𝔫, M) of an 𝔫-module M we have to take a free resolution P_{•} ↠ ℂ of the trivial representation ℂ of 𝔫 and compute the cohomology of the complex Hom_{𝔫}(P_{•}, M). If M is a 𝔤-module this coincides with the complex Hom_{𝔤}(U(𝔤) ⊗_{U(𝔫)} P_{•}, M). If χ ⊂ Z is an ideal with χM = 0, we can identify our complex also with the complex Hom_{U_{χ}}(U_{χ} ⊗_{U(𝔫)} P_{•}, M).

Let W be the Weyl group of 𝔤. Then by Chevalley’s theorem we can find a homogeneous subspace H ⊂ U(𝔥) such that the multiplication gives an isomorphism U(𝔥)^{W} ⊗ H →^{∼} U(𝔥), and by Kostant’s theorem for any such H the multiplication gives an isomorphism U(\overline{𝔫}) ⊗ Z ⊗ H ⊗ U(𝔫) →^{∼} U(𝔤). In particular U_{χ} is a free right U(𝔫)-module, thus the complex U_{χ} ⊗_{U(𝔫)} P_{•} is a resolution of U_{χ} ⊗_{U(𝔫)} ℂ by free U_{χ}- modules. Hence the cohomology of the complex Hom_{U_{χ}}(U_{χ}⊗_{U(𝔫)}P_{•}, M) is Ext_{U_{χ}}^{•}(U_{χ} ⊗_{U(𝔫)} ℂ, M).

Now we are interested in the 𝔫-cohomology of limits of discrete series. So consider for any χ ∈ MaxZ the category

M_{χ} = {M ∈ 𝔤-mod | ∀ m ∈ M, ∃n ≫ 0 such that χ^{n}m = 0}.

If ψ, χ ∈ MaxZ are integral we define the translation functor T_{ψ}^{χ} : M_{ψ} → M_{χ} as usual. It comes with an adjointness (T_{ψ}^{χ}, T_{χ}^{ψ}). Recall also the category

O = \{M ∈ 𝔤-mod | \begin{matrix}M is finitely generated, \\ locally finite over 𝔟 and \\ semisimple over 𝔥. \end{matrix} \}

We put O_{ψ} = O ∩M_{ψ} for ψ ∈ MaxZ. We will also need its Z-diagonal version

O′_{ψ} = \{M ∈ 𝔤-mod | \begin{matrix}M is finitely generated, \\ locally finite over 𝔟 and \\ ψM = 0. \end{matrix} \}

Recall the Verma modules M(λ) = U(𝔤) ⊗_{U(𝔟)} ℂ_{λ} for λ ∈ 𝔥^{∗}. We have Ann_{Z}M(λ) = ξ(λ). Now by [Soe89], 2.1 there exists for ψ ∈ MaxZ regular an equivalence of categories

S = S_{ψ} : O_{ψ} →^{∼} O′_{ψ}

such that SM(λ) ≅ M(λ) for every λ ∈ ξ^{−1}(ψ). Let us fix such equivalences. The following theorem will permit us to translate our problem of computing the 𝔫-cohomology of limits of discrete series into geometry.

Theorem 3.2. Suppose ψ, χ ∈ MaxZ are integral and ψ is regular. Then for any N ∈ U_{ψ}-mod and λ ∈ ξ^{−1}(χ) we have

H^{i}(𝔫, T_{ψ}^{χ}N)_{λ} ≅ Ext_{U_{ψ}}^{i}(ST_{χ}^{ψ}M(λ), N).

Remark 3.3. For χ = ψ regular this boils down to the well-known formula

H^{i}(𝔫, N)_{λ} ≅ Ext_{U_{χ}}^{i}(M(λ), N).

Proof. We start with the equality

H^{i}(𝔫, T_{ψ}^{χ}N) = Ext_{U_{χ}}^{i}(U_{χ} ⊗_{U(𝔫)} ℂ, T_{ψ}^{χ}N)  (1)

from Lemma 3.1. Now we want to use the adjointness (T_{χ}^{ψ}, T_{ψ}^{χ}). For this we need

Lemma 3.4. Let T be any translation functor. Then the inclusion Ann_{U}M ⊂ Ann_{U}N implies Ann_{U}T M ⊂ Ann_{U}T N.

Proof. Left to the reader.

Lemma 3.5. Let ψ, χ ∈ MaxZ be integral and suppose ψ is regular. Then Ann_{U}T_{χ}^{ψ}M(λ) = UAnn_{Z}T_{χ}^{ψ}M(λ) for all λ with χM(λ) = 0.

Proof. The inclusion ⊃ is obvious. For the other inclusion remark first that by the previous lemma we can assume without restriction M(λ) simple. Then T_{χ}^{ψ}M(λ) has some simple Verma module as its unique simple quotient, call it M(μ). Consider now a Verma flag T_{χ}^{ψ}M(λ) = F_{r} ⊃ … ⊃ F_{1} ⊃ F_{0} = 0. We prove by induction from above on i that Ann_{U}(F_{r}/F_{i}) = UAnn_{Z}(F_{r}/F_{i}). For i = r − 1 we find F_{r}/F_{r−1} ≅ M(μ) and our statement follows from a theorem of Duflo. Now we have z ∈ Ann_{Z}(F_{r}/F_{i}) if and only if zF_{r} ⊂ F_{i}. But it is clear that Hom_{𝔤}(F_{r}, F_{i}/F_{i−1}) is onedimensional for every i. Since Ann_{Z}(F_{r}/F_{i−1}) is the kernel of the obvious map Ann_{Z}(F_{r}/F_{i}) → Hom_{𝔤}(F_{r}, F_{i}/F_{i−1}), there exists z_{0} ∈ Z such that

Ann_{Z}(F_{r}/F_{i}) = Ann_{Z}(F_{r}/F_{i−1}) + ℂz_{0}.

If z_{0}(F_{r}) ⊂ F_{i−1} we are done. Otherwise z_{0} has to induce an isomorphism of F_{r}/F_{r−1} onto the socle of F_{i}/F_{i−1}. Any u ∈ Ann_{U}(F_{r}/F_{i−1}) annihilates F_{r}/F_{i}, hence can be written by induction in the form u = u_{1} + u_{0}z_{0} with u_{1} ∈ UAnn_{Z}(F_{r}/F_{i−1}), u_{0} ∈ U. But we deduce that u_{0} annihilates the socle M(μ) of F_{i}/F_{i−1}, thus u_{0} = ũ_{0}z̃_{0} with z̃_{0} ∈ ψ and u = u_{1} + ũ_{0}z̃_{0}z_{0} ∈ UAnn_{Z}(F_{r}/F_{i−1}).

Remarks 3.6. 1. This proof shows that the codimension in Z of the annihilator Ann_{Z}(T_{χ}^{ψ}M(λ)) is at most r alias the cardinality of the isotropy group W_{λ} of λ under the dot-action, W_{λ} = {w ∈ W | w(λ + ρ) = λ + ρ}.

2. The lemma and its proof use in fact neither integrality nor regularity. We just need that ψ lies on less walls than χ.

Let us put ψ̃ = Ann_{Z}T_{χ}^{ψ}M(λ) for any λ ∈ ξ^{−1}(χ). So ψ̃ ⊂ Z depends on χ, but by Lemma 3.4 it does not depend on the choice of λ ∈ ξ^{−1}(χ).

Lemma 3.7. The translations T_{χ}^{ψ}, T_{ψ}^{χ} restrict to functors

T_{χ}^{ψ} : U_{χ}-mod → Uψ̃-mod
T_{ψ}^{χ} : Uψ̃-mod → U_{χ}-mod.

Proof. We only prove the second statement, the proof of the first being analogous and easier. Remark that T_{ψ}^{χ}T_{χ}^{ψ}M(λ) is just a direct sum of some copies of M(λ), for every λ ∈ ξ^{−1}(χ). So

N ∈ Uψ̃-mod ⇒ Ann_{U}N ⊃ Ann_{U}T_{χ}^{ψ}M(λ) = Uψ̃
⇒ Ann_{U}T_{ψ}^{χ}N ⊃ Ann_{U}T_{ψ}^{χ}T_{χ}^{ψ}M(λ) = Uχ
⇒ T_{ψ}^{χ}N ∈ U_{χ}-mod.

Now if P ∈ U_{χ}-mod is projective, so is T_{χ}^{ψ}P ∈ Uψ̃-mod. Indeed, Hom_{Uψ̃}(T_{χ}^{ψ}P, N) = Hom_{U_{χ}}(P, T_{ψ}^{χ}N) is exact in N ∈ Uψ̃-mod. So we can continue our equality from the beginning and get

\begin{aligned}H^{i}(𝔫, T_{ψ}^{χ}N) & = & Ext_{U_{χ}}^{i}(U_{χ} ⊗_{U(𝔫)} ℂ, T_{ψ}^{χ}N) & (1) \\ & = & Ext_{Uψ̃}^{i}(T_{χ}^{ψ}(U_{χ} ⊗_{U(𝔫)} ℂ), N). & (2) \end{aligned}

To proceed further, we need

Lemma 3.8. T_{χ}^{ψ}(U_{χ} ⊗_{U(𝔫)} ℂ) is free over Z/ψ̃.

Proof. As explained in [MS95], 2.4 we have an isomorphism

U_{χ} ⊗_{U(𝔫)} ℂ ≅ U ⊗_{U(𝔟)} (U(𝔥)/U(𝔥)ξ^{♯}(χ))

since both objects share a universal property. In addition by invariant theory we find that

U(𝔥)/U(𝔥)ξ^{♯}(χ) = ⨁_{ξ(λ)=χ} U(𝔥)/λ̃

where λ̃ ⊂ U(𝔥) are suitable ideals with radical \sqrt{λ̃} = ker λ ⊂ U(𝔥) and codimension dim_{ℂ}(U(𝔥)/λ̃) = |W_{λ}| the cardinality of the isotropy group of λ under the dot-action. Let us define for any ideal I ⊂ U(𝔥) the “thick Verma” M(I) = U(𝔤) ⊗_{U(𝔟)} U(𝔥)/I. Clearly Ann_{Z}M(I) = (ξ^{♯})^{−1}(I). For any 𝔤-module E we have E ⊗ M(I) = U(𝔤) ⊗_{U(𝔟)} (E ⊗ U(𝔥)/I). If E is finite dimensional, its restriction to 𝔟 admits a filtration with onedimensional subquotients k_{ν} where ν runs over the multiset P (E) ⊂ 𝔥^{∗} of weights of E. Thus E ⊗ M(I) has a filtration with subquotients M(I + ν) for ν running over P (E), in obvious notations. In these terms we find

U_{χ} ⊗_{U(𝔫)} ℂ = ⨁_{ξ(λ)=χ} M(λ̃)

and we deduce

T_{χ}^{ψ}(U_{χ} ⊗_{U(𝔫)} ℂ) = ⨁_{ξ(λ)=χ} T_{χ}^{ψ}M(λ̃).

In addition each of the summands has a filtration with subquotients of the form M(μ̃) for suitable ideals μ̃ ⊂ U(𝔥) of codimension |W_{λ}| with radical \sqrt{μ̃} ∈ ξ^{−1}(ψ). Now it is certainly sufficient to show that these “thick Vermas” M(μ̃) are free over Z/ψ̃. For this remark first that we have ψ̃ ⊂ Ann_{Z}M(μ̃) = (ξ^{♯})^{−1}(μ̃). Now ψ̃ ⊂ Z has codimension at most |W_{λ}| by remark (1) following the proof of Lemma 3.5. On the other hand (ξ^{♯})^{−1}(μ̃) ⊂ Z has the same codimension as μ̃ ⊂ U(𝔥), since ξ is étale in \sqrt{μ}, and this codimension we know already to be |W_{λ}|. Comparing we find ψ̃ = (ξ^{♯})^{−1}(μ̃). Now it is clear from the isomorphism

M(μ̃) = U(\overline{𝔫}) ⊗ U(𝔥)/μ̃

that this module is free our Z/ψ̃.

But this lemma means that a resolution of T_{χ}^{ψ}(U_{χ} ⊗_{U(𝔫)} ℂ) by free Uψ̃- modules stays exact when we apply ⊗_{Z}Z/ψ. So we can go on and write

H^{i}(𝔫, T_{ψ}^{χ}N) = Ext_{U_{χ}}^{i}(U_{χ} ⊗_{U(𝔫)} ℂ, T_{ψ}^{χ}N)  (1)
= Ext_{Uψ̃}^{i}(T_{χ}^{ψ}(U_{χ} ⊗_{U(𝔫)} ℂ), N)  (2)
= Ext_{U_{ψ}}^{i}(T_{χ}^{ψ}(U_{χ} ⊗_{U(𝔫)} ℂ) ⊗_{Z} Z/ψ, N)  (3)

if ψN = 0, as assumed in our theorem. So to finish the proof of the theorem, we ought just to show (in the notations from above):

Proposition 3.9. ST_{χ}^{ψ}M(λ) ≅ T_{χ}^{ψ}(M(λ̃)) ⊗_{Z} Z/ψ.

Proof. We first show that the dimensions of (generalized) 𝔥-weight spaces on both sides coincide. Indeed, both sides have a Verma flag with subquotients M(μ) where μ runs over the set

Λ = {μ ∈ ξ^{−1}(ψ) | T_{ψ}^{χ}M(μ) ≅ M(λ)}.

For the left hand side this is evident, for the right it follows from the existence of a filtration of T_{χ}^{ψ}M(λ̃) by certain M(μ̃) which appeared in the proof of lemma 3.8. So we have just to establish a surjection T_{χ}^{ψ}M(λ̃) ↠ ST_{χ}^{ψ}M(λ). Now ST_{χ}^{ψ}M(λ) has a unique simple quotient L, and T_{ψ}^{χ}L is the unique simple quotient L(λ) of M(λ). But by definition of M(λ̃) we can identify for all M ∈ U_{χ}-mod our Hom_{𝔤}(M(λ̃), M) with the generalized 𝔥-weight space in M^{𝔫} of weight λ. Since T_{ψ}^{χ}ST_{χ}^{ψ}M(λ) is just a sucessive extension of copies of M(λ), this means that the map

Hom_{𝔤}(M(λ̃), T_{ψ}^{χ}ST_{χ}^{ψ}M(λ))
↓
Hom_{𝔤}(M(λ̃), T_{ψ}^{χ}L)

has to be a surjection, and taking the adjoint we find the surjection T_{χ}^{ψ}M(λ̃) ↠ ST_{χ}^{ψ}M(λ) needed.

This finishes the proof of the proposition and the theorem.

4. Localization of our problem

Let Z^{+} ⊂ Z be the annihilator of the trivial representation ℂ of 𝔤. By the famous localization theorem of [BB81], the action of G on its flag variety X induces an isomorphism U/Z^{+}U →^{∼} Γ(X, D_{X}) from U/Z^{+}U to the global (algebraic) differential operators on X, and the functor of global sections

Γ : D_{X}-mod → U/Z^{+}U-mod

is an equivalence of categories. The inverse functor is denoted Δ and called “localization”. To apply this theory to our problem we have to understand the localization of the objects ST_{χ}^{ψ}M(λ) appearing in Theorem 3.2 for ψ = Z^{+}. Recall that here λ is an integral weight and χ = ξ(λ) its image in MaxZ. Let us abreviate T_{χ}^{ψ} = T_{χ}, T_{ψ}^{χ} = T^{χ} for ψ = Z^{+}. Let us take D = D(λ) ⊂ W is as in Theorem 2.1, O = BDB/B, A = A(D) ∈ P_{B}(O) the projective cover of the unique simple standard object of P_{B}(O) and j : O ↪ G/B the inclusion.

We will freely use the Riemann-Hilbert correspondence to pass between regular holonomic D-modules and perverse sheaves. We want to prove Proposition 4.1. ΔST_{χ}M(λ) ≅ j_{!}A.

Proof. This is a remodeling of the proof of Lemma 10 in [Soe89]. Remark first that j_{!} : P_{B}(O) → P_{B}(G/B) is a fully faithful exact functor by [BBD82], and it is even fully faithful on higher Ext, since by e.g. [BGS96], 3.3 these coincide with the Ext in the derived categories of all D-modules (or all sheaves, in the other picture). Let for {M_{i} | i ∈ I} a collection of objects from some abelian category denote ⟨M_{i} | i ∈ I⟩ the smallest full subcategory closed under kernels, cokernels and extensions containing the M_{i}. For x ∈ W let M_{x} ∈ P_{B}(G/B) denote the localization of the Verma module M_{x} = ΔM_{x} where we put M_{x} = M(−xρ−ρ). This is just the dual standard object M_{x} = j_{x!}O_{BxB/B} corresponding to the cell j_{x} : BxB/B ↪ G/B. It is then clear that we have j_{!}P_{B}(O) = ⟨M_{x} | x ∈ D⟩ as subcategories of P_{B}(G/B).

Let y ∈ D be the shortest element. Clearly j_{!}A is the projective cover of M_{y} in ⟨M_{x} | x ∈ D⟩. We will be done if we show that T_{χ}M(λ) is the projective cover of M_{y} in the subcategory ⟨M_{x} | x ∈ D⟩ of O, since under the functor ΔS this corresponds to the category ⟨M_{x} | x ∈ D⟩ considered above.

Now T_{χ}M(λ) admits a filtration with its subquotients among the M_{x}, x ∈ D (each appearing once), so it already lies in ⟨M_{x} | x ∈ D⟩. For any M ∈ ⟨M_{x} | x ∈ D⟩ we know that T^{χ}M is a direct sum of copies of M(λ), since T^{χ}M_{x} ≅ M(λ) for all x ∈ D, EndM(λ) = ℂ and the M(λ) admit no selfextensions in O. For M ∈ ⟨M_{x} | x ∈ D⟩ we have

[M : M_{y}] = dim Hom(M(λ), T^{χ}M)
= dim Hom(T_{χ}M(λ), M)

since this holds for M = M_{x}, x ∈ D and since both sides are additive on short exact sequences. Thus indeed T_{χ}M(λ) is the projective cover of M_{y} in ⟨M_{x} | x ∈ D⟩ and the proposition is proved.

Corollary 4.2. A is selfdual.

Proof. This is in fact rather a corollary of the preceding proof. Let us denote by 𝔻 the Verdier duality functor on any variety. It is certainly sufficient to prove j_{!}𝔻A ≅ j_{!}A. Now clearly the left hand side is the injective hull of M_{y} in ⟨M_{x} | x ∈ D⟩. So we should prove the same thing for the right hand side, or equivalently prove that T_{χ}M(λ) is the injective hull of M_{y} in the subcategory ⟨M_{x} | x ∈ D⟩ of O. But as before we find for any M ∈ ⟨M_{x} | x ∈ D⟩ the equality

[M : M_{y}] = dim Hom(T^{χ}M, M(λ))
= dim Hom(M, T_{χ}M(λ))

and the statement follows.

The proposition allows us to rewrite Theorem 3.2 in more geometric form.

Corollary 4.3. Let χ ∈ MaxZ be integral, λ ∈ ξ^{−1}(χ) and N a representation of 𝔤 such that Z^{+}N = 0. Then

H^{n}(𝔫, T^{χ}N)_{λ} ≅ Ext_{D}^{n}(j_{!}A, ΔN).

Here Ext_{D} stands for extensions in the category of all D-modules and A is the D-module depending on λ considered above.

If N is a discrete series or more generally N = Γ(X, i_{∗}O_{Y}) for i : Y ↪ G/B the embedding of a smooth closed subvariety, we can rewrite this still further. First let us pass to the language of perverse sheaves. Let \underline{Y} denote the constant sheaf on Y. Then the D-module O_{Y} corresponds to the perverse sheaf \underline{Y} [dim Y] and letting now Hom_{D} denote homomorphisms in the derived category of sheaves we find

H^{n}(𝔫, T^{χ}N)_{λ} = Hom_{D}(j_{!}A, i_{∗}\underline{Y} [n + dim Y])
= Hom_{D}(i_{!}\underline{Y} , j_{∗}A[n − dim Y])
= Hom_{D}(\underline{Y} , i^{!}j_{∗}A[n − dim Y])
= ℍ^{n−dim Y} (Y, i^{!}j_{∗}A)

where we used the selfduality of A (Corollary 4.2) in the second step. For easy reference, we reformulate our overall findings as a

Theorem 4.4. For Y ⊂ G/B an irreducible smooth closed subvariety, χ an integral central character, λ ∈ ξ^{−1}(χ) and A as above we have

H^{n}(𝔫, L(Y, χ))_{λ} ≅ ℍ^{n−dim Y} (Y, i^{!}j_{∗}A).

Remark 4.5. In fact this holds more generally for arbitrary affine embeddings i : Y ↪ G/B of smooth varieties Y, when we interpret L(Y, χ) = T^{χ}Γ(X, i_{∗}O_{Y}).

5. Decomposition by codimension

To better understand the right hand side of the above Theorem 4.4, we break it into pieces. Let Y ⊂ X be a closed K-orbit and i : Y ↪ X the inclusion. By our assumptions B ∩ K is a Borel in K, each closed K-orbit Y in X is isomorphic to the flag manifold of K, and each B-orbit in X meets Y in a (B ∩ K)-orbit or not at all. For x ∈ W let us define c_{Y} (x) = c(x) ∈ ℤ_{≥0} ∪ {∞} to be the codimension of Y ∩ \overline{BxB/B} in \overline{BxB/B} resp. ∞ if Y ∩ \overline{BxB/B} = ∅. We obtain a filtration of Y by the closed subsets Y_{c} = ⋃_{c(x)≤c} Y ∩ \overline{BxB/B}, and it is clear that we have as well Y_{c} = ⋃_{c(x)≤c} Y ∩ BxB/B. Consider now the decomposition Y_{c} = Y_{c−1} ∪ (Y_{c} − Y_{c−1}) into a closed subset and its open complement. Let i_{c}, j_{c} be the corresponding closed and open immesions and let a_{c} resp. u_{c} denote the embedding of any Y_{c} resp. Y_{c} − Y_{c−1} into Y. For a variety Y let D(Y) denote the bounded derived category with algebraically constructible cohomology of sheaves of ℂ-vectorspaces on Y^{an}. Certainly for F ∈ D(Y) we have distinguished triangles

(i_{c!}i_{c}^{!}a_{c}^{!}F, a_{c}^{!}F, j_{c∗}j_{c}^{∗}a_{c}^{!}F).

If we apply k_{∗} (where k : Y → pt is the constant map to a point) and rewrite a bit the result, we get distinguished triangles

(k_{∗}a_{c−1}^{!}F, k_{∗}a_{c}^{!}F, k_{∗}u_{c}^{!}F)

in D(pt). The point of this section is to prove that in case F = i^{!}j_{∗}A with A = A(D) as above all these triangles are in fact split short exact sequences, thus leading to a direct sum decomposition

Lemma 5.1. ℍ^{•}i^{!}j_{∗}A = ⨁_{c} ℍ^{•}u_{c}^{!}i^{!}j_{∗}A.

Proof. To see this, we need weights. As is proved in [BGS96], the antidominant projective A(W) on the whole of G/B admits a mixed structure (in the sense of mixed Hodge modules, say) such that it admits a filtration where the sucessive subquotients are isomorphic to

j_{z∗}\underline{BzB/B}[l(z)]

with z running over the Weyl group. Here for Z a variety we let \underline{Z} denote the constant sheaf with trivial mixed structure. From the geometric constructions in [Soe89], Lemmas 8 and 9 we deduce that similarily j_{∗}A admits a mixed structure with filtrations by subquotients j_{z∗}\underline{BzB/B}[l(z)] where z runs over D. In general, if a : Y → Z is a morphism of smooth varieties, then a^{!}\underline{Z} = \underline{Y} [−2c](−c) with c = dim Z − dim Y the codimension.

Using base change, we see that for z ∈ D all

ℍ^{ν}u_{c}^{!}i^{!}j_{z∗}\underline{BzB/B}[l(z)]

are pure of weight 2c (or zero). Whence all ℍ^{ν}u_{c}^{!}i^{!}j_{∗}A are pure of weight 2c, and the Lemma follows.

6. Investigation of some spectral sequences

To investigate the pieces ℍ^{•}u_{c}^{!}i^{!}j_{∗}A we begin with some generalities. Let Z be a variety, and Z = Z_{−∞} ⊃ … Z_{p} ⊃ Z_{p+1} ⊃ … ⊃ Z_{n} = ∅ a (finite) filtration by closed subvarieties. Let i_{p} : Z_{p} − Z_{p+1} ↪ Z denote the inclusion. Let F be a complex of sheaves on Z. As explained in [BGS96], 3.4 the hypercohomology ℍ^{n}F is the limit of a spectral sequence with E_{1}-term E_{1}^{p,q} = ℍ^{p+q}i_{p}^{!}F. Its differential E_{1}^{p,q} → E_{1}^{p+1,q} can be described as follows: Consider the decomposition of Z_{p} − Z_{p+2} into an open and a closed subset

Z_{p} − Z_{p+1} ↪^{u} Z_{p} − Z_{p+2} ↩^{a} Z_{p+1} − Z_{p+2}

and let j : Z_{p} − Z_{p+2} ↪ Z be the inclusion. Then we have a distinguished triangle (a_{!}a^{!}j^{!}F, j^{!}F, u_{∗}u^{∗}j^{!}F) = (a_{∗}i_{p+1}^{!}F, j^{!}F, u_{∗}i_{p}^{!}F) giving rise to boundaries ℍ^{n}i_{p}^{!}F → ℍ^{n+1}i_{p+1}^{!}F which are the boundaries of our spectral sequence.

Now we go to a much more special situation. Namely Z should admit a stratification by affine spaces Z_{v} ≅ 𝔸^{l(v)}, v ∈ I for I some index set such that Z_{p} = ⋃_{l(v)≤−p} Z_{v}. We suppose furthermore that F is actually a perverse sheaf and admits a filtration with subquotients isomorphic to the standard objects j_{v∗}\underline{Z}_{v}[l(v)] where j_{v} : Z_{v} ↪ Z is the inclusion. To simplify the exposition let us assume in addition that each standard module occurs at most once as a subquotient of F, and let V ⊂ I be the parameters of those standard modules which actually occur. By rearranging our filtration if necessary, we can find a filtration

F = F_{−∞} ⊃ … ⊃ F_{0} ⊃ F_{1} = 0

such that F_{p}/F_{p+1} = ⨁_{l(v)=−p} N_{v} where each N_{v} is isomorphic to j_{v∗}\underline{Z}_{v}[l(v)]. Remark that since between different standard objects with same l(v) there are no nonzero homomorphisms, the N_{v} are well defined as subobjects of F_{p}/F_{p+1}. There is however no canonical isomorphism between N_{v} and j_{v∗}\underline{Z}_{v}[l(v)].

The object F determines for every v, w ∈ V with l(v) = l(w) + 1 an extension of perverse sheaves e_{w,v}(F) ∈ Hom_{D}(N_{v}, N_{w}[1]), namely the extension “realized” in the short exact sequence F_{p+1}/F_{p+2} ↪ F_{p}/F_{p+2} → F_{p}/F_{p+1}. Let us get back to our spectral sequence. Remark that

i_{p}^{!}F = i_{p}^{!}(F_{p}/F_{p+1}) = ⨁_{l(v)=−p} i_{p}^{!}N_{v}.

Now ℍ^{n}i_{p}^{!}N_{v} = ℍ^{n}N_{v} is onedimensional if n = l(v), and zero else. Thus ℍ^{p+q}i_{p}^{!}F = 0 for q ≠ 0 whereas ℍ^{p}i_{p}^{!}F = ⨁_{l(v)=−p} ℍ^{p}N_{v}. So our spectral sequence lives just on one line and therefore degenerates at the E_{2}-stage. Furthermore we see that its differential

⨁_{l(v)=−p} ℍ^{p}N_{v} → ⨁_{l(w)=−p−1} ℍ^{p+1}N_{w}

is just given by the matrix e_{w,v}(F) defined above.

7. Restricting the spectral sequence

In addition to the assumptions of the previous section suppose now we are given a locally closed subvariety u : U ↪ Z (think of U = Y_{c} − Y_{c−1}) such that the nonempty intersections U_{v} = Z_{v} ∩ U form a stratification of U by affine spaces U_{v} = 𝔸^{l(v)−c} for some fixed c ≥ 0. We let J = {v ∈ I | Z_{v} ∩ U ≠ ∅} be the subset of I parametrizing this stratification of U, and let ĵ_{v} : U_{v} ↪ U be the inclusion of the strata. Then by base change

u^{!}j_{v∗}\underline{Z}_{v}[l(v)] = \{ \begin{matrix}ĵ_{v∗}\underline{U}_{v}[l(v) − 2c](−c) & if U ∩ Z_{v} ≠ ∅; \\ 0 & else. \end{matrix}

In particular, if F is as before a perverse sheaf on Z which can be written as a sucessive extension of standard objects N_{v} with v ∈ V, then u^{!}F[c] is a perverse sheaf on U and a sucessive extenion of the standard objects u^{!}N_{v} with v ∈ V ∩ J. Furthermore ℍ^{p}N_{v} = ℍ^{p+c}u^{!}N_{v}[c](c) canonically for v ∈ V ∩ J. Indeed any choice of an isomorphism N_{v} ≅ j_{v∗}\underline{Z}_{v} leads to an identification of both sides with ℂ, and the composition of these two identifications is independent of the chosen isomorphism. The problem now is to determine for v, w ∈ V ∩ J with l(v) = −p, l(w) = −p − 1 the factor a_{w,v} ∈ ℂ such that the diagram

ℍ^{p}N_{v}  ⟶  ℍ^{p+1}N_{w}
∥  ∥
ℍ^{p+c}u^{!}N_{v}[c](c) ⟶ ℍ^{p+1+c}u^{!}N_{w}[c](c)

commutes when we take as the upper horizontal e_{w,v}(F) and as the lower one a_{w,v}e_{w,v}(u^{!}F[c]). I want to explain why a_{w,v} is one, if U is in a suitable sense transversal to the stratification on Z. More precisely we need

Proposition 7.1. Let v, w ∈ V ∩ J be given with l(v) = l(w) + 1. Suppose that Z_{v,w} := Z_{v} ∪ Z_{w} and U_{v,w} := U_{v} ∪ U_{w} are open smooth subvarieties of \overline{Z_{v}} and \overline{U_{v}} respectively. Suppose further that Z_{w} and U_{v,w} meet transversally in some point x ∈ U_{w}, i.e. T_{x}U_{w} = T_{x}Z_{w} ∩ T_{x}U_{v,w}. Then a_{w,v} = 1.

Remark 7.2. Since the codimension of T_{x}U_{w} in T_{x}U_{v,w} is one, the transversality condition could as well be written T_{x}U_{v,w} ⊄ T_{x}Z_{w}.

Proof. Take E = {z ∈ ℂ | |z| < 1}. It is possible to find a holomorphic embedding d : E ↪ U_{v,w} such that d(0) = x and that the image of T_{0}E and T_{x}U_{w} together span T_{x}U_{v,w}, in words the image of E in U_{v,w} is transversal to U_{w}. By our assumptions on the geometry the composition E ↪ U_{v,w} ↪ Z_{v,w} has analogously an image in Z_{v,w} which is transversal to Z_{w}. We will also denote by d the composition E ↪ U_{v,w} ↪ U. Put p = −l(v) and q = −p − 1. Then I claim a commutative diagram

ℍ^{p}N_{v}  ⟶^{e_{w,v}(F)}  ℍ^{p+1}N_{w}
∥  ∥
ℍ^{−1}d^{!}u^{!}N_{v}[q](q) ⟶^{e_{w,v}(d^{!}u^{!}F[q])} ℍ^{0}d^{!}u^{!}N_{w}[q](q)
∥  ∥
ℍ^{p+c}u^{!}N_{v}[c](c) ⟶^{e_{w,v}(u^{!}F[c])} ℍ^{p+c+1}u^{!}N_{w}[c](c)

where all four vertical equalities are analogs of our previous canonical isomorphism in an analytic situation, and the composition of two vertical equalities is just exactly our previous canonical isomorphism. To check the commutativity of both little squares then is work in local coordinates, and commutativity of the big square is just our claim. I have good reasons to leave the details to the reader.

8. Proof of the main result

In the first sections we transformed the problem of computing the 𝔫-cohomology of limits of discrete series representations into the geometric problem of computing the hypercohomology groups ℍ^{•}(u_{c}^{!}i^{!}j_{∗}A) for A = A(D) as in Proposition 4.1. Now we attack this problem with the methods developed in the two preceding sections.

Let us first have a look at the spectral sequence computing ℍ^{•}(j_{∗}A). Remark that j_{∗}A is an object as the F considered in Section 6, with corresponding N_{x} for x ∈ D. I claim that for x, y ∈ D with x < y, l(y) = l(x) + 1 the induced map e_{x,y} = e_{x,y}(j_{∗}A) from ℍ^{−l(y)}N_{y} to ℍ^{−l(x)}N_{x} doesn’t vanish. Indeed, consider the locally closed subset V = ByB/B ∪ BxB/B of the flag manifold and let a : V → G/B be the inclusion. Since Schubert varieties are normal (see e.g. [Jan83] II, 14.15), our V is smooth. Then a_{!}\underline{V} [l(y)] is just the cokernel of the inclusion of the !-standard objects corresponding to the Bruhat cells of x and y, thus

\begin{aligned}0 & = & Hom_{D}(a_{!}\underline{V} [l(y)], j_{∗}A) \\ & = & ℍ^{−l(y)}a^{!}j_{∗}A. \end{aligned}

On the other hand, if we try to compute the hypercohomology of a^{!}j_{∗}A by restricting the spectral sequence as in section 7, we find that it is the cohomology of the complex

… 0 ⟶ ℍ^{−l(y)}N_{y} ⟶^{e_{x,y}} ℍ^{−l(x)}N_{x} ⟶ 0 …

and hence e_{x,y} ≠ 0.

It is furthermore clear that for every small square (x; y, w; z) from D we have e_{x,y} ∘ e_{y,z} = −e_{x,w} ∘ e_{w,z} as maps from ℍ^{−l(z)}N_{z} to ℍ^{−l(x)}N_{x}, since the square of the differential of our spectral sequence computing the hypercohomology of j_{∗}A has to vanish. These two properties already determine our spectral sequence up to isomorphism. In fact, we have more generally

Lemma 8.1. Let (W, S) be a finite crystallographic Coxeter group, l : W → ℕ its length function. Let W(i) = {x ∈ W | l(x) = i} and let ℂW(i) be the free vector space with basis W(i) over ℂ. Suppose we are given for any x < y with l(y) = l(x) + 1 a constant e_{x,y} ∈ ℂ^{×} such that e_{x,y}e_{y,w} = −e_{x,z}e_{z,w} for every small square (x; y, z; w) from W.

Then the matrix of all e_{y,x} determines a differential on the graded space ⨁ ℂW(i) making it into a complex, and any other choice of constants e′_{y,x} leads to an isomorphic complex.

Proof. I am quite unhappy to know only a silly overkill proof of this purely combinatorial result. But anyhow, one can argue as follows:

Any choice of e_{x,y} determines a differential for a BGG-resolution of the trivial one-dimensional 𝔤-module

… → C^{1} → C^{0} → ℂ

with C^{i} = ⨁_{x∈W(i)} M(x · 0). But any two such resolutions are isomorphic to a canonical one, where the C^{i} are rather given inductively as sums of Verma modules with highest weight spaces certain onedimensional weight spaces of ker(C^{i−1} → C^{i−1}).

Now an isomorphism between the complexes coming from the e_{x,y} and certain other e′_{x,y} is necessarily given by scalars a_{x} ∈ ℂ^{x} on M(x·0). These scalars then give the isomorphisms of complexes claimed by the Lemma.

Now we restrict the spectral sequence calculating ℍ^{•}(j_{∗}A) via u_{c}^{!}i^{!}. We want to apply Proposition 7.1 and have to show some transversality of K-orbits with respect to the stratification by Bruhat cells. Let for v ∈ W denote X_{v} = BvB/B the corresponding Bruhat cell. If w ∈ W is given such that v > w and l(v) = l(w) + 1, then X_{v,w} := X_{v} ∪ X_{w} is open smooth in \overline{X_{v}} , as follows from the normality of Schubert varieties. If U = Y_{c} − Y_{c−1} meets both X_{v} and X_{w}, the intersections U_{v} = U ∩ X_{v} and U_{w} = U ∩ X_{w} are again neighbouring Bruhat cells in Y (which is isomorphic to the flag variety of K), whence U_{v,w} = U_{v} ∪ U_{w} is also open smooth in \overline{U_{v}}. So to apply Proposition 7.1 to our situation we just need

Lemma 8.2. T_{x}U_{v,w} is not contained in T_{x}X_{w}, for some x ∈ U_{w}.

Proof. Certainly T_{x}U_{v,w} lies in T_{x}Y, so it will be sufficient to show T_{x}X_{w} ∩ T_{x}Y = T_{x}U_{w}. Here X_{w}, Y and U_{w} are respectively the orbit of x under B, K and B ∩ K. We may choose x as the fixed point of our maximal torus T which lies in B and K. Then the isotropy group G_{x} of x in G is also a Borel subgroup above T. We can identify T_{x}(G/B) = LieG/LieG_{x} and our tangent spaces above become just the images of LieB, LieK and Lie(B ∩ K) = LieB ∩ LieK under the projection LieG ↠ LieG/LieG_{x}. We can thus check the equality T_{x}X_{w} ∩ T_{x}Y = T_{x}U_{w} root space by root space, and the Lemma is proven.

We now prove the main Theorem 2.1.

Proof. We have

ℍ^{n}(𝔫, L(Y, χ))_{λ} = ℍ^{n−dim Y} (Y, i^{!}j_{∗}A) by 4.4,
= ⨁_{c} ℍ^{n−dim Y −c}(u_{c}^{!}i^{!}j_{∗}A[c]) by 5.1,

and the spectral sequence computing ℍ^{n}(u_{c}^{!}i^{!}j_{∗}A[c]) is just the complex C^{c−n}V(A, D, c) by the last three sections.

References

[BB81] Alexander A. Beilinson and Joseph N. Bernstein, Localisation de 𝔤modules, C. R. Acad. Sci. Paris, Sér. 1 292 (1981), 15–18.

[BBD82] Alexander A. Beilinson, Joseph N. Bernstein, and Pierre Deligne, Faisceaux pervers, Astérisque 100 (1982), 1–172.

[BGG75] Joseph N. Bernstein, Israel M. Gelfand, and Sergei I. Gelfand, Differential operators on the base affine space and a study of 𝔤-modules, Lie groups and their Representations (I. M. Gelfand, ed.), Halsted: New York, 1975, pp. 21–64.

[BGS96] Alexander A. Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, JAMS 9 (1996), no. 2, 473–527.

[Car96] Henri Carayol, Limites dégénérées de séries discrètes, formes automorphes et variétés de Griffiths-Schmid: Le cas du groupe U(2, 1), Prépublication IRMA 1996/10 (Strasbourg), 1996.

[Jan83] Jens C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Springer, 1983.

[Mir90] Ivan Mirković, On the localization for a singular infinitesimal character, Preprint, University of Massachusetts at Amherst, 1990.

[MS95] Dragan Miličić and Wolfgang Soergel, The composition series of modules induced from Whittaker modules, Preprint, 1995.

[Sch76] Wilfried Schmid, L^{2}-cohomology and the discrete series, Annals of Mathematics 103 (1976), 375–394.

[Soe89] Wolfgang Soergel, 𝔫-cohomology of simple highest weight modules on walls and purity, Inventiones 98 (1989), 565–580.

[Wil88] Floyd L. Williams, The 𝔫-cohomology of limits of discrete series, Journal of functional analysis 80 (1988), 451–461.

Universität Freiburg, Mathematisches Institut, Eckerstraße 1, D-79104 Freiburg, Germany

E-mail address: soergel@mathematik.uni-freiburg.de