COMBINATORICS OF HARISH-CHANDRA MODULES

WOLFGANG SOERGEL

Abstract. These lectures survey recent work on the combinatorics of certain infinite dimensional representations of complex semisimple Lie algebras. Their focus is not on understanding the irreducible objects but rather on understanding the structure of suitable representation categories. They concentrate on the relation of these representation categories with categories of modules over the coinvariant algebra associated to the action of the Weyl group on a Cartan subalgebra. We also discuss conjectural generalizations to the representation theory of real Lie groups.

1. Introductory dream

Let k be a field. Let G × X → X be the action of a complex algebraic group G on a complex algebraic variety X. We suppose that there are only finitely many orbits and associate to this situation a ℤ-graded k-algebra Ext_{G}^{•}X, the “equivariant extension algebra” in the following way: Let D_{G}(X) be the bounded derived equivariant category, with algebraically constructible cohomology and coefficients in k, see [BL94]. Let P_{G}(X) ⊂ D_{G}(X) be the abelian subcategory of equivariant perverse sheaves. Let L ∈ P_{G}(X) be the sum of all simple objects of P_{G}(X) (take one from each isomorphism class). We put

Ext_{G}^{•}X = ⨁_{i} Hom_{D_{G}(X)}(L, L[i]).

Next let us look at respresentation theory. If G is a complex algebraic group and γ : G → G an antiholomorphic involution, we let G(ℝ, γ) denote the fixed points of γ, a real Lie group. Once we fix a real form γ, all real forms of G (up to isomorphism) are parametrized by H^{1}(Γ; AutG), with Γ = {1, γ} = Gal(ℂ/ℝ). Now conjugation gives a Γ-equivariant group homomorphism G → AutG. So every δ ∈ H^{1}(Γ; G) will give rise to a real form G(ℝ, δ) of G.

Let now G be a connected reductive complex algebraic group and γ : G → G an antiholomorphic involution stabilizing a Borel subgroup of G. Let χ ∈ MaxZ be a character of the center Z = Z(𝔤) ⊂ U(𝔤) of the enveloping algebra U(𝔤) of the complex Lie algebra 𝔤 = LieG. To these data (G, γ, χ) Adams, Barbasch and Vogan [ABV92] associate a complex algebraic variety X(χ), the “modified Langlands parameter space” on which the dual group G^{∨}, i.e. the connected reductive complex algebraic group with the dual root datum, acts with finitely many orbits. The big picture motivating this series of talks is the following

Conjecture 1.1. [Soe92b] There is an equivalence of categories

⨁_{δ∈H^{1}(Γ;G)} M(G(ℝ, δ))_{χ} ≅ Ext_{G^{∨}}^{•}X(χ) − Nil.

Here M(G(ℝ, δ))_{χ} denotes the category of Harish-Chandra-modules for G(ℝ, δ) killed by some power of χ, and for a graded ring A = ⊕_{i≥0}A^{i} the category A − Nil consists of all finite dimensional A-modules killed by A^{i} for i ≫ 0.

Remark 1.2. I expect the analogous conjecture to hold for p-adic groups as well. Results of Lusztig and Ginzburg point in that direction, see Ginzburg’s talks [Gin98] at this school.

Evidence and Motivation 1.3. 1. The conjecture works for tori. 2. It works for SL(2, ℝ).

3. It works for complex groups, regarded as real Lie groups.

4. The simple objects on both sides are in “natural” bijection.

5. It works for generic χ.

6. I don’t know a counterexample.

7. It fits together with results of [ABV92] in a bigger conjectural framework.

In these talks I will concentrate on 3 and explain how one can get his hands on both sides sufficiently well to see they are equal. But first let’s discuss some of the other points.

About 1. Choose G = ℂ^{×} and let γ be the complex conjugation. Then H^{1}(Γ; ℂ^{×}) consits just of one element δ and G(ℝ, δ) = ℝ^{×}. The space X(χ) consists just of two points, for every χ, and this is acted upon by G^{∨} = ℂ^{×}. We find

Ext_{G^{∨}}^{•}X(χ) = H_{ℂ^{×}}^{•}(two points) = ℂ[t] × ℂ[t]

where t is a basis of LieG and is given degree deg t = 2. It is clear that

M(ℝ^{×})_{χ} ≅ (ℂ[t] × ℂ[t]) − Nil

for every χ. The case of arbitray tori is a straightforward generalization of this example.

About 4. The simple objects of P_{G}(X) are parametrized by the set Par = Par(G, X) = {(Y, τ) | Y is a G-orbit in X and τ a Gequivariant local system on Y, alias an irreducible representation of the component group of the isotropy group G_{y} of some fixed y ∈ Y }.

For π = (Y, τ) ∈ Par the corresponding simple object will be denoted L^{π} = IC(Y, τ), so L = ⨁_{π∈Par} L^{π}. Certainly the degree zero part of A = Ext_{G}^{•}X is just A^{0} = ∏_{π∈Par} k1_{π} with 1_{π} ∈ EndL the projection onto L^{π}. So the irreducibles of A − Nil are parametrized by Par.

Let IrrA denote the set (or class) of isomorphism classes of irreducibles in any abelian category A. Adams, Barbasch and Vogan [ABV] construct a bijection

\begin{matrix}Par(G^{∨}, X(χ)) & ⟶^{∼} & ⨿_{δ∈H^{1}(Γ;G)}IrrM(G(ℝ, δ))_{χ} \\ π & ↦ & L_{π} \end{matrix}

depending on additional choices, which has the following wonderful property: If we write L_{π} as the unique simple quotient of an induced from tempered M_{π}, we get a (Par × Par)-matrix of Jordan-Hölder multiplicities JH_{π,ψ} = [M_{π} : L_{ψ}], which is triangular with ones on the diagonal. Dually, we have also two bases of the Grothendieck group [D_{G^{∨}}X(χ)], given by [L^{π}] and [M^{π}] with M^{π} = i_{!}(τ[dim_{ℂ} Y]) for i : Y ↪ X(χ) the injection. From the geometry we then get another (Par×Par)-matrix IC_{π,ψ} = [M^{π} : L^{ψ}] given by [M^{π}] = ∑_{ψ} IC_{π,ψ}[L^{ψ}]. The bijection of [ABV] is such that up to signs these matrices are inverse transpose of one another, more precisely

∑_{π}(−1)^{d(π)+d(ψ)}IC_{π,ψ}JH_{π,ξ} = δ_{ψ,ξ}

where d(π) = dim_{ℂ} Y for π = (Y, τ).

Let me finish this first lecture with some comments on the equivariant Ext-algebras. I expect these algebras to be more natural then they look like, there should be a fully faithful embedding D_{G}(X) ↪ K^{+}(Ext_{G}^{•}X) of the constructible equivariant derived category D_{G}(X) into the bounded below homotopy category of differential graded modules over Ext_{G}^{•}X, where Ext_{G}^{•}X is considered as a differential graded algebra with zero differential.

2. Endomorphisms of antidominant projectives

Let 𝔤 ⊃ 𝔟 ⊃ 𝔥 be a semisimple complex Lie algebra, a Borel and a Cartan. Put O^{∞} = {M ∈ 𝔤-mod | M is finitely generated over 𝔤 and locally finite for 𝔟}. By work of [BG80], certain direct summands of M(G(ℂ)) are equivalent to O^{∞}, where G is a complex reductive group with LieG = 𝔤 and we let M(G(ℂ)) be the Harish-Chandra-modules for G(ℂ) considered as a real Lie group. So we want to understand O^{∞}.

Restrict to the easier task to understand

O = {M ∈ O^{∞} | 𝔥 acts diagonally on M} .

This category is studied extensively in the literature, we will build on [BG80, Jan83] and won’t give proofs for results which can be found in these two sources. For λ ∈ 𝔥^{∗}, let M(λ) = U ⊗_{U(𝔟)} ℂ_{λ} be the Verma module and L(λ) its unique simple quotient. Then every irreducible object of O is isomorphic to exactly one of the L(λ), every object of O has finite length, and we have O = ⨁_{χ∈MaxZ} O_{χ} where O_{χ} ⊂ O denotes the subcategory consisting of all objects killed by some power of χ. Let pr_{χ} : O ↠ O_{χ} be the corresponding projection functors. Certainly we have a map ξ : 𝔥^{∗} → MaxZ such that L(λ) ∈ O_{ξ(λ)}. This gives in fact an isomorphism 𝔥^{∗}/(W ·) →^{∼} MaxZ with corresponding comorphism Z →^{∼} S^{(W ·)}, where S = S(𝔥) is the symmetric algebra over 𝔥 alias the regular functions on 𝔥^{∗} and where the Weyl group W acts on 𝔥^{∗} by the dot action x · λ = x(λ + ρ) − ρ, for ρ the half-sum of the roots R^{+} from 𝔟.

Theorem 2.1. 1. There are enough projective objects in O.

2. For every λ ∈ 𝔥^{∗} there exists a unique up to isomorphism indecomposable projective P (λ) ∈ O surjecting onto L(λ).

3. P (λ) admits a Verma flag, and for the multiplicities (P (λ) : M(μ)) of Vermas in such a flag we have the reciprocity formula (P (λ) : M(μ)) = [M(μ) : L(λ)].

4. M(λ) = P (λ) iff ⟨λ + ρ, α^{∨}⟩ ∉ {−1, −2, … } ∀α ∈ R^{+}.

5. M(λ) = L(λ) iff ⟨λ + ρ, α^{∨}⟩ ∉ {1, 2, … } ∀α ∈ R^{+}.

6. If P ∈ O is projective, then so is E⊗P for every finite dimensional representation E of 𝔤.

7. E ⊗ M(λ) admits a Verma flag with M(ν + λ) appearing dim_{ℂ} E_{ν} times, for every finite dimensional representation E of 𝔤.

Proof. [Jan83]

If A is an artinian category and P ∈ A is a projective generator, i.e. a projective object surjecting onto every irreducible object, then we have an equivalence of categories Hom_{A}(P, ) : A →^{∼} mof-End_{A}(P), denoting by mof- the category of all right modules of finite length. So to understand O_{triv} for triv = ξ(0) ∈ MaxZ the trivial central character we need to understand End_{𝔤}P with P = ⊕_{x∈W} P (x · 0). The first step here is to understand End_{𝔤}P (−2ρ).

Let S_{+} ⊂ S be the elements of positive degree and let (S_{+}^{W}) ⊂ S denote the ideal of S generated by the W -invariants in S_{+}. Be careful that here we consider invariants for the usual “linear” action, not for the dot-action!

Theorem 2.2. [Soe90] 1. The multiplication induces a surjection Z ↠ End_{𝔤}P (−2ρ).

2. The composition Z →^{∼} S^{(W ·)} ↪ S ↠ S/(S_{+}^{W}) also is a surjection.

3. Both these surjections have the same kernel, so we have a unique isomorphism End_{𝔤}P (−2ρ) ≅ S/(S_{+}^{W}) making commute the diagram

Z ↠ End_{𝔤}P (−2ρ)
↓  ∥
S ↠ S/(S_{+}^{W}).

The proof I give due to Bernstein [Ber90]. It will take up the rest of this section. Let us abbreviate the coinvariant algebra by S/(S_{+}^{W}) = C. It is known [Bou81] that dim_{ℂ} C = |W |. Let us check first that the algebras we want to be isomorphic at least have the same dimension, i.e. that also dim_{ℂ} End_{𝔤}P (−2ρ) = |W |. But indeed [M(x·0) : L(−2ρ)] = 1 for all x ∈ W by [Jan83], and then (P (−2ρ) : M(x · 0)) = 1 by reciprocity. Thus

dim_{ℂ} End_{𝔤}P (−2ρ) = [P (−2ρ) : L(−2ρ)]
= ∑_{x}(P (−2ρ) : M(x · 0))[M(x · 0) : L(−2ρ)]
= |W |.

Next we explain why our map Z → C is a surjection. This is a special case of general results in invariant theory. Namely let a finite group G act on a finite dimensional complex vector space V. Let R be the regular functions on V and for v ∈ V let 𝔪_{v} ⊂ R be the ideal of functions vanishing at v. If v ∈ V has trivial isotropy group, then R^{G} → R/I is a surjection for every 𝔪_{v}-primary ideal I, i.e. for any ideal I such that I ⊃ 𝔪_{v}^{n} for n ≫ 0. Indeed, for n fixed and x ∈ R given we find f ∈ R such that f ∈ x + 𝔪_{v}^{n} and f ∈ 1 + 𝔪_{gv}^{n} if g ∈ G, g ≠ e. Then necessarily the product of all G-conjugates of f maps to x̄ ∈ R/I.

From there we get easily that Z → C is a surjection, and by comparison of codimension we only need to show that Ann_{Z}P (−2ρ) is contained in the kernel of Z ↠ C. Put in other words, we should show that if z ∈ Z doesn’t go to zero under Z ↠ C, then it doesn’t kill P (−2ρ) either. Now it is known [Bor53] that C is the cohomology ring of a smooth connected compact complex manifold of dimension l(w_{0}), the “flag variety” consisting of all Borel subalgebras in 𝔤, and thus by Poincaré duality we know that if we take a nonzero homogeneous element ω ∈ C of maximal possible degree l(w_{0}), then for every nonzero c_{1} ∈ C there exists c_{2} ∈ C such that c_{1}c_{2} = ω. Hence we are reduced to proving

Claim 2.3. All z ∈ Z which map to ω under Z ↠ C do not kill P (−2ρ).

To prove this we need another description of P (−2ρ).

Lemma 2.4. P (−2ρ) = pr_{triv}(L(ρ) ⊗ M(−ρ)).

Proof. It is not hard to check that both sides have the same Verma flag multiplicities and are both projective in O.

Let us now take n so big that χ^{n}pr_{χ}(L(ρ) ⊗ M(−ρ)) = 0 for all χ ∈ MaxZ. (It is not hard to see that n = |W | will do.) To establish our Claim 2.3, it will then be sufficient to show that for some n which is sufficiently big in this sense we have

Lemma 2.5. If z ∈ Z maps to ω ∈ C and is contained in χ^{n} for all χ ∈ MaxZ with χ ≠ triv but pr_{χ}(L(ρ) ⊗ M(−ρ)) ≠ 0, then z ∉ Ann(L(ρ) ⊗ M(−ρ)).

So the point is to show that certain elements of Z do not kill L(ρ) ⊗ M(−ρ). For this we may use a beautiful construction of Bernstein. Remark that whenever V, M are vector spaces and dim V < ∞, we obtain a canonical linear map

tr_{V} : End(V ⊗ M) → EndM

as the composition

End(V ⊗ M) → End(V^{∗} ⊗ V ⊗ M) → EndM

where the first map is tensoring with the identity on V^{∗}, and the second is induced from the canonical maps ℂ → V^{∗} ⊗ V → ℂ. Another way to define tr_{V} would be to use a basis of V to identify End(V ⊗M) with the matrix algebra of (dim V × dim V) square matrices with entries in EndM, and then take the usual trace of a matrix. This is why tr_{V} might be called the “relative trace”. It is clear that for dim M < ∞ the following diagram commutes:

End(V ⊗ M) →^{tr_{V}} EndM
∥  ↓ tr
End(V ⊗ M) →^{tr} ℂ.

If V, M are representations of a Lie algebra 𝔤, then certainly tr_{V} restricts to a linear map

tr_{V} : End_{𝔤}(V ⊗ M) → End_{𝔤}M

which is so natural that it determines in fact a map

tr_{V} : End_{F} (V ⊗) → End_{F} (id)

where V ⊗, id : 𝔤-mod → 𝔤-mod are considered as functors and End_{F} denotes the natural transformations of a functor to itself.

Now for any unitary ring A the endomorphism ring of the identity functor id on A-mod is canonically isomorphic to the center Z(A) of A, in formulas

Z(A) →^{∼} End_{F} (id).

Thus in our representation theoretic context we get for any finite dimensional representation V of 𝔤 a linear map Tr_{V} : Z → Z such that the following diagram commutes:

Z  →^{Tr_{V}}  Z
↓  ∥
End_{F} (V ⊗) →^{tr_{V}} End_{F} (id)

To prove our Lemma 2.5, it will be sufficient to show for z as in the Lemma that Tr_{L(ρ)}(z) doesn’t kill M(−ρ). We will deduce this from an explicit formula for the map Tr_{V}, which we explain next.

For M ⊂ 𝔥^{∗} a finite multiset and f ∈ S = R(𝔥^{∗}) define (M ∗ f) ∈ R(𝔥^{∗}) by (M ∗ f)(λ) = ∑_{μ∈M} f(λ + μ). Define Λ ∈ S by Λ(λ) = ∏_{α∈R^{+}}⟨λ+ρ, α^{∨}⟩. Then Λ is (W ·)-antiinvariant, and the set of all (W ·)- antiinvariant elements of S is precisely ΛS^{(W ·)}. Consider Z = S^{(W ·)} as a subring of S, so that z acts on L(λ) by the scalar z(λ). Let P_{V} denote the multiset of weights of V counted with their multiplicities.

Theorem 2.6. [Ber90] Tr_{V} (z) = Λ^{−1}(P_{V} ∗ Λz) for all z ∈ Z.

Remark 2.7. Here z is viewed as a (W ·)-invariant element of S, then Λz is (W ·)-antiinvariant, so is P_{V} ∗ Λz, and thus we may divide by Λ to get again an element of S^{(W ·)} = Z.

Proof. Both sides are regular functions on 𝔥^{∗}, thus we only need to show that they take the same values on all integral weights λ ∈ P which lie in the dominant chamber far from the walls. But for those we know V ⊗ L(λ) ≅ ⨁_{μ∈P_{V}} L(λ + μ), and using the commutative diagram

Z ⟶^{Tr_{V}} Z
↓  ↓
End(V ⊗ L(λ)) ⟶^{tr_{V}} EndL(λ)
∥  ↓ tr
End(V ⊗ L(λ)) ⟶^{tr}  ℂ

we find (dim L(λ))(Tr_{V} z)(λ) = ∑_{μ∈P_{V}} (dim L(λ + μ))z(λ + μ). Now we can use the Weyl character formula dim L(λ) = Λ(λ)/Λ(0) to conclude.

Finally, we get what we really wanted.

Proof. [of Lemma 2.5 and Theorem 2.2] It will be sufficient to show that for z as in the Lemma we have (Tr_{L(ρ)}z)(−ρ) ≠ 0. To see this we have to calculate a little while. Let

\begin{aligned}Sgn & : & S → S \\ & & f ↦ |W |^{−1} ∑_{w∈W} (−1)^{l(w)}f^{(w·)} \end{aligned}

be the antisymmetrisation with respect to the dot-action of the Weyl group. For μ ∈ 𝔥^{∗} define

\begin{aligned}Tr_{μ} : & S & → & S \\ & f & ↦ & Λ^{−1}Sgn(μ ∗ Λf). \end{aligned}

It is not hard to check that Tr_{V} = ∑_{μ∈P_{V}} Tr_{μ} on Z and Tr_{μ}(z) = Tr_{wμ}(z) for all z ∈ Z. Now if f vanishes of sufficiently high degree at (μ−ρ), then clearly (Tr_{μ}f)(−ρ) = 0. Furthermore (Tr_{ρ}f)(−ρ) depends only on the image of f in C. Indeed, for f = gh with g ∈ S^{W} we have (Tr_{ρ}f)(λ) = g(λ + ρ) · (Tr_{ρ}h)(λ) and if we put here λ = −ρ we see that (Tr_{ρ}f)(−ρ) = 0 for f ∈ (S_{+}^{W}). Thus we can just take any preimage f ∈ S of ω ∈ C, why not f = (−ρ) ∗ Λ, and need only for this f to check (Tr_{ρ}f)(−ρ) ≠ 0. And indeed

(Tr_{ρ}f) = Λ^{−1}Sgn((ρ ∗ Λ)Λ)
= |W |^{−1} ∑_{x∈W} (ρ ∗ Λ)^{(x·)}

and thus

\begin{matrix}(Tr_{ρ}f)(−ρ) = & |W |^{−1} ∑_{x∈W} (ρ ∗ Λ)(−ρ) \\ = & Λ(0) \\ ≠ & 0. \end{matrix}

3. Reduction to coinvariants

Last time we proved C = End_{𝔤}P (−2ρ). Let us consider the functor

𝕍 : O_{triv} → C-mod
M ↦ Hom_{𝔤}(P (−2ρ), M).

Theorem 3.1. The functor 𝕍 is fully faithful on projective objects, it gives even for all M, Q ∈ O_{triv} with Q projective an isomorphism

Hom_{𝔤}(M, Q) →^{∼} Hom_{C}(𝕍M, 𝕍Q).

Proof. Step 1. Q = P (−2ρ). In this case both P (−2ρ) = pr_{triv}(L(ρ) ⊗ M(−ρ)) and C (by Poincaré duality) are injective in O_{triv} and C-mod respectively. Thus both sides of our presumed isomorphism are exact in M ∈ O_{triv}, and we only have to check equality for M simple. This is easy.

Step 2. Q = M(0). Let C_{+} ⊂ C the ideal of all elements of positive degree.

Lemma 3.2. M(0) = {v ∈ P (−2ρ) | C_{+}v = 0}

Proof. Let us call the right hand side N. It is not difficult to get a short exact sequence M(0) ↪ P (−2ρ) ↠ K where K has a Verma flag. Thus the socle of K is a sum of copies of L(−2ρ). Since anyhow M(0)⊂N, we see that M(0) ≠ N would imply [N : L(−2ρ)] ≥ 2. So I = Hom_{𝔤}(P (−2ρ), N) ⊂ C = End_{𝔤}P (−2ρ) would be an ideal with C_{+}I = 0 and dim_{ℂ} I ≥ 2. But the subspace of elements C killed by C^{+} is just the line through ω, and this contradiction shows that necessarily M(0) = N.

To complete step 2 we consider the diagram

Hom_{𝔤}(M, M(0)) = {φ ∈ Hom_{𝔤}(M, P (−2ρ)) | C_{+}φ = 0}
↓  ↓
Hom_{C}(𝕍M, ℂ) = {ψ ∈ Hom_{C}(𝕍M, C) | C_{+}ψ = 0},

and since here by step 1 the second vertical is an isomorphism, the first one is an isomorphism too.

Step 3. Q arbitrary. Without restriction we may assume Q = pr_{triv}(E ⊗ M(0)) for some finite dimensional E. Consider the functor F = F_{E} : O_{triv} → O_{triv}, M ↦ pr_{triv}(E ⊗ M). Its right and left adjoint is F_{E^{∗}} = G. We need some general category theory.

Call an exact functor between abelian categories

𝕍 : A ⟶ B

a “quotient functor” iff it has the obvious universal property. For A artinian and P ∈ A projective, always Hom(P, ) : A → mof-End_{A}P is a quotient functor. In our case of 𝕍 : O_{triv} → C-mof, the kernel

ker 𝕍 = {M | [M : L(−2ρ)] = 0}

is stable under both F and G. Thus we have a commutative diagram

O_{triv}  ⟶^{𝕍} C-mof
↓ F, G  ↓ F̄, Ḡ
O_{triv}  ⟶^{𝕍} C-mof

for suitable F̄, Ḡ. One can check without much difficulty that F̄ and Ḡ are adjoints of each other, too. And then we just calculate

Hom_{𝔤}(M, Q) = Hom_{𝔤}(M, F M(0)) since Q = F M(0)
= Hom_{𝔤}(GM, M(0)) by adjointness
= Hom_{C}(𝕍GM, 𝕍M(0)) by step 2
= Hom_{C}(Ḡ𝕍M, 𝕍M(0)) since 𝕍G = Ḡ𝕍
= Hom_{C}(𝕍M, F̄𝕍M(0)) by adjointness
= Hom_{C}(𝕍M, 𝕍F M(0)) since 𝕍F = F̄𝕍
= Hom_{C}(𝕍M, 𝕍Q) since Q = F M(0).

We deduce

O_{triv} ≅ mof-End_{𝔤}(⊕_{x}P (x · 0))
≅ mof-End_{C}(⊕_{x}𝕍P (x · 0))

and 𝕍P (x·) ∈ C-mod is indecomposable, as there are no idempotents in its endomorphism ring except 0 and 1.

So what are these strange indecomposable C-modules? Our next goal is a geometric description of the form 𝕍P (x · 0) ≅ ℍ^{•}IC\overline{(Bx^{−1}B/B}. But first we need some preliminaries. Put P^{−} = {λ ∈ 𝔥^{∗} | ⟨λ + ρ, α^{∨}⟩ ∈ {0, −1, −2, … } ∀α ∈ R^{+}}. For λ ∈ P^{−} put O_{λ} = O_{ξ(λ)}. For λ, μ ∈ P^{−} define the translation functor T_{λ}^{μ} : O_{λ} → O_{μ} by T_{λ}^{μ} = pr_{ξ(μ)}(L(ν) ⊗ M) where ν is the unique integral dominant weight in W (μ − λ), hence dim L(ν) < ∞. Put W_{λ} = {w ∈ W | w · λ = λ}.

Theorem 3.3. For all λ ∈ P^{−} we have T_{λ}^{−2ρ}P (λ) ≅ P (−2ρ), and the induced map End_{𝔤}P (λ) → End_{𝔤}P (−2ρ) is an inclusion with image C^{W_{λ}} =: C_{λ}.

Proof. Omitted, see [Soe90].

Let us define 𝕍 = 𝕍_{λ} : O_{λ} → C_{λ}-mof as 𝕍_{λ} = Hom_{𝔤}(P_{λ}, ).

Corollary 3.4. We have a commutative square

O_{triv}  ⟶^{𝕍} C-mof
T_{−2ρ}^{λ} ↓  ↓ res
O_{λ}  ⟶^{𝕍} C_{λ}-mof.

Proof. Hom_{𝔤}(P_{λ}, T_{−2ρ}^{λ}M) = Hom_{𝔤}(T_{λ}^{−2ρ}P_{λ}, M).

Corollary 3.5. We have a commutative square

O_{triv}  ⟶^{𝕍} C-mof
T_{λ}^{−2ρ} ↑  ↑ C⊗_{C_{λ}}
O_{λ}  ⟶^{𝕍} C_{λ}-mof.

Proof. Take the adjoint on both verticals of the preceding diagram.

If W_{λ} = {e, s} with s a simple reflection, then Θ_{s} = T_{λ}^{−2ρ}T_{−2ρ}^{λ} : O_{triv} → O_{triv} is called a “wall-crossing functor”, and we get finally

Proposition 3.6. We have a commutative square

O_{triv} ⟶^{𝕍} C-mof
Θ_{s} ↓  ↓ C⊗_{C^{s}}
O_{triv} ⟶^{𝕍} C-mof.

Proof. Clear.

Now remark that we may describe P (x · 0) inductively as follows: Write x = s_{1} ⋯ s_{l} as a product of minimal length of simple reflections. Then P (x · 0) is the unique indecomposable direct summand of Θ_{s_{l}} … Θ_{s_{1}}M(0) not isomorphic to some P (y · 0) with y < x. Thus 𝕍P (x · 0) is the unique direct summand of the C-module C ⊗_{C^{s_{l}}} C ⊗ … ⊗_{C^{s_{1}}} ℂ which is not isomorphic to 𝕍P (y · 0) for some y < x.

4. Relations with geometry

Let G be a semisimple complex algebraic group with Lie algebra 𝔤, and let B ⊂ G be its Borel subgroup with Lie algebra 𝔟. We will explain first why 𝕍P (x · 0) is as a C-module just the intersection cohomology group of a suitable Schubert variety, 𝕍P (x · 0) ≅ IC\overline{(Bx^{−1}B/B)}. Remark first that for any bounded below complex of sheaves F^{•} on a topological space X the total hypercohomology ℍ^{•}F = Hom^{•}(ℂ_{X}, F) is in a natural way a module over the cohomology ring H^{•}X = End^{•}ℂ_{X}. Now let s be a simple reflection, P_{s} ⊂ G the corresponding minimal parabolic subgroup containing B, and π = π_{s} : G/B → G/P_{s} the projection onto the corresponding partial flag variety.

Lemma 4.1. Let F^{•} be a bounded below complex of sheaves on G/B. Then ℍ^{•}(π^{∗}π_{∗}F^{•}) ≅ C ⊗_{C^{s}} ℍ^{•}F are isomorphic as C-modules.

Proof. [Soe90].

To get our argument running, we still need another fact. Let IC_{x} denote the intersection cohomology complex of \overline{(Bx^{−1}B/B)}.

Theorem 4.2. For x, y ∈ W the hypercohomology functor induces an isomorphism

Hom^{•}(IC_{x}, IC_{y}) →^{∼} Hom_{C}^{•}(ℍ^{•}IC_{x}, ℍ^{•}IC_{y}).

Proof. [Gin91].

Let now x ∈ W be given and let x = s_{1} … s_{l} be a reduced decomposition. Then π_{s_{l}}^{∗}π_{s_{r}∗} … π_{s_{1}}^{∗}π_{s_{1}∗}IC_{e} decomposes into a direct sum of objects of the form IC_{y}[i] for suitable y and i, here the biggest y occuring is x itself, and it occurs just once. So ℍ^{•}IC_{x} can be described inductively as the one direct summand of the C-module

ℍ^{•}π_{s_{l}}^{∗}π_{s_{l}∗} … π_{s_{1}}^{∗}π_{s_{1}∗}IC_{e} ≅ C ⊗_{C^{s_{l}}} … C ⊗_{C^{s_{1}}} ℂ

that is not isomorphic to ℍ^{•}IC_{y} for y < x. Comparing this description with the description of 𝕍P (x · 0) given at the end of the preceding section shows that indeed ℍ^{•}IC_{x} ≅ 𝕍P (x · 0).

Let us put L = ⨁_{x∈W} IC_{x}. The following theorem was a conjecture of Beilinson and Ginzburg [BG86]. Its ramifications are explored in [BGS96].

Theorem 4.3. There is an equivalence of categories

O_{triv} ≅ mof-End^{•}(L, L).

Proof. Let P = ⨁_{x∈W} P (x · 0) be a minimal projective generator of O_{triv}. We get

O_{triv} ≅ mof-End_{𝔤}P by category theory
≅ mof-End_{C}𝕍P by Theorem 3.1
≅ mof-End_{C}ℍ^{•}L since 𝕍P ≅ ℍ^{•}L
≅ mof-End^{•}(L, L) by Theorem 4.2.

This theorem is almost the statement of Conjecture 1.1 in the case of complex groups. A proof of the conjecture in the case of trivial central character is given in [Soe92b] based on combinatorial work from [Soe92a].

References

[ABV92] Jeffrey Adams, Dan Barbasch, and David A. Vogan, Jr. The Langlands classification and irreducible characters for real reductive groups. Progress in Mathematics. Birkhäuser, 1992.

[Ber90] Joseph N. Bernstein. Trace in categories. In Alain Connes, Michel Duflo, Anthony Joseph, and Rudolph Rentschler, editors, Operator Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory, pages 417–423. Birkhäuser, 1990.

[BG80] Joseph N. Bernstein and Sergei I. Gelfand. Tensor products of finite and infinite representations of semisimple Lie algebras. Compositio Math., 41:245–285, 1980.

[BG86] Alexander A. Beilinson and Victor Ginsburg. Mixed categories, Extduality and representations (results and conjectures). Preprint, 1986.

[BGS96] Alexander A. Beilinson, Victor Ginzburg, and Wolfgang Soergel. Koszul duality patterns in representation theory. JAMS, 9(2):473–527, 1996.

[BL94] Joseph N. Bernstein and Valery Lunts. Equivariant Sheaves and Functors, volume 1578 of Lecture Notes in Mathematics. Springer, 1994.

[Bor53] Armand Borel. Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des groupes de Lie compacts. Ann. of Math., 57(2):115– 207, 1953.

[Bou81] Nicolas Bourbaki. Groupes et algèbres de Lie, volume 4-6. Masson, 1981.

[Gin91] Victor Ginsburg. Perverse sheaves and ℂ^{∗}-actions. JAMS, 4(3):483–490, 1991.

[Gin98] Victor Ginzburg. Geometric methods in representation theory of Hecke algebras and quantum groups. Preprint, 1998.

[Jan83] Jens Carsten Jantzen. Einhüllende Algebren halbeinfacher Lie-Algebren, volume 750 of Lecture Notes in Mathematics. Springer, 1983.

[Soe90] Wolfgang Soergel. Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. Journal of the AMS, 3:421–445, 1990.

[Soe92a] Wolfgang Soergel. The combinatorics of Harish-Chandra bimodules. Journal für die reine und angewandte Mathematik, 429:49–74, 1992.

[Soe92b] Wolfgang Soergel. Langlands’ philosophy and Koszul duality. Preprint, 1992.

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

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