LANGLANDS’ PHILOSOPHY AND KOSZUL DUALITY

WOLFGANG SOERGEL

Contents

1. Introduction 1 1.1. The basic conjecture 1 1.2. The easiest example 4 1.3. The geometry of Adams-Barbasch-Vogan 4 1.4. Thanks 6 2. Notations and conventions 6 2.1. Groups and varieties 6 2.2. Grothendieck groups 7 2.3. Equivariant derived categories 8 2.4. Geometric extension algebras 8 3. Examples 11 3.1. Tori 11 3.2. Generic central character 12 3.3. The case G = SL_{2} 13 3.4. Complex groups 18 3.5. Compact groups 22 3.6. Tempered representations 23 3.7. The center and geometry 24 4. Further conjectures 25 4.1. Generalities on Koszul duality 25 4.2. Application to our situation 25 4.3. Motivation 28 References 28

1. Introduction

In this article I will formulate conjectures relating representation theory to geometry and prove them in some cases. I am trying to extend work of Adams-Barbasch-Vogan [ABV92] and some joint work of myself with Beilinson and Ginzburg [BGS96]. All this should be regarded as a contribution to (the local case of) Langlands’ philosophy.

The localization of Beilinson-Bernstein also establishes a relation between representation theory and geometry. This is however very different from the relation to be investigated in this article: Whereas localization leads to geometry on the group itself (or its flag manifold), the results in [ABV92] to be extended in this article lead to geometry on the Langlands dual group.

1.1. The basic conjecture. Let G be (only for a short moment) a real reductive Zariski-connected algebraic group. We want to study admissible representations of the associated Lie group G(ℝ) of real points of G. More precisely, we want to study the category M(G(ℝ)) of smooth admissible representations of G(ℝ). Remark that by a theorem of Casselman-Wallach this is nothing else but the category of Harish-Chandra modules for G(ℝ), modulo the choice of a maximal compact subgroup K ⊂ G(ℝ). Let 𝔤 be the complexification of the Lie algebra of G(ℝ), U − mod − U = U − mod − U(𝔤) the enveloping algebra of 𝔤 and Z ⊂ U − mod − U the center of U − mod − U. Then M(G(ℝ)) decomposes according to infinitesimal character into

M(G(ℝ)) = ⨁_{χ} M(G(ℝ))_{χ}

where χ runs over the set MaxZ of maximal ideals of Z.

There are several classifications of the simple objects in M(G(ℝ)). The most popular ones are by Langlands [Lan73], Vogan-Zuckerman [Vog81], and Beilinson-Bernstein [BB81]. In this article I will exploit a version of the first classification scheme, due to Langlands and Shelstad, and more precisely the form given to it by Adams-Barbasch-Vogan [ABV92].

Let me roughly explain its structure. First of all, we have to forget about the real algebraic group G and just remember the inner class of real forms for G ×_{ℝ} ℂ given by G. In other words and other notations, we start out with a complex connected reductive algebraic group G endowed with an inner class of real forms. To simplify the exposition let us furthermore choose a finite central subgroup F of G. To these data one associates in a more or less unique way (and to detail this “more or less” takes most of the time):

1. a finite family S_{F} of antiholomorphic involutions δ : G → G, all from the fixed inner class.

2. a finite connected algebraic covering G_{F}^{∨} of the dual group G^{∨} (i.e. of the complex connected reductive algebraic group G^{∨} whose root system is dual to the root system of G).

3. a collection {X(χ)}_{χ∈MaxZ} of complex algebraic G^{∨}−varieties.

I will give details in section 1.3. For every antiholomorphic involution δ : G → G let G(ℝ, δ) denote the fixed point set of the induced map δ : G(ℂ) → G(ℂ), and for any χ ∈ MaxZ let G(ℝ, δ)_{χ}^{∧} denote the set of (isomorphism classes of) simple objects in M(G(ℝ, δ))_{χ}. Then Adams-Barbasch-Vogan [ABV92] establish a bijection (depending on additional choices) between

(a): the set ˙⋃_{δ∈S_{F}} G(ℝ, δ)_{χ}^{∧} and

(b): the set of (isomorphism classes of) pairs (Y, τ), where Y is a G^{∨}- orbit on X(χ) and τ an irreducible G_{F}^{∨}-equivariant local system on Y.

Strictly speaking this classifies the representations of a whole collection of real forms simultaneously rather then representations of a single group at a time. To explain which geometric parameters (Y, τ) belong to which real form δ ∈ S_{F} wouldn’t make good reading for this introduction (compare [Vog93], 4.16). Let me mention however that any real form in the given inner class is equivalent to G(ℝ, δ) with δ ∈ S_{F} if we choose F big enough, hence the above classification captures all real forms.

One point of setting up the above classification scheme is the very beautiful and miraculous fact that the KL-data on the representation theoretic side are mirrored by the IC-data on the geometric side. This is the essence of Vogan’s character duality [Vog82]. I conjecture that this duality lifts to the level of categories. To give more details, I have to introduce more notation.

Let H be a complex algebraic group and X a complex algebraic Hvariety. Suppose H decomposes X into finitely many orbits. Then we can form a graded ℂ-algebra Ext_{H}(X) as follows: Take the equivariant derived category D_{H}(X) as defined in [BL92]. This is a triangulated ℂ-category containing the category P_{H}(X) of equivariant perverse sheaves as a full abelian subcategory. For F, G ∈ D_{H}(X) let us denote by Hom_{H}(F, G) the space of homomorphisms in D_{H}(X) and put Hom_{H}^{n}(F, G) = Hom_{H}(F, G[n]). Since H was supposed to act with finitely many orbits, there are (up to isomorphism) only finitely many simple objects in P_{H}(X). Let L ∈ P_{H}(X) denote the direct sum of “all” simple objects (i.e. take one from each isomorphism class). Then put

Ext_{H}(X) = ⊕_{n≥0}Hom_{H}^{n}(L, L) = End_{H}^{•}(L).

There is our graded ℂ-algebra. We will refer to it as the “geometric extension algebra”.

For any ring R let R−Mod be the category of all left R-modules. For a graded ℂ-algebra A = ⊕_{n≥0}A^{n} define the full subcategory A − Nil ⊂ A − Mod by

A − Nil = {M ∈ A − Mod | dim_{ℂ}M < ∞, A^{n}M = 0 for n ≫ 0}.

So objects in A − Nil are not assumed to be graded, but only “continous along the graduation”. Recall now the classification scheme from [ABV92]. The main objective of this article is to formulate, motivate and discuss the following

Basic conjecture 1. There exists an equivalence of categories

⨁_{δ∈S_{F}} M(G(ℝ, δ))_{χ} ≅ Ext_{G_{F}^{∨}} (X(χ)) − Nil.

Remarks:

1. Take again our algebraic group H acting on our variety X with finitely many orbits. Then the simple objects of Ext_{H}(X) − Nil are in an obvious bijection with the simple objects of P_{H}(X), and these in turn are parametrized by pairs (Y, τ) with Y an H-orbit on X and τ an irreducible H-equivariant local system on Y, c.f. [BL92]. Thus we see that the simple objects on the left and on the right of our conjectured equivalence are parametrized by the same set. Certainly our equivalence should identify simple objects which have he same parameter.

2. In this article I will check the conjecture in the following cases: (1) if G is a torus, (2) for arbitrary G and “almost all” central characters, under the additional assumption F = 1, (3) for G = SL_{2}, (4) for complex groups, more precisely when G = G_{c}×G_{c} for a semisimple simply connected complex algebraic group G_{c} and F = 1 and the inner class of real forms is such that S_{F} consists just of one δ and G(ℝ, δ) = G_{c}(ℂ), under the additional assumption that the central character χ is integral regular.

3. I will explain in the last section 4 how I expect this conjecture to be related to Vogan’s character duality formulas [Vog82, ABV92].

4. It seems reasonable to make analogous conjectures for nonarchimedean fields as well. In these cases X(χ) should be replaced by spaces of quasiadmissible morphisms of the Weil-Deligne group to the Langlands dual group. I expect that one may identify in special situations the geometric extension algebras with Lusztig’s graded affine Hecke algebras.

1.2. The easiest example. To give the reader a more concrete idea of what is going on, let me treat the easiest example. Let us take G = ℂ^{×} endowed with the inner class of real forms consisting of the split form δ so that G(ℝ, δ) = ℝ^{×}, and choose F = 1.

Then S_{F} will consist of the single element δ, the covering G_{F}^{∨} → G^{∨} will be the identity ℂ^{×} → ℂ^{×}, and X(χ) will consist just of two points, for every χ ∈ MaxZ. We find that

Ext_{G_{F}^{∨}} (X(χ)) = Ext_{ℂ^{×}}(two points)
= Ext_{ℂ^{×}}(point) × Ext_{ℂ^{×}}(point)
= H^{•}(Bℂ^{×}; ℂ) × H^{•}(Bℂ^{×}; ℂ)
= H^{•}(ℙ^{∞}ℂ; ℂ) × H^{•}(ℙ^{∞}ℂ; ℂ)
= ℂ[t] × ℂ[t]

where Bℂ^{×} = ℙ^{∞}ℂ is the classifying space of ℂ^{×} and deg(t) = 2. On the other hand, M(ℝ^{×}) is just the category of continuous finite dimensional representations of ℝ^{×}. I leave it to the reader to check for every χ ∈ MaxZ the equivalence

M(ℝ^{×})_{χ} ≅ (ℂ[t] × ℂ[t]) − Nil.

This proves our conjecture in this special case.

1.3. The geometry of Adams-Barbasch-Vogan. I promised at the beginning that given a complex connected reductive algebraic group G, endowed with an inner class of real forms and a finite central subgroup F, I would construct (1) a multiset S_{F} of real forms, (2) a finite connected covering G_{F}^{∨} of the dual group and (3) a collection {X(χ)}_{χ∈MaxZ} of G^{∨}-varieties.

To fullfill this promise I have to make some choices. In [ABV92] the authors investigate in detail how these choices affect the final outcome.

1. Let us choose an antiholomorphic involution γ : G → G in the given inner class that stabilizes a Borel subgroup. Then form the (real) Lie group ^{Γ}G = G×^{|}Γ(= G(ℂ)×^{|}Γ if we wouldn’t abuse notation), where Γ = Gal(ℂ/ℝ) = {1, γ} acts on G via this antiholomorphic involution γ. For any δ ∈ ^{Γ}G − G with δ^{2} ∈ Z(G) the conjugation int(δ) : G → G is an antiholomorphic involution, inner to γ. Let S_{F} ⊂ ^{Γ}G − G be a system of representatives for the G-conjugacy classes of δ ∈ ^{Γ}G − G with δ^{2} ∈ F. As explained we may consider S_{F} as a multiset of antiholomorphic involutions of G. For example S_{1} = H^{1}(Γ; G).

I warn the reader that for a complex algebraic group I quite often write just G for the complex points of G, to be denoted G(ℂ) in more careful notation. I hope this does not lead to confusion.

2. Let us choose in addition T ⊂ B ⊂ G a γ-stable maximal torus and Borel subgroup of G. This gives rise to a based root system (𝔛, R, Δ; 𝔛^{∨}, R^{∨}, Δ^{∨}) with an involution γs : 𝔛 → 𝔛 stabilizing R and Δ. In more detail, 𝔛 = 𝔛(T) = {homomorphisms of complex algebraic groups λ : T → ℂ^{×}} and γs : 𝔛 → 𝔛 is characterized by the formula (γs(λ))(t) = \overline{λ(γ(t))} ∀t ∈ T. (The notation γs is explained in more detail in 3.1). Then choose a triple (G^{∨}, T^{∨}, ϕ^{∨}) consisting of a connected reductive complex algebraic group G^{∨}, a maximal torus T^{∨} ⊂ G^{∨} and an isomorphism ϕ^{∨} : 𝔛^{∨} → 𝔛(T^{∨}) that identifies the root system of G^{∨} with the dual of the root system of G.

Now the finite central subgroups F of G are in one-to-one correspondence with finite quotients of 𝔛/ℤR, via F ↦ 𝔛(F). But the finite connected coverings π : H → G^{∨} of G^{∨} are also in one-to-one correspondence with finite quotients of 𝔛/ℤR, via H ↦ 𝔛/𝔛^{∨}(π^{−1}(T^{∨})). In this way any finite central subgroup F of G gives rise to a finite connected covering G_{F}^{∨} of G^{∨}.

3. This is the most subtle part. We start introducing some notations.

Let H be any complex connected reductive algebraic group, 𝔥 = LieH its Lie algebra and λ ∈ 𝔥 a semisimple element. Put

\begin{aligned}𝔥(λ)_{n} & = & \{μ ∈ 𝔥 | [λ, μ] = nμ\} \\ 𝔫(λ) & = & ⊕_{n=1,2,...}𝔥(λ)_{n} \\ e(λ) & = & exp(2πiλ) ∈ H. \end{aligned}

The “canonical flat through λ” is the affine subspace F(λ) = λ + 𝔫(λ) ⊂ 𝔥. One proves (see [ABV92]) that the canonical flats partition the set 𝔥_{s} of semisimple elements of 𝔥 and even partition each conjugacy class of semisimples. Furthermore the function e is constant on each canonical flat. Let F(𝔥) denote the space of all canonical flats (through semisimple elements of 𝔥) and for a semisimple orbit O ⊂ 𝔥 put F(O) = {Λ ∈ F(𝔥) | Λ ⊂ O}. Then e is a function e : F(𝔥) → H.

Now recall G^{∨} and put 𝔤^{∨} = LieG^{∨}. We may and will identify Lieℂ^{×} = ℂ in such a way that the exponential map of Lie groups corresponds to the usual exponential map. This determines an embedding 𝔛 = Hom(T, ℂ^{×}) → Hom(LieT, Lieℂ^{×}) = (LieT)^{∗}, hence an isomorphism 𝔛⊗_{ℤ}ℂ = (LieT)^{∗}. On the other hand ϕ^{∨} determines an embedding 𝔛 = Hom(ℂ^{×}, T^{∨}) → Hom(Lieℂ^{×}, LieT^{∨}) = LieT^{∨}, hence an isomorphism 𝔛 ⊗_{ℤ} ℂ = LieT^{∨}.

Let W be the Weyl group. We have bijections

MaxZ = (LieT)^{∗}/W = (𝔛 ⊗_{ℤ} ℂ)/W = (LieT^{∨})/W = 𝔤_{s}^{∨}/G^{∨}

induced respectively by the Harish-Chandra homomorphism, our above isomorphisms and the embedding LieT^{∨} ⊂ 𝔤_{s}^{∨}. Denote by χ ↦ O(χ) their composition.

Now choose a collection {x_{α^{∨}}}_{α∈Δ} of isomorphisms x_{α^{∨}} : ℂ → U_{ϕ^{∨}(α^{∨})} of the additive group ℂ onto the simple root subgroups of G^{∨}. The involution γs : 𝔛 → 𝔛 has an adjoint (γs)^{∨} : 𝔛^{∨} → 𝔛^{∨}. We will abreviate (γs)^{∨} = γ : 𝔛^{∨} → 𝔛^{∨}. This gives rise to an involutive automorphism γ : G^{∨} → G^{∨} of the complex algebraic group G^{∨} characterized by γ(T^{∨}) = T^{∨}, ϕ^{∨}γ = γϕ^{∨} : 𝔛^{∨} → 𝔛(T^{∨}) and γ ∘ x_{α^{∨}} = x_{γ(α^{∨})}∀α ∈ Δ. Thus we may form the semidirect product ^{Γ}G^{∨} = G^{∨}×^{|}Γ, the so-called L-group.

Now for any χ ∈ MaxZ we define

X(χ) = {(y, Λ) | y ∈ ^{Γ}G^{∨} − G^{∨}, Λ ∈ F(O(χ)) such that y^{2} = e(Λ)}.

This is in a natural way a complex algebraic variety on which G^{∨} acts via conjugation. For more details see [ABV92]. This fullfills my promise.

It is useful to remark, that by [ABV92], (12.11)(e) the component groups of isotropy groups of G_{F}^{∨} acting on X(χ) are all commutative. Hence all local systems τ appearing in our geometric parameters are of rank one.

1.4. Thanks. I thank David Vogan, Victor Ginsburg, and Valery Lunts for very helpful discussions. I thank the DFG, MPI, and CNRS for financial support, and the latter two for providing a stimulating research atmosphere.

2. Notations and conventions

2.1. Groups and varieties. Let G, H be complex algebraic groups. A holomorphic (resp. antiholomorphic) morphism from G to H is a morphism of group schemes ϕ : G → H such that the diagram

G →^{ϕ} H
↓  ↓
Specℂ → Specℂ

with the lower horizontal the identity (resp. the complex conjugation) commutes.

Let G be a complex algebraic group. Two antiholomorphic involutions δ, δ′ of G are said to belong to the same inner class if and only if δδ′ is an inner automorphism of G, i.e. δ = δ′int(x) with x ∈ G. Two antiholomorphic involutions δ, δ′ of G are said to be equivalent if and only if they are conjugate by an inner automorphism of G, i.e. δ = int(x^{−1})δ′int(x) with x ∈ G. Any two equivalent antiholomorphic involutions belong to the same inner class.

Let A, B be groups and σ : B → AutA a group homomorphism. Let us write (σ(b))(a) = a^{b} ∀a ∈ A, b ∈ B. We define a new group A×^{|}_{σ}B = A×^{|}B as follows: As a set A×^{|}B = A×B and the multiplication is given by (a, b)(a_{1}, b_{1}) = (a(a_{1}^{b}), bb_{1}).

Let G be a complex algebraic group. Then we denote by LieG its Lie algebra. This is a Lie algebra over ℂ. Let H be a Lie group. Then we denote by Lie_{ℝ}H the Lie algebra of H. This is a Lie algebra over ℝ. We denote its complexification by Lie_{ℂ}H = ℂ ⊗_{ℝ} Lie_{ℝ}H. Let G be a complex algebraic group. Then LieG = Lie_{ℝ}(G(ℂ)) canonically. This is an isomorphism of Lie algebras over ℝ, where on the left we quietly restrict scalars from ℂ to ℝ. Let δ be an antiholomorphic involution on G. Then LieG = Lie_{ℂ}(G(ℝ; δ)) canonically.

Let X be a complex algebraic variety. We often abuse notation and write X instead of X(ℂ). Sometimes we also abuse notation in the other direction and write for example ℂ^{×}, ℂ instead of 𝔾_{m,ℂ}, 𝔾_{a,ℂ}.

2.2. Grothendieck groups. Let A be an abelian (resp. triangulated) category. Then we let [A] be the Grothendieck group of A, generated by the objects with the usual relations for all short exact sequences (resp. distinguished triangles). Any object A ∈ A gives an element [A] ∈ [A].

Let A be an additive category. Then we let ⟨A⟩ be the split Grothendieck group of A, generated by the objects with the usual relations for all split short exact sequences. Any object A ∈ A gives an element ⟨A⟩ ∈ ⟨A⟩.

2.3. Equivariant derived categories. My basic reference for what follows is [BL92]. Let G be a complex algebraic group and X a complex algebraic G-variety. To X we may associate a triangulated ℂcategory D_{G}(X), the bounded constructible G-equivariant derived category, called D_{G,c}^{b}(X) in [BL92], 4.1. We note Hom_{G} the morphisms in D_{G}(X), put Hom_{G}^{•} = ⊕_{n}Hom_{G}^{n} and use analogous notation End_{G}, End_{G}^{•} for endomorphisms. If G = 1 we come back to our usual triangulated categories D_{1}(X) = D_{c}^{b}(X) = D(X) and since Hom_{1} looks so stupid, we write Hom_{D} in this case.

I cannot repeat the details of the definition of D_{G}(X). To give the reader an idea of what D_{G}(X) is like, let me just remark that it embeds as a full triangulated subcategory in the bounded derived category D^{b}(Sh(EG ×_{G} X)) of sheaves of ℂ-vectorspaces on EG ×_{G} X, where EG is a contractible space with a free right G-action, e.g. the total space of the universal bundle on the classifying space BG of G.

The usual formalism of the six operations of Grothendieck extends to this equivariant situation: For F, G ∈ D_{G}(X) we may form F ⊗ G, RHom(F, G) ∈ D_{G}(X) and any equivariant morphism f : X → Y of G-varieties gives rise to adjoint pairs of triangulated functors (f^{∗}, f_{∗}) and (f_{!}, f^{!}) relating D_{G}(X) and D_{G}(Y). The categories D_{G}(X) also come equipped with a (perverse) t-structure, and we let P_{G}(X) be its core, i.e. the G-equivariant perverse sheaves for the middle perversity, and H^{i} : D_{G}(X) → P_{G}(X) the perverse cohomology functors. Let \underline{pt} ∈ P_{G}(pt) be the constant object ℂ, and put \underline{X} = p^{∗}\underline{pt} ∈ D_{G}(X) for any X, where p : X → pt is the constant map.

The irreducible objects of P_{G}(X) are parametrized by pairs (Y, τ) with Y a G-stable locally closed subset of X and τ an irreducible Gequivariant local system on Y. Namely to such a pair one associates the intersection cohomology complex with twisted coefficients IC(Y, τ) ∈ P_{G}(X).

Let ϕ : H → G be a morphism of algebraic groups. Then we may define the restriction functors Res : D_{G}(X) → D_{H}(X). They are triangulated and compatible with all our “internal” operations. If ϕ is either (1) a finite covering or (2) the inclusion of a reductive Levi factor, then Res induces isomorphisms Hom_{G}(F, G) → Hom_{H}(ResF, ResG) for all F, G ∈ D_{G}(X).

2.4. Geometric extension algebras. Assume now again that X is a H-variety decomposing into finitely many orbits. Remember we defined the graded ℂ-algebra Ext_{H}(X) = ⊕_{n≥0}Hom_{H}^{n}(L, L) where L = ⊕_{(Y,τ)}IC(Y, τ) is the sum over all simple objects of P_{H}(X), so that Y runs over all H-orbits in X and τ over all irreducible H-equivariant local systems on Y.

Suppose H ⊂ G is a closed subgroup. Then the “induction equivalence” gives an isomorphism Ext_{H}(X) = Ext_{G}(G ×_{H} X).

Suppose H is connected. Then Ext_{H}(pt) = H^{•}(BH; ℂ) is just the cohomology ring of the classifying space of H. If in particular T is a torus, then Ext_{T} (pt) = S((LieT)^{∗}) is just the ring of regular functions on its Lie algebra, graded in such a way that deg(LieT)^{∗} = 2. More generally, if H is any connected algebraic group, then Ext_{H}(pt) = H^{•}(BH; ℂ) = S((LieT)^{∗})^{W} consists of the invariants of the Weyl group in the regular functions on the Lie algebra of a maximal torus T ⊂ G.

Let N be a complex algebraic group, H ⊂ N its identity component and W = N/H the quotient. Then N acts on H by conjugation, hence it acts on H^{•}(BH; ℂ) and since H is connected this N-action factorizes to give an action of W on H^{•}(BH; ℂ). Let us denote this action by σ : W → Aut_{ℂ}(H^{•}(BH; ℂ)).

Now whenever we are given a ℂ-algebra A together with an action σ : W → Aut_{ℂ}A of a group W on A by algebra automorphisms, we may form the “twisted group ring” A^{σ}[W] = A⊗_{ℂ}ℂ[W] with multiplication (a ⊗ w)(b ⊗ v) = a(σ(w)(b)) ⊗ wv. In the above notations, we prove:

Theorem 2.4.1. Let N be a complex algebraic group, H ⊂ N its identity component and W = N/H the quotient. If W is commutative, then

Ext_{N}(pt) = H^{•}(BH; ℂ)^{σ}[W].

If not these two rings are still Morita-equivalent.

In fact we will prove even a more general theorem to prepare for the calculations concerning SL_{2}. Namely let H ↪ N ↠ W be as above, let X be an N-variety and L ∈ D_{N}(X) an object in the equivariant derived category. We have the restriction functor Res : D_{N}(X) → D_{H}(X). Its right and left adjoints coincide in our situation, so we denote them just by Ind : D_{H}(X) → D_{N}(X). See [BL92], 3.7 for the definitions of Ind_{∗}, Ind_{!}. Let me first give the statement I want, before I explain and prove it.

Theorem 2.4.2. End_{N}^{•}(Ind ResL) = End_{H}^{•}(ResL)^{σ}[W].

Explanation: I ought to explain the action σ of W on End_{H}^{•}(ResL). Let E = EN denote the universal bundle over the classifying space of N. Let π : E ×_{H} X → E ×_{N} X be the projection. We have the following two commutative squares

D_{N}(X) ↪ D^{+}(Sh(E ×_{N} X))
Res ↓↑ Ind  π^{∗} ↓↑ π_{∗}
D_{H}(X) ↪ D^{+}(Sh(E ×_{H} X))

where π^{∗}, π_{∗} stand for the functors usually denoted Lπ^{−1}, Rπ_{∗}. The horizontal arrows are fully faithful embeddings. In fact there is a right W-action on on E ×_{H} X given by [w] : E ×_{H} X → E ×_{H} X, (e, x) ↦ (en, n^{−1}x) for any n ∈ N representing w ∈ W. Our π : E ×_{H} X → E ×_{N} X is just the quotient by this right W-action.

Let the image of L by the upper horizontal in the above diagram be denoted L̃. Thus End_{H}^{•}(ResL) = End^{•}(π^{∗}L̃). But any w ∈ W gives

σ(w) : End^{•}(π^{∗}L̃) → End^{•}([w]^{∗}π^{∗}L̃) → End^{•}(π^{∗}L̃)

where the first map is induced by [w]^{∗}, the second by π ∘[w] = π. There is our W-action on End^{•}(π^{∗}L̃). Proof: We procede in four steps.

1. Define a morphism End_{H}^{•}(ResL) → End_{N}^{•}(Ind ResL).

2. Define a morphism ℂ[W] → End_{N}^{•}(Ind ResL).

3. Prove that they are “σ-compatible”, i.e. fit together to define a morphism

End_{H}^{•}(ResL)^{σ}[W] → End_{N}^{•}(Ind ResL).

4. Prove that this morphism is an isomorphism.

So let’s go to work!

(1) The first step is clear by functoriality of Ind. Translating into a different picture, the morphism is π_{∗} : End^{•}(π^{∗}L̃) → End^{•}(π_{∗}π^{∗}L̃).

(2) Translating via our diagram we see that we need a morphism ℂ[W] → End^{•}(π_{∗}π^{∗}L̃). Indeed any w ∈ W defines a morphism ⟨w⟩ : π_{∗}π^{∗}L̃ → π_{∗}π^{∗}L̃ as the composition π_{∗}π^{∗}L̃ → π_{∗}[w]_{∗}[w]^{∗}π^{∗}L̃ → π_{∗}π^{∗}L̃ where the first map comes from the adjointness transformation in → [w]_{∗}[w]^{∗} and the second from π ∘[w] = π. One may check that ⟨w⟩⟨v⟩ = ⟨wv⟩. Thus we defined a morphism ℂ[W] → End(π_{∗}π^{∗}L̃).

(3) Let f ∈ End^{•}(π^{∗}L̃) and w ∈ W. We have to establish in End^{•}(π_{∗}π^{∗}L̃) the equation ⟨w⟩(π_{∗}f) = (π_{∗}(σ(w)f))⟨w⟩. For this we consider the commutative diagram

π_{∗}π^{∗}L̃ → π_{∗}[w]_{∗}[w]^{∗}π^{∗}L̃ → π_{∗}[w]^{∗}π^{∗}L̃ → π_{∗}π^{∗}L̃
π_{∗}f ↓ π_{∗}[w]_{∗}[w]^{∗}f ↓ π_{∗}[w]^{∗}f ↓ π_{∗}σ(w)f ↓
π_{∗}π^{∗}L̃ → π_{∗}[w]_{∗}[w]^{∗}π^{∗}L̃ → π_{∗}[w]^{∗}π^{∗}L̃ → π_{∗}π^{∗}L̃

which gives us the required equation.

(4) We have to prove that the product

End^{•}(π^{∗}L̃) ⊗_{ℂ} ℂ[W] → End^{•}(π_{∗}π^{∗}L̃), f ⊗ w ↦ π_{∗} ∘ id ∘ ⟨w⟩

is an isomorphism or, equivalently, that the product

Hom^{•}(π^{∗}L̃, π^{∗}L̃)⊗_{ℂ}ℂ[W] → Hom^{•}(π^{∗}π_{∗}π^{∗}L̃, π^{∗}L̃), f⊗w ↦ π_{∗}∘adj∘π^{∗}⟨w⟩

is an isomorphism where adj is the map corresponding to the identity id via the adjointness (π^{∗}, π_{∗}). But π^{∗}π_{∗}π^{∗}L̃ ≅ ⊕_{w∈W}π^{∗}L̃ and under this isomorphism the W-action on the left by the π^{∗}⟨w⟩ corresponds to the “regular action” on the right. So indeed we get an isomorphism. q.e.d. Proof of the first theorem: Apply the second theorem to X = pt, L = \underline{pt}. q.e.d.

3. Examples

3.1. Tori. To establish the conjecture for G a torus T is not to hard. We begin with some definitions. As is well known there is a contravariant equivalence of abelian categories

\{ \begin{matrix}diagonalizable \\ complex algebraic \\ groups \end{matrix} \} \begin{matrix}𝔛 \\ → \\ ← \\ Diag \end{matrix} \{ \begin{matrix}finitely generated \\ commutative \\ groups \end{matrix} \}

For morphisms γ of the left (resp. right) category we will often write just γ instead of 𝔛(γ) (resp. Diag(γ)). The reason that this does not lead us into a catastrophe is that we only have to work with pairwise commuting involutions.

Any diagonalizable complex algebraic group G admits a unique antiholomorphic involution s = s_{G} : G → G such that ϕ(s(g)) = \overline{ϕ(g)} ∀g ∈ G, ϕ ∈ 𝔛(G). This antiholomorphic involution gives rise to what one calls the split real form of G. For every (holomorphic or antiholomorphic) morphism of complex diagonalizable groups f : G → H we have f ∘ s_{G} = s_{H} ∘ f.

Both the holomorphic and antiholomorphic homomorphisms from one complex diagonalizable group to another form an abelian group. We write these groups additively. For example, if f : G → H is holomorphic (or antiholomorphic), we have (−f)(g) = f(g)^{−1} ∀g ∈ G.

Now let us get at our torus T. It comes with an inner class of real forms, i.e. with an antiholomorphic involution γ. This gives us an involution γs = 𝔛(γs) on 𝔛 = 𝔛(T) and thus an involution γ = (γs)^{∨} on the dual lattice 𝔛^{∨}. Let us describe the varieties X(χ). They are either empty or isomorphic to X(0), which we henceforth abreviate to X. The isotropy group I is the same for all points of X and is in fact the kernel of (id − γ) : T^{∨} → T^{∨}. Applying our exact contravariant functor 𝔛 we get 𝔛(I) = 𝔛^{∨}/(id − γ)𝔛^{∨}. Thus (LieI)^{∗} = 𝔛(I) ⊗_{ℤ} ℂ = (𝔛^{∨} ⊗_{ℤ} ℂ)^{γ} and Ext_{T_{F}^{∨}}X is just a cartesian product of say n_{g} copies of the symmetric algebra S = S((𝔛^{∨} ⊗_{ℤ} ℂ)^{γ}) over the dual of the Lie algebra of the isotropy group.

On the other hand, T (ℝ, γ) admits a Cartan decomposition T (ℝ, γ) = K ×A with K compact and A a vector group. In fact, A is just the onecomponent of T (ℝ, γ) ∩ T (ℝ, s), and this intersection in turn consists just of all fixed points for some antiholomorphic involution on the kernel of (id−sγ) : T → T. Thus Lie_{ℂ}A = Lie(ker(id−sγ)) = (𝔛^{∨} ⊗_{ℤ}ℂ)^{γ}. In other words, the S from above is also the enveloping algebra of Lie_{ℂ}A. So for each χ ∈ MaxZ either

⨁_{δ∈S_{F}} M(T (ℝ, δ))_{χ} ≅ ⨁_{n_{r} copies} S − Nil

for some n_{r}, or the left hand side vanishes. So to establish the conjecture we just have to show that X(χ) ≠ ∅ ⇔ M(T (ℝ, γ))_{χ} ≠ 0 and that if both sides do not vanish, then n_{g} = n_{r}. But this is calculated in detail in [ABV92] and essentially even in [Lan73] (for F = 1).

3.2. Generic central character. For most χ ∈ MaxZ the situation is very simple. Namely for χ outside a countable union of proper closed subvarieties of MaxZ only the split real form admits representations with infinitesimal character χ, and the irreducible ones are just the principal series induced from a Borel. I now test the conjecture on this example.

Let (LieT)_{ni}^{∗} = {λ ∈ (LieT)^{∗} | ⟨λ, α̌⟩ ∉ ℤ ∀α ∈ R} be the set of all “not at all integral” weights and (MaxZ)_{ni} ⊂ MaxZ its image under the Harish-Chandra homomorphism.

Lemma 3.2.1. If G is endowed with the inner class of real forms containing the split form and χ ∈ (MaxZ)_{ni}, then X(χ) = G^{∨}/T^{∨} × {t ∈ T^{∨} | t^{2} = 1} as a G^{∨}-variety.

Proof: Since G is endowed with the split inner class, ^{Γ}G^{∨} = G^{∨} × Γ. For χ ∈ (MaxZ)_{ni} each element of O(χ) is its own canonical flat, hence X(χ) = {(y, λ) | y ∈ G^{∨}, λ ∈ O(χ) such that y^{2} = exp(2πiλ)}. Let us determine for one λ_{0} ∈ O(χ) all possible y. Assume to simplify λ_{0} ∈ LieT^{∨}. Since χ was supposed to be not at all integral, the centralizer of exp(2πiλ_{0}) in G^{∨} is just T^{∨} and we find that necessarily y ∈ T^{∨}. Thus a bijection {t ∈ T^{∨} | t^{2} = 1} → {y ∈ G^{∨} | y^{2} = exp(2πiλ_{0})}, t ↦ t exp(πiλ_{0}). But every χ ∈ (MaxZ)_{ni} is in particular regular, so O(χ) = G^{∨}/T^{∨} and we find indeed X(χ) = G^{∨}/T^{∨} × {t ∈ T^{∨} | t^{2} = 1} with G^{∨} acting only on the first factor. q.e.d.

Let us now assume for simplicity that F = 1.

Lemma 3.2.2. Suppose G is endowed with the inner class of real forms containing the split form. Then only one of the δ ∈ S_{1} gives a split form of G.

Proof: If we identify S_{1} = H^{1}(Γ; G), then δ ∈ S_{1} gives the split form if and only if it maps to the identity element in H^{1}(Γ; G/Z(G)). However consider the exact sequence

H^{1}(Γ; Z(G)) → H^{1}(Γ; G) → H^{1}(Γ; G/Z(G))

of nonabelian cohomology. Since the first map factors over the cohomology of a split torus T ⊂ G and H^{1}(Γ; T) = 1, the preimage of the identity element in H^{1}(Γ; G/Z(G)) is just the identity element in H^{1}(Γ; G). q.e.d.

Let s ∈ S_{1} represent the split form. For χ ∈ (MaxZ)_{ni} we know that M(G(ℝ, δ))_{χ} = 0 if δ ∈ S_{1}, δ ≠ s. So we have for χ ∈ (MaxZ)_{ni} just to establish an equivalene of categories

M(G(ℝ, s))_{χ} ≅ Ext_{G^{∨}}X(χ) − Nil.

But both sides are just Nil-modules over a Cartesian product of 2^{rankG} copies of the symmetric algebra S = S(LieT) of the Lie algebra of the maximal torus. For the right hand side this follows from Ext_{G^{∨}}(G^{∨}/T^{∨}) = Ext_{T^{∨}}(pt) = H^{•}(BT^{∨}; ℂ) = S(LieT) where the grading is determined by deg(LieT) = 2. For the left hand side it follows from the fact that for any λ ∈ (LieT)_{ni}^{∗} the functor

M(T (ℝ, s))_{λ} → M(G(ℝ, s))_{ξ(λ)}

of extending trivially to a Borel and inducing (with the shift that ensures that unitary representations go to unitary ones) is an equivalence of categories.

3.3. The case G = SL_{2}. There is only one inner class of real forms. Certainly G^{∨} = P GL(2; ℂ). Let us compute the X(χ). We distinguish three cases.

χ not integral.: Then X(χ) = G^{∨}/T^{∨}×{two points} as we showed

already more generally. χ singular.: Then O(χ) = {0} and X(χ) = {y ∈ G^{∨} | y^{2} = 1}.

But any element of finite order is semisimple, hence contained in a maximal torus and in a torus there are precisely two roots of one. The stabilizers in G^{∨} of these two are G^{∨} and N_{G^{∨}}(T^{∨}), which we will henceforth abreviate N. Thus X(χ) = pt ˙∪ (G^{∨}/N).

χ regular integral.: Then exp(2πiλ) = 1 for all λ ∈ O(χ), hence X(χ) is just the Cartesian product of F(O(χ)) with {y ∈ G^{∨} | y^{2} = 1}. But if χ ∈ MaxZ is regular integral, we have quite generally F(O(χ)) = G^{∨}/B^{∨}. Hence in our situation {F(λ) | λ ∈ O(χ)} = G^{∨}/B^{∨} and

X(χ) = G^{∨}/B^{∨} ˙∪ G^{∨}/B^{∨} × G^{∨}/N

where G^{∨} acts diagonally on the second part.

3.3.1. The subcase F == 1. Let us first check the conjecture for F = 1. Then S_{1} = H^{1}(Γ; SL(2, ℂ)) consists just of one element δ with G(ℝ, δ) = SL(2, ℝ). For any χ ∈ MaxZ let us abreviate M(SL(2, ℝ))_{χ} = M_{χ}. These categories break down according to K-types into M_{χ} = M_{χ}^{ev} ⊕ M_{χ}^{odd}. Now we procede case by case.

χ not integral. This has already been treated in general.

χ singular. It is known that M_{χ}^{ev} is isomorphic to the category of all finite dimensional continuous modules over the completion Z_{χ}^{∧} of Z at χ. Via the Harish-Chandra homomorphism we find M_{χ}^{ev} ≅ S^{W} − Nil, with S = S(LieT) the symmetric algebra over LieT. But on the other hand Ext_{G^{∨}}(pt) = H^{•}(BG^{∨}; ℂ) = H^{•}(BT^{∨}; ℂ)^{W} = S^{W} so that indeed

M_{χ}^{ev} ≅ Ext_{G^{∨}}(pt) − Nil.

The structure of M_{χ}^{odd} is more complicated. The irreducible objects of this category are the two limits of principal series, and these extend among each other. More precisely the picture is as follows: Choose in 𝔤 a “standard” basis X, H, Y with [H, X] = 2X, [H, Y] = −2Y, [X, Y] = H and such that ℂH ⊂ 𝔤 is the complexification of the Lie algebra of the compact torus SO(2) ⊂ SL(2, ℝ). For M ∈ M, n ∈ ℤ put M_{n} = {v ∈ M | Hv = nv}. Then

M ↦ { [figure: M_{−1}; M_{+1}; X; Y] }

is an equivalence of M_{χ}^{odd} to finite dimensional representations of the quiver

[figure: •; •; ϕ; ψ]

with relations ϕψ nilpotent. This is not difficult, and follows for example from the more general results in [Kho80]. I leave it to the reader to identify the quiver algebra with the twisted group ring S^{σ}[W], where σ : W → Aut_{ℂ}S is the obvious action. Putting things together, we get M_{χ}^{odd} ≅ S^{σ}[W] − Nil.

But on the other hand

Ext_{G^{∨}}(G^{∨}/N) = H^{•}(BN; ℂ) by ??
= H^{•}(BT^{∨}; ℂ)^{σ}[W] by theorem 2.4.1 of section ??
= S^{σ}[W].

So indeed M_{χ}^{odd} ≅ Ext_{G^{∨}}(G^{∨}/N) − Nil and with our previous result we obtain finally M_{χ} ≅ Ext_{G^{∨}}X(χ) − Nil for singular χ.

χ regular integral. By the translation principle we may and will henceforth assume χ = Ann_{Z}ℂ is just the annihilator of the trivial representation. Then M_{χ}^{odd} can be identified with the category of all finite dimensional continuous modules over the completion Z_{χ}^{∧} of Z at χ, and via the Harish-Chandra homomorphism we find M_{χ}^{odd} ≅ S−Nil. But Ext_{G^{∨}}(G^{∨}/B^{∨}) = H^{•}(BB^{∨}; pt) = S, so M_{χ}^{odd} ≅ Ext_{G^{∨}}(G^{∨}/B^{∨}) − Nil.

The category M_{χ}^{ev} is much more interesting. Using the preceding notations, we may describe it as follows: The functor

M ↦ { [figure: M_{−2}; M_{0}; M_{+2}; X; Y; X; Y] }

establishes an equivalence of M_{χ}^{ev} to finite dimensional representations of the quiver

[figure: •; •; •; (∗); −2; 0; +2; ϕ^{−}; ψ^{−}; ϕ^{+}; ψ^{+}]

with relations ϕ^{−}ψ^{−} = ψ^{+}ϕ^{+} and ψ^{+}ϕ^{+} nilpotent. This is not difficult, and follows for example from the more general results in [Kho80]. Now let us attack the geometric extension algebra. Since

G^{∨}/B^{∨} × G^{∨}/N = G^{∨} ×_{N} (G^{∨}/B^{∨})

we have (using the induction identity) just to understand Ext_{N}(G^{∨}/B^{∨}). There are two N-orbits on (G^{∨}/B^{∨}) = ℙ^{1}ℂ = X, namely Y = {0, ∞} and U = ℂ^{×}. Here Y supports just one irreducible N-equivariant local system \underline{Y} , but U supports two of them, with trivial and nontrivial N-action, say \underline{U}^{+} and \underline{U}^{−}. The irreducibles in P_{N}(X) are correspondingly i_{∗}\underline{Y} , \underline{X}^{+}[1] and \underline{X}^{−}[1] where i : Y → X is the inclusion and \underline{X}^{+} (resp. \underline{X}^{−}) denotes just the constant sheaf ℂ on X with trivial (resp. nontrivial) N-action. By definition, Ext_{N}X = End_{N}^{•}(i_{∗}\underline{Y} ⊕ \underline{X}^{+}[1] ⊕ \underline{X}^{−}[1]). But since we only need to know this ring up to Morita equivalence, let us rather compute the equivariant selfextensions of i_{∗}\underline{Y} ⊕ i_{∗}\underline{Y} ⊕ \underline{X}^{+}[1] ⊕ \underline{X}^{−}[1]. This object has the advantage to be isomorphic to IndRes(i_{∗}\underline{Y} ⊕ \underline{X}^{+}[1]) where we consider N as the semidirect product N = T^{∨}×^{|}W and use the notations of ??, so we may apply theorem 2.4.2 of section ?? to L = i_{∗}\underline{Y} ⊕ \underline{X}^{+}[1]. The theorem tells us that we should first compute End_{T^{∨}}^{•}(ResL) with its W-action. This is not to hard, so I just give the result. Proposition 3.3.1. End_{T^{∨}}^{•}(ResL) is the quiver algebra of the quiver

[figure: •; •; •; (∗∗); l; m; r; (ml); (lm); (rm); (mr)]

with relations (rm)(ml) = 0 = (lm)(mr) where every arrow has degree one. The reflection of the quiver along the vertical symmetry axis induces the looked-for action σ of W on the quiver algebra.

Proof: Left to the reader. Perhaps it would have been more suggestive to call the extremal points 0 and ∞ instead of r and l.

The finite dimensional representations of Ext_{N}X can be identified by Morita equivalence with finite dimensional representations of

End_{N}^{•}(Ind ResL) = End_{T^{∨}}^{•}(ResL)^{σ}[W],

and these in turn are just finite dimensional representations of the quiver with relations from the proposition equipped with an “action of W on the representation compatible with the above action of W on the quiver”. Such an object consists of a diagram of finite dimensional spaces

[figure: W_{l}; W_{m}; W_{r}; (ml); (lm); (rm); (mr)]

together with isomorphisms s : W_{r} → W_{l}, s : W_{m} → W_{m}, s : W_{l} → W_{r} such that the relations s^{2} = id, (lm)(mr) = 0 = (rm)(ml), s(mr) = (ml)s, s(lm) = (rm)s are satisfied. We got to identify the category of such objects with the category of finite dimensional representations of the related quiver (∗).

This can be done as follows: To a representation

[figure: M_{−2}; M_{0}; M_{+2}; ϕ^{−}; ψ^{−}; ϕ^{+}; ψ^{+}]

of the related quiver (∗) we associate the representation

[figure: M_{0}; M_{+2} ⊕ M_{−2}; M_{0}; \binom{ϕ^{+}}{ϕ^{−}}; (ψ^{+},ψ^{−}); (ψ^{+},−ψ^{−}); \binom{ϕ^{+}}{−ϕ^{−}}]

of the related quiver (∗∗) with W-action given by

s = id : M_{0} → M_{0}, s = ( \begin{matrix}id & 0 \\ 0 & −id \end{matrix} ) : M_{+2} ⊕ M_{−2} → M_{+2} ⊕ M_{−2}.

This can be seen to be an equivalence of categories and induces an equivalence of categories M_{χ}^{ev} ≅ Ext_{N}X − Nil. So putting things together we indeed established the desired equivalence of categories M_{χ} ≅ Ext_{G^{∨}}X(χ) − Nil for regular integral χ.

3.3.2. The subcase F = Z(G). Now take for F the full center of SL(2, ℂ), i.e. F = {+1, −1}. Then S_{F} can be seen to consist of three elements, giving rise to the three real forms SU(2, 0), SU(1, 1) and SU(0, 2). Certainly two of them are just the compact form SU(2) and the middle one is the split form SL(2, ℝ). So on the representation theoretic side of our conjecture only for χ regular integral there are some changes, and we have to consider instead of M(SL(2, ℝ)_{χ} = M_{χ} the category

⨁_{i=−1,0,1} M(SU(1 + i, 1 − i))_{χ} ≅ (ℂ − Nil) ⊕ M_{χ} ⊕ (ℂ − Nil).

On the geometric side we get G_{F}^{∨} = SL(2, ℂ). By ?? homomorphisms between objects in our equivariant derived categories do not change when we pass from G^{∨} to G_{F}^{∨}. However there might be more simple objects in P_{G_{F}^{∨}} X(χ) than in P_{G^{∨}}X(χ), and this happens precisely when the isotropy groups of points of X(χ) with respect to P_{G_{F}^{∨}} have more components than the isotropy groups with respect to P_{G^{∨}}. In our situation, this happens only for χ regular integral and for points in the open orbit on G^{∨}/B^{∨} × G^{∨}/N ⊂ X(χ). To verify our conjecture for F = {+1, −1} we just have to check that

Ext_{G_{F}^{∨}} (G^{∨}/B^{∨} × G^{∨}/N) ≅ (Ext_{G^{∨}}(G^{∨}/B^{∨} × G^{∨}/N)) × ℂ × ℂ.

Via the “induction identity” this can be reduced to showing

Ext_{N_{F}} X ≅ (Ext_{N}X) × ℂ × ℂ

where N_{F} = N_{G_{F}^{∨}} (T^{∨}). Let L ∈ P_{N}X be the sum of all simple objects, L = i_{∗}\underline{Y} ⊕ \underline{X}^{+}[1] ⊕ \underline{X}^{−}[1]. In P_{N_{F}} X there are an additional two simple objects. In more detail, the isotropy group in N_{F} of a point in the open orbit U ⊂ X is just ℤ/4ℤ, thus there are four simple objects F_{0}, F_{1}, F_{2}, F_{3} in P_{N_{F}} X, corresponding to the four irreducible characters on ℤ/4ℤ. Let j : U → X be the inclusion. Then j_{!∗}F_{0} = \underline{X}^{+}[1], j_{!∗}F_{2} = \underline{X}^{−}[1] and our two new irreducible objects in P_{N_{F}} X are j_{!}F_{1} = j_{∗}F_{1} and j_{!}F_{3} = j_{∗}F_{3}. I leave it to the reader to check that

End_{N_{F}}^{•} (L ⊕ j_{∗}F_{1} ⊕ j_{∗}F_{3}) ≅ End_{N}^{•}(L) × ℂ × ℂ.

This gives the conjecture for G = SL(2, ℝ), F = {+1, −1}.

3.4. Complex groups. Here comes one of the main motivations for the conjecture. Namely we check that it correctly predicts the structure of part of the representation theory of G_{c}(ℂ) for a semisimple connected complex algebraic group, where we consider G_{c}(ℂ) as a real Lie group. In fact we will only give details for G_{c} simply connected and assume this from now on. To get in the set-up of the conjecture, choose a split antiholomorphic involution s : G_{c} → G_{c} and put G = G_{c} × G_{c} with the quasisplit antiholomorphic involution γ : G → G, γ(x, y) = (sy, sx). Then G(ℝ, γ) = G_{c}(ℂ) and 1 = H^{1}(Γ; G) = S_{1} = {γ}. We prove under the above assumptions:

Theorem 3.4.1. Suppose χ is a regular integral central character. Then there is an equivalence of categories

M(G(ℝ, γ))_{χ} ≅ Ext_{G^{∨}}X(χ) − Nil.

Proof[Theorem]: We begin by simplifying the right hand side. Abreviate G_{c}^{∨} = G and let B ⊂ G be a Borel subgroup.

Lemma 3.4.2. Suppose χ is regular integral. Then X(χ) ≅ G^{∨} ×_{B} (G/B) for a suitable embedding B ↪ G^{∨}.

Proof: Left to the reader. Next we simplify the right hand side. Let HCh denote the category of “Harish-Chandra bimodules” for 𝔤_{c} = LieG_{c} investigated in [Soe92]. Let Z_{c} ⊂ U − mod − U(𝔤_{c}) be the center and Z_{c}^{+} = Ann_{Z_{c}}ℂ the central annihilator of the trivial representation of 𝔤_{c}. Define H = {M ∈ HCh | (Z_{c}^{+})^{n}M = 0 = M(Z_{c}^{+})^{n} for n ≫ 0} the category of Harish-Chandra bimodules with trivial central character from both sides.

Lemma 3.4.3. For χ regular integral there is an equivalence of categories

M(G(ℝ, γ))_{χ} = M(G_{c}(ℂ))_{χ} ≅ H.

Proof: This is contained in the literature [BG80]. Modulo these two lemmas we see that we need to establish an equivalence of categories H ≅ Ext_{B}(G/B) − Nil. This was a conjecture in [BG86]. It was essentially solved in [Soe92], however I did not write down there the topological part. Merely I constructed in a combinatorial way a graded ℂ-algebra A and proved the equivalence of categories H ≅ A − Nil. So we just have to establish an isomorphism A ≅ Ext_{B}(G/B) of graded ℂ-algebras.

Let T ⊂ B be a maximal torus and W = N_{G}T /T the Weyl group. Then the irreducible objects of P_{B}(G/B) are just the intersection cohomology complexes IC_{x} of the closures of the Schubert varieties \overline{Bx^{−1}B/B}, for x ∈ W. Now by definition Ext_{B}(G/B) = End_{B}^{•}(⊕_{x∈W}IC_{x}). We got to understand this algebra better.

The B-equivariant cohomology ring of G/B is well known to be End_{B}^{•}(\underline{G/B}) = S⊗_{S^{W}} S where S = S((LieT )^{∗}) are the regular functions on LieT , graded in such a way that (LieT )^{∗} is the degree two part of S. Now equivariant hypercohomology is a functor

ℍ_{B} = Hom_{B}^{•}(\underline{G/B}, ) : D_{B}(G/B) → S ⊗_{S^{W}} S − mod

from the equivariant derived category to the category of graded modules over the equivariant cohomology ring.

Proposition 3.4.4. For all x, y ∈ W, equivariant hypercohomology induces an isomorphism

ℍ_{B} : Hom_{B}^{•}(IC_{x}, IC_{y}) → Hom_{S⊗S}(ℍ_{B}IC_{x}, ℍ_{B}IC_{y}).

Proof: Postponed to ??. In particular we deduce an isomorphism

Ext_{B}(G/B) ≅ End_{S⊗S}(⊕_{x∈W}ℍ_{B}IC_{x}).

But the algebra A was defined as

A = End_{S⊗S}(⊕_{x∈W}𝔹_{x})

where the 𝔹_{x} are certain graded S ⊗ S-modules described in a combinatorial way in [Soe92]. So to prove our theorem we need just Lemma 3.4.5. For all x ∈ W there is an isomorphism of graded S⊗Smodules ℍ_{B}IC_{x} ≅ 𝔹_{x}.

Proof: The proof of this lemma is very similar to the proof of its non-equivariant analogon in [Soe90], so I will rather give a sketch of proof.

Let (W, S) be the Coxeter system associated to G ⊃ B and let H = H(W, S) = ⊕_{x∈W}ℤ[t, t^{−1}]T_{x} be the corresponding Hecke algebra. For s ∈ S let P_{s} ⊃ B be the associated parabolic. Let π_{s} : G/B → G/P_{s} be the projection. One may make ⟨D_{B}(G/B)⟩ into a H-module by the prescription t⟨F⟩ = ⟨F[−1]⟩ and (T_{s} +1)⟨F⟩ = ⟨π_{s}^{∗}π_{s∗}F⟩ for all s ∈ S. If {C′_{x}}_{x∈W} is the selfdual basis of the Hecke algebra defined in [KL79], then C′_{x}⟨IC_{e}⟩ = ⟨IC_{x}⟩ for all x ∈ W.

Now let us specify that we choose the identification H_{B}^{•}(G/B) = S ⊗_{S^{W}} S such that the pullback via π_{s} : G/B → G/P_{s} corresponds to the inclusion S^{s} ⊗_{S^{W}} S ↪ S ⊗_{S^{W}} S on equivariant cohomology rings. Furthermore, let us identify graded S ⊗ S-modules with graded S-bimodules in the obvious way. Let S − mof − S denote the category of all graded S-bimodules which are finitely generated as left and as right modules. Then we may interpret ℍ_{B} as a functor

ℍ_{B} : D_{B}(G/B) → S − mof − S,

and use essentially the same arguments as in [Soe90] to prove ℍ_{B}π_{s}^{∗}π_{s∗}F = S ⊗_{S^{s}} (ℍ_{B}F) for s ∈ S, F ∈ D_{B}(G/B).

Now recall from [Soe92] how we made ⟨S−mof −S⟩ into a H-module. The above formula says then just that ℍ_{B} induces a morphism of Hmodules ℍ_{B} : ⟨D_{B}(G/B)⟩ → ⟨S − mof − S⟩. Certainly ℍ_{B}⟨IC_{e}⟩ = ⟨S⟩, thus ℍ_{B}⟨IC_{x}⟩ = ℍ_{B}C′_{x}⟨IC_{e}⟩ = C′_{x}ℍ_{B}⟨IC_{e}⟩ = C′_{x}⟨S⟩ = ⟨𝔹_{x}⟩ by definition of 𝔹_{x}. q.e.d.[Lemma and theorem]

We still have to prove proposition 3.4.4. Let us start with some general considerations. Let X be a topological space filtered by closed subspaces X = X_{0} ⊃ X_{1} ⊃ … ⊃ X_{r} = ∅ and let F ∈ D^{b}(ShX) be an object in the bounded derived category of sheaves on X. To calculate its hypercohomology ℍ^{•}F we have to choose an injective resolution I^{•} of F and take cohomology of the complex of global sections Γ(I^{•}). Certainly this is filtered by supports, 0 ⊂ Γ_{X_{r}}(I^{•}) ⊂ … ⊂ Γ_{X_{0}}(I^{•}) = Γ(I^{•}). So its cohomology is the limit of a spectral sequence with E_{1}- term E_{1}^{p,q} = ℍ_{X_{p}−X_{p+1}}^{p+q}(F) the local hypercohomology of F along X_{p} − X_{p+1}. Let j : Y ↪ X be the embedding of a locally closed subset. Then we can write the local hypercohomology as ℍ_{Y}^{•} (F) = ℍ^{•}v^{!}F, i.e. as the hypercohomology of the derived local cohomology sheaf v^{!}F ∈ D^{+}(ShY). Let j_{p} : X_{p} − X_{p+1} ↪ X denote the inclusion. We may then write our E_{1}-term as E_{1}^{p,q} = ℍ^{p+q}j_{p}^{!}F.

Suppose next that X is a complex algebraic B-variety and that all X_{p} are closed B-stable subvarieties. Applying the above considerations to EB×_{B}X we find that for F ∈ D_{B}(X) the equivariant hypercohomology ℍ_{B}^{•}F is the limit of a spectral sequence with E_{1}-term E_{1}^{p,q} = ℍ_{B}^{p+q}j_{p}^{!}F. In particular, if G ∈ D_{B}(X) is another object the equivariant extensions Hom_{B}^{•}(F, G) = ℍ_{B}^{•}RHom(F, G) are the limit of a spectral sequence with E_{1}-term E_{1}^{p,q} = ℍ_{B}^{p+q}j_{p}^{!}RHom(F, G) = ℍ_{B}^{p+q}RHom(j_{p}^{∗}F, j_{p}^{!}G).

Suppose that X is a B-variety with a B-stable stratification X = ˙⋃_{w∈W}X_{w} into smooth irreducible subvarieties. Put |w| = dimX_{w} and let C_{w} ∈ P_{B}(X_{w}) be the constant perverse sheaf C_{w} = \underline{X_{w}}[|w|]. Let IC_{w} ∈ P_{B}(X) denote the intersection cohomology complex of the closure of X_{w}, extended by zero to X. Let j_{w} : X_{w} ↪ X be the inclusions of the strata and suppose there exist isomorphisms

j_{v}^{∗}IC_{w} ≅ ⊕_{ν}n_{v,w}^{ν}C_{v}[ν] (∗)

in D_{B}(X_{v}), for all v, w ∈ W, with suitable integers n_{v,w}^{ν} defined by these isomorphisms.

Proposition 3.4.6. Suppose we have in addition to the above assumptions parity vanishing, i.e. (1) n_{v,w}^{ν} = 0 unless ν has the same parity as |v| + |w| and (2) H_{B}^{i}(X_{w}) = 0 for all w unless i is even. Then

i.) ℍ_{B}^{•} induces an injection Hom_{B}^{•}(IC_{x}, IC_{y}) → Hom_{ℂ}(ℍ_{B}^{•}IC_{x}, ℍ_{B}^{•}IC_{y}). ii.) dim_{ℂ}Hom_{B}^{ν}(IC_{x}, IC_{y}) = ∑_{a+i+j=ν, z∈W} n_{z,v}^{i}n_{z,w}^{j}dim_{ℂ}H_{B}^{a}(X_{z}).

Proof: ii.) Consider the filtration on X given by X_{p} = ⋃_{|x|+p≤dimX} X_{x}. If we calculate Hom_{B}^{ν}(IC_{x}, IC_{y}) by the associated spectral sequence, we find as E_{1}-term

E_{1}^{p,q} = ℍ_{B}^{p+q}RHom(j_{p}^{∗}IC_{x}, j_{p}^{!}IC_{y})
= ⊕_{|z|+p=dimX}ℍ_{B}^{p+q}RHom(j_{z}^{∗}IC_{x}, j_{z}^{!}IC_{y})
= ⊕_{|z|+p=dimX}ℍ_{B}^{p+q}RHom(⊕_{i}n_{z,x}^{i}C_{z}[i], ⊕_{j}n_{z,y}^{j}C_{z}[−j])
= ⊕_{|z|+p=dimX} ⊕_{i,j} n_{z,x}^{i}n_{z,y}^{j}H_{B}^{p+q−i−j}(X_{z}).

By our parity assumptions this vanishes unless p + q has the same parity as |x| + |y|. In particular, our spectral sequence degenerates at this E_{1}-term and ii.) follows.

i.) Also ℍ_{B}^{i}IC_{x} is the limit of a spectral sequence with E_{1}-term E_{1}^{p,q} = ℍ_{B}^{p+q}j_{p}^{!}IC_{x}, and again by parity vanishing it degenerates at this term. It is sufficient to show that any nonzero morphism f : IC_{x} → IC_{y}[ν] in D_{B}(X) induces for some p a nonzero morphism j_{p}^{!}f : j_{p}^{!}IC_{x} → j_{p}^{!}IC_{y}[ν]. Let i_{p} : X_{p} ↪ X be the inclusion. We may decompose X_{p} into an open and a closed subset

X_{p+1} ↪^{i} X_{p} ↩^{u} (X_{p} − X_{p+1}).

Then we have a distinguished triangle (i_{!}i^{!}, id, u_{∗}u^{∗})i_{p}^{!} = (i_{∗}i_{p+1}^{!}, i_{p}^{!}, u_{∗}j_{p}^{!}) which says that i_{p+1}^{!}f = 0 and j_{p}^{!}f = 0 imply i_{p}^{!}f = 0. So if all j_{p}^{!}f = 0 then i_{0}^{!}f = f = 0. q.e.d. Proof[Proposition 3.4.4]: It will be sufficient to (1) prove injectivity of the map and (2) check that the Dimensions coincide on both sides, where for a graded vector space V = ⊕V^{i} we understand its Dimension to be the polynomial DimV = ∑ t^{i}dim_{ℂ}V^{i}. First we check that the preceding proposition applies in our situation, i.e. for the stratification of X = G/B by the Bruhat cells X_{w} = Bw^{−1}B/B. We need

Lemma 3.4.7. In D_{B}(X_{v}) the j_{v}^{∗}IC_{w} decompose into the direct sum of their perverse cohomology, i.e. j_{v}^{∗}IC_{w} ≅ ⊕_{ν}n_{v,w}^{ν}C_{v}[ν] for suitable integers n_{v,w}^{ν}.

Proof: In the nonequivariant situation this follows from pointwise purity: Namely the j_{v}^{∗}IC_{w} are “pure”, hence the sum of their perverse cohomology groups.

We may reduce our equivariant situation to a nonequivariant situation as follows: First choose a finite interval I ⊂ ℤ such that IC_{w} ∈ D_{B}^{I}(X). Then we have to prove the lemma for the j_{v}^{∗}IC_{w} ∈ D_{B}^{I}(X_{v}). Now choose a smooth n-acyclic free right B-space M, where n is bigger then the length of I. Then by [BL92] we have a commutative diagram

D_{B}^{I}(X) → D^{I}(M ×_{B} X)
j_{v}^{∗} ↓  ↓ (id×j_{v})^{∗}
D_{B}^{I}(X_{v}) → D^{I}(M ×_{B} X_{v})

where the horizontal maps are fully faithful embeddings. But the nature of the singularities is the same for M ×_{B} \overline{X_{w}} and \overline{X_{w}}. Hence the intersection cohomology complex of M ×_{B} \overline{X_{w}} is also pointwise pure and the lemma follows. q.e.d.

Now we may in fact use proposition 3.4.6 to show that our map

ℍ_{B} : Hom_{B}^{•}(IC_{x}, IC_{y}) → Hom_{S⊗S}(ℍ_{B}IC_{x}, ℍ_{B}IC_{y}).

is an injection. So we just have to check that the Dimensions are equal. But

DimHom_{B}^{•}(IC_{x}, IC_{y}) = DimHom_{D}^{•}(IC_{x}, IC_{y}) ⊗_{ℂ} S

by proposition 3.4.6 and

DimHom_{S⊗S}(ℍ_{B}IC_{x}, ℍ_{B}IC_{y}) = DimHom_{S⊗S}(𝔹_{x}, 𝔹_{y}) by
= DimHom_{S⊗S}(𝔹_{x}, 𝔹_{y}) ⊗_{S} ℂ ⊗_{ℂ} S by
= DimHom_{S}(𝔹_{x} ⊗_{S} ℂ, 𝔹_{y} ⊗_{S} ℂ) ⊗_{ℂ} S by
= DimHom_{S}(ℍ^{•}IC_{x}, ℍ^{•}IC_{y}) ⊗_{ℂ} S.

But we know already from [Soe90] that

Hom_{D}^{•}(IC_{x}, IC_{y}) = Hom_{S}(ℍ^{•}IC_{x}, ℍ^{•}IC_{y}).

q.e.d. [Proposition 3.4.4]

3.5. Compact groups. Suppose under the ABV-parametrization some geometric parameter (Y, τ) corresponds to a representation of a compact form of G. Our basic conjecture would say that

Homj_{G_{F}^{∨}} (IC(Y, τ), IC(Y′, τ′)[n]) = 0

unless (Y, τ) = (Y′, τ′) and n = 0. The results of [ABV92] on the duality between IC-data and KL-data tell us already that the above space of homomorphisms vanishes unless (Y, τ) = (Y′, τ′). We will verify in addition that the isotropy group of points in Y is finite. This implies that there are now higher selfextensions of IC(Y, τ).

So let χ ∈ MaxZ be integral regular. Put I = {y ∈ ^{Γ}G^{∨} − G^{∨} | y^{2} = 1}. Clearly X(χ) = I × G^{∨}/B^{∨} with the diagonal G^{∨}-action. In [ABV92] it is shown that I decomposes into a finite number of G^{∨}-orbits, that these orbits are all closed and that the corresponding isotropy groups are just complexifications of maximal compact subgroups of real forms of G^{∨}. More precisely the real forms appearing here are just all real forms of one inner class of real forms on G^{∨}, and this inner class corresponds to the involution (−w_{∘}γ) of our based root system, where γ is the involution of our based root system coming from the choosen inner class of real forms on G and w_{∘} is the longest element of the Weyl group.

Hence if G is endowed with the inner class of real forms containing the compact one, then γ = −w_{∘} so that the corresponding inner class of real forms on G^{∨} contains the split form. In particular some components of X(χ) are isomorphic to G^{∨}/K × G^{∨}/B^{∨} where K is the complexification of a maximal compact subgroup of a split form of G^{∨}. Now the representations of compact forms of G correspond just to open G^{∨}-orbits on these G^{∨}/K × G^{∨}/B^{∨} ⊂ X(χ). And clearly the isotropy groups of points in these open orbits are finite.

3.6. Tempered representations. I want to suggest a link of my basic conjecture with the results of A. Guichardet [Gui92]. He considers an L-packet of irreducible tempered representations and the category T of all Harish-Chandra modules with composition factors only from this Lpacket. He then proves that this category T is equivalent to a category S^{σ}[R] − Nil, for a suitable polynomial ring S with a suitable action σ : R → Aut_{ℂ}S of a suitable finite group R on it.

In the ABV-parametrization a tempered L-packet corresponds to a collection of geometric parameters (Y, τ_{i}) for a fixed open orbit Y ⊂ X(χ). If my basic conjecture is true, there should be an equivalence of categories

T ≅ Ext_{G_{F}^{∨}} X(χ)/I − Nil

where the twosided ideal I is generated by all idempotents which belong to parameters other than the (Y, τ_{i}). Now restriction gives us a map

Ext_{G_{F}^{∨}} X(χ) → Ext_{G_{F}^{∨}} Y

which annihilates I, and by 2.4 the ring Ext_{G_{F}^{∨}} Y has the form S^{σ}[R]. I didn’t check however that these S, σ, R are indeed those of Guichardet.

3.7. The center and geometry. To each additive category A we may associate a commutative ring Z(A), the center of A, as follows: Z(A) consists just of all natural transformations of the identity functor id : A → A to itself, i.e. Z(A) = End(id). Let R be a commutative ring. By an “action of R on A” we mean a ring homomorphism R → Z(A).

Now consider our basic conjecture. On the representation theoretic side of the conjectured equivalence there acts certainly the completion Z_{χ}^{∧} at χ of the center Z ⊂ U − mod − U. I want to explain what this should correspond to on the geometric side. Remark that the construction of X(χ) gives us a G^{∨}-equivariant map X(χ) → F(O(χ)). This leads to a morphism

H_{G^{∨}}^{•}(F(O(χ))) → Ext_{G_{F}^{∨}} X(χ)

of the equivariant cohomology ring of F(O(χ)) to our geometric extension algebra, and this morphism has central image. It should correspond to the action of Z_{χ}^{∧} on the representation theoretic side of the conjectured equivalence via the morphism H_{G^{∨}}^{•}(F(O(χ))) → Z_{χ}^{∧} defined as the composition

H_{G^{∨}}^{•}(F(O(χ))) =^{(1)} H_{G^{∨}}^{•}(O(χ)) =^{(2)} H^{•}(BG_{λ}^{∨}; ℂ) =^{(3)} S^{W_{λ}} →^{(4)} (S_{0}^{∧})^{W_{λ}} →^{(5)} Z_{χ}^{∧}

which we will now explain step by step.

1. This is clear since O(χ) ↠ F(O(χ)) is a fibration with affine spaces as fibres.

2. We choose λ ∈ O(χ) ∩ LieT^{∨} and let G_{λ}^{∨} be the isotropy group of λ. Then this is the induction equivalence.

3. The Weyl group of G_{λ}^{∨} is the stabilizer W_{λ} of λ in W. We let S = S(LieT) be the regular functions on the Lie algebra LieT^{∨} of the maximal torus T^{∨} ⊂ G_{λ}^{∨}. For any linear complex algebraic group the cohomology ring of its classifying space can be identified with the regular functions on the Lie algebra of a maximal torus which are invariant under the Weyl group.

4. We let S_{0}^{∧} be the completion of S at the origin of LieT^{∨}.

5. We will rather construct the inverse morphism. The Harish-Chandra homomorphism ξ : Z → S, ξ(Z) = S^{W} certainly induces a morphism between the respective completions Z_{χ}^{∧} → S_{λ}^{∧}. If we compose with the isomorphism S_{λ}^{∧} → S_{0}^{∧} defined by translation, we get a morphism Z_{χ}^{∧} → S_{0}^{∧}. By some invariant theory this factorizes over an isomorphism Z_{χ}^{∧} → (S_{0}^{∧})^{W_{λ}}. The inverse of this isomorphism is the last map of our above composition.

4. Further conjectures

To provide additional motivation for my basic conjecture, I want to explain how it should be related to character duality [Vog82, ABV92]. Let me cite from [ABV92]. “It is natural to look for a functorial relationship between a category of representations of strong real forms of G, and one of our geometric categories. We do not know what form such a relationship should take, or how one may hope to establish it directly.” I conjecture that the relationship in question should take the form of Koszul duality. However I have no idea how to establish such a relationship directly. I want now to give more details and start by explaining some Koszul duality.

4.1. Generalities on Koszul duality. Let A = ⊕_{i≥0}A_{i} be a graded ℂ-algebra with A_{0} = ⊕_{x∈W}ℂ1_{x}, where W is a suitable finite parameter set. Let A-mod be the category of graded A-modules M = ⊕_{i}M_{i}. We define M⟨1⟩ ∈ A-mod by (M⟨1⟩)_{i} = M_{i−1}. For M, N ∈ A-mod we let hom_{A}(M, N) (resp. Homj_{A}(M, N)) denote morphisms in A-mod (resp. A-Mod). So hom_{A}(M, N) = {f ∈ Hom_{A}(M, N) | f(M_{i}) ⊂ N_{i}}.

Let us form the bounded from above derived category D = D^{−}(A-mod). It comes with three automorphisms [1] “shift of a complex”, ⟨1⟩ “shift of the grading on all terms of a complex” and (1/2) = [−1]⟨−1⟩. The strange notation comes from the fact that later (1/2) should correspond to a “root of the Tate twist”. We will denote morphisms in D by hom_{D}.

We consider two t-structures on D. The “representation theoretic” t-structure (D^{≤0,r}, D^{≥0,r}) is the obvious t-structure with heart A-mod.

The “geometric” t-structure (D^{≤0,g}, D^{≥0,g}) is its “Koszul dual”. More precisely, D^{≤0,g} (resp. D^{≥0,g}) consists of all complexes which are quasiisomorphic to complexes … → X^{i} → X^{i+1} → … of projective objects X^{i} of A-mod such that X^{i} is generated by its parts of degree ≤ −i (resp. ≥ −i).

The twists ⟨1⟩ (resp. (1/2)) respect the representation theoretic (resp. geometric) t-structure. Up to twist the simple objects in the hearts of the respective t-structures are the L_{x} = A_{0}1_{x} (resp. the L^{x} = A1_{x}). Let D_{r} (resp. D_{g}) be the full triangulated subcategories of D generated by the L_{x} (resp. L^{x}) and their twists. The respective t-structures on D induce t-structures (D_{r}^{≤0}, D_{r}^{≥0}) and (D_{g}^{≤0}, D_{g}^{≥0}) on D_{r} and D_{g} respectively.

4.2. Application to our situation. Let us now go back to the situation of our basic conjecture. There we conjectured an equivalence of categories

⨁_{δ∈S_{F}} M(G(ℝ, δ))_{χ} ≅ Ext_{G_{F}^{∨}} (X(χ)) − Nil.

Let us abreviate the left hand side category by HCh and put G_{F}^{∨} = G, X(χ) = X. Let {L_{x}}_{x∈W} and {L^{x}}_{x∈W} represent the simple isomorphism classes in HCh and P_{G}(X) respectively, where the indices are choosen in such a way that objects with the same index correspond to each other under the ABV-parametrization (a) ↔ (b) from the beginning. In particular W is not at all the Weyl group. Finally put A = Ext_{G}X = End_{G}^{•}(⊕L^{x}). This is a graded ℂ-algebra and A_{0} = ⊕_{x∈W}ℂ1_{x} in an obvious way. The basic conjecture predicts an equivalence of categories HCh ≅ A − Nil.

Now construct to A as explained before the triangulated category D = D^{−}(A-mod) with its twists [1], ⟨1⟩, (1/2) and the two t-structures giving rise to two full triangulated subcategories

D_{r} ↪ D ↩ D_{g}.

Conjecture 4.2.1. There exists a t-exact functor v_{r} : D_{r} → D^{b}(HCh) together with canonical isomorphisms v_{r}(M⟨1⟩) = v_{r}M such that

i.) Hom_{D^{b}(HCh)}(v_{r}M, v_{r}N) = ⊕_{i}hom_{D}(M, N⟨i⟩) for all M, N ∈ D_{r}. ii.) v_{r}L_{x} ≅ L_{x} ∀x ∈ W.

Conjecture 4.2.2. There exists a t-exact functor v_{g} : D_{g} → D_{G}(X) together with canonical isomorphisms v_{g}(M(1/2)) = v_{g}M such that

i.) Hom_{G}(v_{g}M, v_{g}N) = ⊕_{i}hom_{D}(M, N(i/2)) for all M, N ∈ D_{g}. ii.) v_{g}L^{x} ≅ L^{x} ∀x ∈ W.

The conjectures mean that the categories D_{r} and D_{g} should be “mixed” or “graded” versions of D^{b}(HCh) and D_{G}(X) respectively, and the functors v_{r}, v_{g} just “forget the grading”. The next conjectures stipulate the existence of graded versions for “standard objects”. Let me explain what I mean. Certainly any IC(Y, τ) is the unique simple quotient of the zero’th perverse cohomology of M^{x} = i_{!}τ[dimY], where i : Y ↪ X is the inclusion of a G-orbit and τ an irreducible G-equivariant local system on Y, hence τ ∈ D_{G}(Y). On the other hand any L_{x} is the socle of a standard object N_{x} ∈ HCh induced from an irreducible tempered representation.

Conjecture 4.2.3. For all x ∈ W there exists M^{x} ∈ D_{g} such that v_{g}M^{x} ≅ M^{x} and hom_{D}(M^{x}, L^{x}) ≠ 0. This M^{x} is unique up to isomorphism.

Conjecture 4.2.4. For all x ∈ W there exists N_{x} ∈ D_{r} such that v_{r}N_{x} ≅ N_{x} and hom_{D}(N_{x}, L^{x}) ≠ 0. This N_{x} is unique up to isomorphism.

The essential ingredient we would need to get the character duality of Vogan’s is the following

Conjecture 4.2.5. We have

hom_{D}(M^{x}(i/2), N_{y}⟨n⟩) = \{ \begin{matrix}ℂ & x = y, i = n = 0 \\ 0 & else. \end{matrix}

Now comes character duality. We make the Grothendieck groups [D_{r}] and [D_{g}] into ℤ[t, t^{−1}]-modules by the prescription t[M] = [M⟨1⟩] for M ∈ D_{r}, respectively t[M] = −[M(1/2)] = [M⟨−1⟩] for M ∈ D_{g}. We obtain a ℤ[t, t^{−1}]-bilinear pairing [D_{g}] × [D_{r}] → ℤ[t, t^{−1}] between these two Grothendieck groups by the prescription

(M, N) ↦ ⟨M, N⟩ = ∑_{ν,i}(−1)^{ν}dim_{ℂ}hom_{D}(M, N[ν]⟨−i⟩)t^{i}.

Certainly [D_{g}] is a free ℤ[t, t^{−1}]-module with two bases {[L^{x}]}_{x∈W} and {[M^{x}]}_{x∈W}, which we call the simple and the standard basis. Also [D_{r}] is a free ℤ[t, t^{−1}]-module with “simple” basis {[L_{x}]}_{x∈W} and “standard” basis {[N_{x}]}_{x∈W}. It is clear from the definitions that ⟨L^{x}, L_{y}⟩ = δ_{x,y} and if we believe conjecture 4.2.5 we find as well ⟨M^{x}, N_{y}⟩ = δ_{x,y}.

Let now P_{r} be the transition matrix from the simple basis {[L_{x}]} to the standard basis {[N_{x}]}. So P_{r} is a W × W-matrix with coefficients (P_{r})_{x,y} ∈ ℤ[t, t^{−1}] given by the equation

[N_{y}] = ∑_{x}(P_{r})_{x,y}[L_{x}].

Similarily we define the geometric base change matrix P_{g} by the equation

[M^{z}] = ∑_{x}(P_{g})_{u,z}[L^{u}]

in [D_{g}]. Now our formulas give us

δ_{z,y} = ⟨M^{z}, N_{y}⟩
= ∑_{u,x}(P_{g})_{u,z}(P_{r})_{x,y}⟨L^{u}, L_{x}⟩
= (P_{g}^{T} P_{r})_{z,y}

where the upper index T transposes the matrix. In other words, we find the equation

P_{g}^{T} P_{r} = 1

of W × W-matrices with coefficients in ℤ[t, t^{−1}].

This equation is essentially Vogan’s character duality. To see this, write (P_{r})_{x,y} = ∑ m_{r}(x, y; i)t^{i} so that [N_{y}] = ∑_{x,i} m_{r}(x, y; i)[L_{x}⟨i⟩]. Applying v_{r} to both sides, we see that if we set t = 1 in P_{r} then P_{r}(1) is just the representation theoretic multiplicity matrix, (P_{r}(1))_{x,y} = m_{r}(x, y) in the notation of [ABV92], (1.21). On the other hand, write (P_{g})_{u,z} = ∑ m_{g}(u, z; j)t^{j} so that [M^{z}] = ∑_{u,j} m_{g}(u, z; j)(−1)^{j}[L^{u}(j/2)]. Applying v_{g} to both sides, we see that if we set t = −1 in P_{g} then P_{g}(−1) is just the intersection cohomology matrix, (P_{g}(−1))_{u,z} = m_{g}(u, z) in the notation of [ABV92], (1.22)(d).

To finish, we have to assume some parity vanishing. More precisely, for x ∈ W let d(x) denote the dimension of the corresponding G-orbit on X.

Conjecture 4.2.6. We have m_{g}(u, z; j) = 0 unless j has the same parity as d(u) + d(z).

From this we can deduce (P_{g}(1))_{u,z} = (−1)^{d(u)+d(z)}(P_{g}(−1))_{u,z} = (−1)^{d(u)+d(z)}m_{g}(u, z) and thus specializing our equation P_{g}^{T} P_{r} = 1 to t = 1 gives precisely Vogan’s character duality

∑_{u}(−1)^{d(u)+d(z)}m_{g}(u, z)m_{r}(u, x) = δ_{z,x}

as formulated in [ABV92], Corollary 1.25. a).

4.3. Motivation. The first of these conjectures is almost our basic conjecture. For the other conjectures there is very little hard evidence. At least for tori everything works, and for complex groups one can find results in the spirit of the above conjectures in [BGS96].

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.

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

[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, Ext-duality 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.

[BL92] Joseph N. Bernstein and Valery Lunts. Equivariant sheaves I (preliminary version, third draft). Preprint, 1992.

[Gui92] Alain Guichardet. Géometrie différentielle locale sur le dual tempéré d’un groupe de Lie semi-simple. Preprint, 1992.

[Kho80] S. M. Khoroshkin. Category of Harish-Chandra modules of the group SU(n, 1). Funct. Anal. Appl., 14:153–155, 1980.

[KL79] David Kazhdan and George Lusztig. Representations of Coxeter groups and Hecke algebras. Inventiones, 53:191–213, 1979.

[Lan73] Robert P. Langlands. On the classification of irreducible representations of real algebraic groups. Preprint (appeared in: Representation theory and harmonic analysis on semisimple Lie groups, P. J. Sally, Jr. and D. A. Vogan, Jr., editors, AMS 1989), 1973.

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