GRADED HECKE ALGEBRAS AND EQUIVARIANT CONSTRUCTIBLE SHEAVES
Maarten Solleveld
IMAPP, Radboud Universiteit Nijmegen
Heyendaalseweg 135, 6525AJ Nijmegen, the Netherlands email: m.solleveld@science.ru.nl
Abstract. The local Langlands correspondence matches irreducible representa- tions of a reductive p-adic group G(F ) with enhanced L-parameters. It is con- jectured by Hellmann and Zhu that it can be categorified. That should make it a fully faithful functor from a derived category of representations to a derived category of equivariant sheaves on some variety of L-parameters. We study this conjecture for finite length G(F )-representations, and prove it for large classes of those.
In this setup the conjecture runs via graded Hecke algebras associated to Bern- stein components or to enhanced L-parameters. We work with graded Hecke alge- bras H on the Galois side, those can be constructed entirely in terms of a complex reductive group, endowed with data from L-parameters.
We fix an arbitrary central character (σ, r) of H (which encodes the image of Frobenius by an L-parameter). That leads to a variety gσ,rN of nilpotent elements in the Lie algebra of G (possibilities for the monodromy operator from an L- parameter) and to a complex of equivariant constructible sheaves KN,σ,r on gσ,rN . We relate the (derived) endomorphism algebra of (gσ,rN , KN,σ,r) to a localization of H, which yields an equivalence between the appropriate categories of finite length modules of these algebras. From there we construct a fully faithful functor between:
• the bounded derived category of finite length H-modules specified by the central character (σ, r),
• the equivariant bounded derived category of constructible sheaves on gσ,rN . Also, we explicitly determine the images of standard modules under this functor.
Combining this with previous work, we obtain similar results for many rep- resentations of reductive p-adic groups. We expect that these results pave the way for more general instances of the aforementioned conjectural extension of the local Langlands correspondence.
Contents
Introduction 2
1. Notations 9
2. Graded Hecke algebras 9
3. Cuspidal local systems and equivariant cohomology 14
Date: June 18, 2021.
2010 Mathematics Subject Classification. 20C08, 22E57, 14F08, 20G20.
The author is supported by a NWO Vidi grant ”A Hecke algebra approach to the local Langlands correspondence” (nr. 639.032.528).
1
4. Localization at a central character 21
5. Standard modules 25
6. Structure of the localized complexes KN,σ,r 31
7. A functor from sheaves to Hecke algebra modules 36 8. A functor from Hecke algebra modules to sheaves 41
9. Twisted graded Hecke algebras with a fixed r 44
Appendix A. Compatibility with parabolic induction 49 Appendix B. Localization in equivariant cohomology 55
References 56
Introduction
The story behind this paper started with with the seminal work of Kazhdan and Lusztig [KaLu]. They showed that an affine Hecke algebra H is naturally isomorphic with a K-group of equivariant coherent sheaves on the Steinberg variety of a complex reductive group. (Here H has a formal variable q as single parameter and the reductive group must have simply connected derived group.) This isomorphism enables one to regard the category of equivariant coherent sheaves on that particular variety as a categorification of an affine Hecke algebra. Later that became quite an important theme in the geometric Langlands program, see for instance [Bez].
This paper is inspired by the quest for a generalization of such a categorification of H to affine Hecke algebras with more than one q-parameter. That is relevant because such algebras arise in abundance from reductive p-adic groups and types [ABPS3,
§2.4]. However, up to today it is unclear how several independent q-parameters can be incorporated in a setup with equivariant K-theory or K-homology. The situation improves when one localizes an affine Hecke algebra with respect to (the kernel of) a central character, as in [Lus2]. Such a localization is Morita equivalent with a localization of a graded Hecke algebra with respect to a central character.
Graded Hecke algebras H with several parameters (now typically called k) do admit a geometric interpretation [Lus1, Lus3]. (Not all combinations of parameters occur though, there are conditions on the ratios between the different k-parameters.) For this reason graded Hecke algebras, instead of affine Hecke algebras, play the main role in this paper.
The appropriate geometric objects are still equivariant sheaves on a variety asso- ciated to a complex reductive group, but now the sheaves are constructible and one considers their equivariant cohomology instead of their K-theory. Results in [Lus3]
strongly suggest that the module category of H is equivalent with some category of equivariant constructible sheaves. We will make this precise by involving derived categories. That can be regarded as a geometric categorification of H, albeit of a different kind as that from [KaLu, Bez].
Motivation from the local Langlands program
Consider a reductive group G defined over a non-archimedean local field F . We de- note its complex dual group by G∨and its Langlands dual group byLG = G∨o WF. Let Rep(G(F )) be the category of smooth complex G(F )-representations and let Irr(G(F )) be its set of irreducible objects. The local Langlands correspondence
(LLC) predicts a nice bijection between Irr(G(F )) and the set of G(F )-relevant en- hanced L-parameters with value in LG. Starting from [Vog], over the years it has become clear that the LLC may admit a categorification, which
(1) relates Rep(G(F )) to coherent sheaves on a variety of L-parameters for G(F ).
It is expected that one should work with a stack of L-parameters and with derived categories of coherent sheaves on that. Then the (bounded) derived category of Rep(G(F )) should admit a fully faithful functor to a (bounded) derived category of such sheaves. We refer to [Hel, Zhu, FaSc] for the technical conjectures and more background.
In [Hel] such an equivalence of derived categories is worked out for Iwahori- spherical representations of GL2(F ), via an affine Hecke algebra. Shortly after that it was proven for GLn(F ) in [BCHN], with techniques from derived algebraic geometry. Of course, this picture calls for generalization. In fact, many of the pieces are already in places for arbitrary L-parameters and arbitrary G(F )-representations.
Let us explain that, and at the same time sketch why Hecke algebras are so ubiqui- tous in these matters.
Galois side
First one defines the cuspidal support Sc(φ, ρ) of an enhanced L-parameter (ψ, ρ) [AMS1, §7]. That gives rise to a partition of the space Φe(LG) of enhanced L- parameters (with values in LG) into Bernstein components Φe(LG)s∨ [AMS1, §8].
To every bounded element (φ, q) of the cuspidal support of Φe(LG)s∨ one associates a (twisted) graded Hecke algebra Hφ,q, defined in terms of equivariant constructible sheaves, see [AMS2, §4] and [AMS3, §3.1]. The family of algebras
{Hφ,q: (φ, q) as above}
can be glued to one (twisted) affine Hecke algebra Hs∨ [AMS3, §3.3]. The construc- tions are such that, upon specializing the parameters of Hs∨ to an array of positive real numbers q, there is a natural bijection
Irr(Hs∨(q)) ←→ Φe(LG)s∨.
Here the subset of Φe(LG)s∨ with one fixed cuspidal support (φ0, q) corresponds to the set of irreducible representations of Hs∨(q) with one fixed central character:
(2) Irrφ0,q(Hs∨(q)) ←→ Sc−1(φ0, q).
The choice of q induces a specialization of the parameters of Hφ,q, which we denote briefly by log q. Then the sets in (2) are also naturally in bijection with the subset Irrφ0,q(Hφ,q(log q)) of Irr(Hφ,q(log q)) picked out by one central character. (Here φ0 determines φ as its ”bounded part”.)
P -adic side
Consider an arbitrary Bernstein block Rep(G(F ))s of Rep(G(F )). Bernstein [BeRu, Ren] exhibited a progenerator Πs of Rep(G)s. By a standard result from cate- gory theory, Rep(G(F ))s is equivalent with EndG(F )(Πs) − Mod. Let M be a Levi subgroup of G such that the supercuspidal support of Rep(G)s can be realized in Irr(M(F )). Every tempered M(F )-representation τ in that supercuspidal support of gives rise, via localizations of EndG(F )(Πs), to a twisted graded Hecke algebra Hτ
[Sol5, §7]. By construction Hτ − Mod describes a well-defined part of Rep(G(F ))s. Under mild conditions, the family of algebras Hτ can be glued to one twisted affine Hecke algebra Hs, which is (almost) Morita equivalent with EndG(F )(Πs) [Sol5, §10].
Local Langlands Correspondence
Suppose now that a LLC for G matches a Bernstein component Irr(G)s with a Bern- stein component Φe(LG)s∨. It can be expected that:
(i) each twisted graded Hecke algebra Hφ,q(log q) is isomorphic with Hτ, where τ has L-parameter (φ, q) ∈ Φe(LM);
(ii) for a suitable choice of q, Hs∨(q) is (almost) Morita equivalent with Hs, and hence with EndG(F )(Πs). In other words, Hs∨(q) − Mod should be equivalent with Rep(G(F ))s.
Indeed, many earlier results about Bernstein components, types and Hecke algebras, e.g. [ABPS2, Hei, Lus4, Roc, S´eSt, Sol4], can be interpreted as confirmations of cases of this expectation. In general all this remains conjectural, because we do not yet have a complete local Langlands correspondence. One may hope that requiring (i) and (ii) may help to determine a LLC in new cases, like for unipotent representations [Lus4, Lus6, Sol3].
Independently, the parameters of the Hecke algebras Hs and Hτ can be investi- gated. In many cases they can be determined [Sol7], and in all those instances they agree with the parameters of some Hecke algebras on the Galois side (or equiva- lently, with the parameters of a Hecke algebra for a Bernstein block of unipotent representations.)
To attack the conjecture (1) from [Hel, Zhu], the above suggests a strategy:
(a) match Hecke algebras on the p-adic and Galois sides of the LLC;
(b) relate Hs to sheaves on a stack of L-parameters coming from Φe(LG)s∨. As a step in that direction, we consider the subcases of (b) obtained by restricting to one infinitesimal central character. In the end, these should account for all finite length G(F )-representations.
Let τ0 ∈ Irr(M(F )) be a twist of τ by an unramified character χ : M(F ) → R>0. The pair (M(F ), τ0) defines a character of the Bernstein centre of G(F ) and a sub- category Rep(G(F ))τ0, consisting of those G(F )-representations all whose irreducible subquotients have supercuspidal support conjugate to (M(F ), τ0). Similarly τ0 de- termines a central character of Hτ, and a subcategory Hτ−Modτ0. The subcategories of finite length objects in Rep(G(F ))τ0 and in Hτ− Modτ0 are equivalent [Sol5].
The Langlands parameter of τ0 should be a twist ( ˆχφ, q) of (φ, q) by the Lang- lands parameter ˆχ of χ. Granting the above expectation (i), Hτ − Modτ0 will be equivalent with Hφ,q(log q) − Modφ0,q. In this way (1) motivates the main goal of the paper:
find a fully faithful functor from Hφ,q(log q)−Modφ0,qto some category of sheaves, maybe with derived categories.
Geometric graded Hecke algebras
Let us describe the involved algebras a bit better, following [AMS2, AMS3]. Con- sider φ ∈ Φe(LM) as a group homomorphism WF × SL2(C) → M∨ o WF. The
group ZG∨(φ(WF)) has a subgroup ZM∨(φ(WF)), both reductive and possibly dis- connected. The data φ|SL2(C), q give rise to an equivariant cuspidal local system qE on a unipotent orbit in ZM∨(φ(WF)).
We change notations, replacing ZG∨(φ(WF)) by an arbitrary complex reduc- tive group G and ZM∨(φ(WF)) by a quasi-Levi subgroup of G. Thus we remove the dependence on WF, and entirely can now be phrased entire complex reduc- tive groups. To the data (G, M, qE ) we attach a twisted graded Hecke algebra H = H(G, M, qE ) = Hφ,q. That brings us, finally, to the actual setting of the pa- per. We have a family of graded Hecke algebras which is defined purely in terms of complex geometry, and likewise our results and results will come in such terms.
At the same time, all these algebras are solidly rooted in the representation theory of reductive p-adic group. To all appearances, our algebras represent the general case of a graded Hecke algebra associated to a Bernstein block for a reductive p-adic group.
Main results
We will work in DG×C×(X), the equivariant (bounded) derived category of con- structible sheaves on a complex variety X [BeLu]. In [Lus1, Lus3, AMS2] an impor- tant object K ∈ DG×C×(g) was constructed from qE , by a process that bears some similarity with parabolic induction. Let gN be the set of nilpotent elements in the Lie algebra g of G and let KN be the pullback of K to gN. Up to degree shifts, both K and KN are direct sums of simple perverse sheaves.
Theorem A. (see Theorem 3.2)
There exist natural isomorphisms of graded algebras H(G, M, qE ) −→ End∗D
G×C×(g)(K) −→ End∗D
G×C×(gN)(KN).
The object KN generates (by the operations cones, degree shifts and taking direct summands) a full subcategory hKNi of DG×C×(gN). Let H(G, M, qE) − Modfgp be the category of finitely generated projective right H(G, M, qE)-modules, and indicate its bounded derived category by a D. It follows quickly from Theorem A that the functor Hom∗D
G×C×(gN)(KN, ?) induces an equivalence of categories (3) hKNi −→ D(H(G, M, qE) − Modfgp).
However, this does not yet fit well with the LLC. The problem is that the variety gN is too small: it sees only nilpotent elements of g, and those only determine a part of an L-parameter. We bring in a semisimple element (related to the image of a Frobenius element under an L-parameter) by localizing H(G, M, qE) with respect to a central character.
As a vector space, H(G, M, qE) is the tensor product of C[WqE] (for a certain finite group WE, generalizing a Weyl group) and O(t ⊕ C), where t = Lie(Z(M )) and O(C) = C[r] comes from the C×-actions. The centre of H(G, M, qE) can be described as O(t)WqE ⊗CC[r].
In these terms, the set of enhanced L-parameters Φe(LG)s∨ becomes a set of triples (σ, y, ρ), where σ ∈ g is semisimple, y ∈ g is nilpotent, [σ, y] = 0 and ρ is a very specific kind of irreducible representation of the component group of (σ, y). One may also replace the condition [σ, y] = 0 by [σ, y] = 2ry for a fixed r ∈ C. We recall from [AMS2] that the conjugacy classes of such triples parametrize both the irreducible and the standard modules of H(G, M, qE)/(r − r), for any r ∈ C.
We fix (σ, r) ∈ t ⊕ C and consider it as a central character of H(G, M, qE). We denote the corresponding completion of Z(H(G, M, qE)) by ˆZ(H(G, M, qE))σ,r. In the process of localization, G × C× will be replaced by ZG(σ) × C× and gN by
gσ,rN := {y ∈ gN : [σ, y] = 2ry}.
A variation on the construction of KN yields an object KN,σ,r ∈ DZ
G(σ)×C×(gσ,rN ).
Since (σ, r) ∈ Lie(ZG(σ) × C×), it defines a character of HZ∗
G(σ)×C×(pt) ∼= O Lie(ZG(σ) × C×)ZG(σ)×C×
, and we can again complete with respect to (σ, r).
Theorem B. (see Theorem 4.4) There is a natural algebra isomorphism
Z(H(G, M, qE))ˆ σ,r ⊗
Z(H(G,M,qE))H(G, M, qE ) −→∼ HˆZ∗
G(σ)×C×(pt)σ,r ⊗
H∗
ZG(σ)×C×(pt)
End∗D
ZG(σ)×C×(gN,σ,r)(KN,σ,r).
This induces an equivalence of categories H(G, M, qE ) − Modfl,σ,r∼= End∗D
ZG(σ)×C×(gσ,rN )(KN,σ,r) − Modfl,σ,r.
When G is connected, Theorem B is due to Lusztig [Lus3]. Let hKN,σ,ri be the full subcategory of DZG(σ)×C×(gN,σ,r) generated by KN,σ,r. Generalizing (3), we obtain:
Theorem C. (see Theorem 8.1) The functor Hom∗D
ZG(σ)×C×(gN,σ,r)(KN,σ,r, ?) gives an equivalence of categories hKN,σ,ri −→ D End∗D
ZG(σ)×C×(gσ,rN )(KN,σ,r) − Modfgp.
Now another problems arises: it is unclear whether all End∗D
ZG(σ)×C×(gσ,rN )(KN,σ,r)- modules, or even those of finite length, admit a finite type projective resolution.
Therefore the target in Theorem C could be much smaller than desired. With substantial effort, we show that at least the objects of End∗D
ZG(σ)×C×(gσ,rN )(KN,σ,r) − Modfl,σ,r have such a resolution. That can be combined with Theorems B and C.
Theorem D. (see Proposition 8.4) There exist derived functors
Hom∗D
ZG(σ)×C×(gσ,rN )(KN,σ,r, ?) : DZG(σ)×C×(gσ,rN ) −→
D ˆZ(H(G, M, qE))σ,r ⊗
Z(H(G,M,qE))H(G, M, qE ) − Mod
⊗ KN,σ,r : D H(G, M, qE) − Modfl,σ,r −→ DZG(σ)×C×(gσ,rN ) such that ⊗KN,σ,r is fully faithful and the composition
Hom∗D
ZG(σ)×C×(gσ,rN )(KN,σ,r, ? ⊗ KN,σ,r)
is naturally equivalent with the identity on D H(G, M, qE) − Modfl,σ,r.
In the context of Theorem D, we also establish a variant of the Kazhdan–Lusztig conjecture [Vog, §8]. It expresses the multiplicity of an irreducible H-module in a standard H-module in terms of the associated equivariant sheaves (Proposition 6.4).
In relation with the LLC it is preferable to compose both functors in Theorem D with the sign automorphism of H = H(G, M, qE), for only then tempered represen- tations correspond to derived sheaves supported on bounded L-parameters. That yields functors
(4)
Fσ,r : DZ
G(σ)×C×(gσ,−rN ) −→ D ˆZ(H)σ,r ⊗
Z(H)H − Mod Fσ,r− : D H − Modfl,σ,r
−→ DZ
G(σ)×C×(gσ,−rN ) with properties as in Theorem D.
Recall that graded Hecke algebras coming from reductive p-adic groups have r specialized to a positive real number. Hence we need a version of the above results for H(G, M, qE)/(r − r), with r ∈ C. As r came from the C×-actions, a natural attempt is to replace G × C×-equivariance by G-equivariance. That does not work well directly in Theorem A or in (2), only after localization.
Theorem E. (see Section 9)
Theorems B, C, D and (4) become valid for H(G, M, qE)/(r − r) once we forget the C×-equivariant structure everywhere.
We note that for r = log(qF)/2:
gσ,−rN = {y ∈ gN : [σ, y] = − log(qF)y} = {y ∈ gN : Ad(exp σ)y = q−1F y}.
Here (exp σ, y) defines an unramified L-parameter φ : WF o C → G, with exp(σ) the image of a Frobenius element of WF. In this way the image of Fσ,log(q−
F)/2 can be translated to ZG(σ)-equivariant derived sheaves on a variety of L-parameters φ with φ|WF fixed.
Finally, we derive consequences for the actual representation theory of reductive p- adic groups. We formulate them with the notations G(F ), M(F ), τ, φ, ρ from earlier in this introduction. Then G arises as a cover of ZG∨(φ(WF)) and (M, qE ) comes from (LM, φ, ρ).
Theorem F. (see Theorem 9.5)
Consider a Bernstein block of Rep(G(F )) in one of the following classes:
• inner forms of general linear groups,
• inner forms of special linear groups,
• symplectic and quasi-split special orthogonal groups,
• principal series representations of split groups (with minor conditions on the residual characteristic of F ),
• unipotent representations (of arbitrary reductive groups over F ).
Let τ ∈ Irr(M(F )) be a tempered representation in the cuspidal support of this Bernstein block. Write r = log(qF)/2 and fix σ ∈ Hom(M(F ), R).
(a) There exists equivalence of categories
Repfl(G(F ))τ ⊗exp(σ)∼= Modfl,σ End∗D
ZG(σ)(gσ,−rN )(KN,σ,−r).
(b) The Kazhdan–Lusztig conjecture [Vog, Conjecture 8.11] holds for irreducible and standard representations in Repfl(G(F ))τ ⊗exp(σ).
(c) There exists a fully faithful functor
Fσ−: D Repfl(G(F ))τ ⊗exp(σ) −→ DZG(σ)(gσ,−rN ) and the image of any standard module can be determined explicitly.
Although it may be superfluous in view of the title, we stress that these sheaves are constructible rather than coherent. Of course this is a consequence of the entire setup of the paper, starting already with the construction of H(G, M, qE). On the other hand, most of the equivariant sheaves we encountered are supported on only finitely many G-orbits, so large and by they are determined by their stalks at finitely many points. In such situations coherence or not does not make much of difference.
We hope that in subsequent work our results can be generalized to finitely gener- ated modules of affine Hecke algebras, and that will most probably involve coherent sheaves.
Structure of the paper
We start with recalling (twisted) graded Hecke algebras in terms of generators and relations. We generalize a few results from [Sol6], which say that the set of irre- ducible representations of a a graded Hecke algebra is essentially independent of the parameters k and r. In Section 3 we recall the construction and fundamental prop- erties of graded Hecke algebras associated to complex reductive groups and cuspidal local systems.
Then (Section 4) we turn to localizing these algebras, generalizing [Lus3]. We show first how to replace the group G × C×by ZG(σ) × C×, and then how to replace gby gσ,r and K by Kσ,r. Our investigations revealed a technical problem in [Lus3], which is resolved in Appendix B.
The next three sections are mainly dedicated to the problem observed after The- orem C: the scarcity of finite type projective resolutions. We will approach this with standard H(G, M, qE)-modules, which is reasonable because those generate the derived category of finite length H(G, M, qE)-modules. After recalling some results about standard (left or right) modules from [AMS2], we discuss more con- venient ways to realize them in Section 5. One involves the sign automorphism of H(G, M, E), the other is in terms close to Theorem B. For every standard right H(G, M, qE )-module Ey,σ,r,ρ∨ with central character (σ, r) and y ∈ gσ,rN , we construct (Section 7) an explicit object
(5) jN ∗Sy( ˜Λ∗y,ρ) ∈ DZG(σ)×C×(gσ,rN ) that is mapped to Ey,σ,r,ρ∨ by the functor Hom∗D
ZG(σ)×C×(gσ,rN )(KN,σ,r, ?). This in- volves a Koszul resolution for the algebra HZ∗◦
G(σ,y)(pt). In our proof of Theorem D we need that jN ∗Sy( ˜Λ∗y,ρ) belongs to hKN,σ,ri. The purpose of Section 6 is to gather a supply of objects of hKN,σ,ri, sufficient to construct jN ∗Sy( ˜Λ∗y,ρ) in that subcategory. With that in order we prove Theorem D and (4) in Section 8.
In the final section we specialize r to r ∈ C. We show that most of the earlier results remain valid if replace H(G, M, qE) by H(G, M, qE)/(r − r) and forget the C×-equivariance of our sheaves. By recalling the appropriate parts of the literature, we derive Theorem F.
Appendix A is dedicated to the relation between standard modules and parabolic induction for graded Hecke algebras. The treatment of this topic in [AMS2] omitted
the most general case (both G and M disconnected) and contained a mistake. We provide the missing proofs. Although these results are not really needed here, we include them because they are used in [AMS3, Sol3], both of which are important in the motivation of this work.
Acknowledgements
We thank George Lusztig, Eugen Hellmann and David Ben-Zvi for some helpful conversations.
1. Notations
Most of our notations can be found in Sections 2 and 3. Here we mention a few conventions that are not so common.
We denote the linear dual of a finite dimensional vector space V by V∨ (we avoid V∗ because * is already used heavily for degrees.) More generally we denote the dual of a local system L by L∨.
We denote the identity component of a topological/algebraic group G by G◦. If G acts on a set X and Y ⊂ X, then we write ZG◦(Y ) for ZG(Y )◦.
For a commutative algebra A and a character σ of A, we let ˆAσ be the (formal) completion of A with respect to the powers of ker σ.
For a differential complex C∗ and m ∈ Z the degree shift C∗[m] is the differential complex with Cn+m in degree n.
All the derived categories in this paper are bounded derived categories. We will indicate this just with a D (no subscript b).
2. Graded Hecke algebras
Let a be a finite dimensional Euclidean space and let W be a finite Coxeter group acting isometrically on a (and hence also on a∗). Let R ⊂ a∗ be a reduced integral root system, stable under the action of W , such that the reflections sα with α ∈ R generate W . These conditions imply that W acts trivially on the orthogonal complement of RR in a∗.
Write t = a ⊗RC and let S(t∨) = O(t) be the algebra of polynomial functions on t. We also fix a base ∆ of R. Let Γ be a finite group which acts faithfully and orthogonally on a and stabilizes R and ∆. Then Γ normalizes W and W o Γ is a group of automorphisms of (a, R). We choose a W o Γ-invariant parameter function k : R → C. Let r be a formal variable, identified with the coordinate function on C (so O(C) = C[r]).
Let \ : Γ2 → C× be a 2-cocycle and inflate it to a 2-cocycle of W o Γ. Recall that the twisted group algebra C[W o Γ, \] has a C-basis {Nw : w ∈ W o Γ} and multiplication rules
Nw· Nw0 = \(w, w0)Nww0. In particular it contains the group algebra of W . Proposition 2.1. [AMS2, Proposition 2.2]
There exists a unique associative algebra structure on C[W o Γ, \] ⊗ O(t) ⊗ C[r] such that:
• the twisted group algebra C[W o Γ, \] is embedded as subalgebra;
• the algebra O(t) ⊗ C[r] of polynomial functions on t ⊕ C is embedded as a subalgebra;
• C[r] is central;
• the braid relation Nsαξ −sαξNsα = k(α)r(ξ −sαξ)/α holds for all ξ ∈ O(t) and all simple roots α;
• NwξNw−1=wξ for all ξ ∈ O(t) and w ∈ Γ.
We denote the algebra from Proposition 2.1 by H(t, W o Γ, k, r, \) and we call it a twisted graded Hecke algebra. It is graded by putting C[W o Γ, \] in degree 0 and t∨\ {0} and r in degree 2. When Γ is trivial, we omit \ from the notation, and we obtain the usual notion of a graded Hecke algebra H(t, W, k, r).
Notice that for k = 0 Proposition 2.1 yields the crossed product algebra (2.1) H(t, W o Γ, 0, r, \) = C[r] ⊗CO(t) o C[W o Γ, \],
with multiplication rule
NwξNw−1 =wξ w ∈ W o Γ, ξ ∈ O(t).
It is possible to scale all parameters k(α) simultaneously. Namely, scalar multipli- cation with z ∈ C× defines a bijection mz : t∨ → t∨, which clearly extends to an algebra automorphism of S(t∨). From Proposition 2.1 we see that it extends even further, to an algebra isomorphism
(2.2) mz : H(t, W o Γ, zk, r, \) → H(t, W o Γ, k, r, \)
which is the identity on C[W o Γ, \] ⊗CC[r]. Notice that for z = 0 the map mz is well-defined, but no longer bijective. It is the canonical surjection
H(t, W o Γ, 0, r, \) → C[W o Γ, \] ⊗CC[r].
One also encounters versions of H(t, W o Γ, k, r, \) with r specialized to a nonzero complex number. In view of (2.2) it hardly matters which specialization, so it suffices to look at r 7→ 1. The resulting algebra H(t, W o Γ, k, \) has underlying vector space C[W o Γ, \] ⊗CO(t) and cross relations
(2.3) ξ · sα− sα· sα(ξ) = k(α)(ξ − sα(ξ))/α α ∈ ∆, ξ ∈ S(t∨).
Since Γ acts faithfully on (a, ∆), and W acts simply transitively on the collection of bases of R, W o Γ acts faithfully on a. From (2.3) we see that the centre of H(t, W o Γ, k, \) is
(2.4) Z(H(t, W o Γ, k, \)) = S(t∨)W oΓ = O(t/W o Γ).
As a vector space, H(t, W o Γ, k, \) is still graded by deg(w) = 0 for w ∈ W o Γ and deg(x) = 2 for x ∈ t∨ \ {0}. However, it is not a graded algebra any more, because (2.3) is not homogeneous in the case ξ = α. Instead, the above grading merely makes H(t, W o Γ, k, \) into a filtered algebra. The graded algebra associated to this filtration is obtained by setting the right hand side of (2.3) equal to 0. In other words, the associated graded of H(t, W o Γ, k, \) is the crossed product algebra (2.1).
Graded Hecke algebras can be decomposed like root systems and reductive Lie algebras. Let R1, . . . , Rdbe the irreducible components of R. Write a∨i = span(Ri) ⊂ a∨, ti= HomR(a∨i, C) and z = R⊥ ⊂ t. Then
(2.5) t= t1⊕ · · · ⊕ td⊕ z.
The inclusions W (Ri) → W (R), t∨i → t∨and z∨ → t∨induce an algebra isomorphism (2.6) H(t1, W (R1), k) ⊗C· · · ⊗CH(td, W (Rd), k) ⊗CO(z) −→ H(t, W, k).
The central subalgebra O(z) ∼= S(z∨) is of course very simple, so the study of graded Hecke algebras can be reduced to the case where the root system R is irreducible.
Now we list some isomorphisms of (twisted) graded Hecke algebras that will be useful later on. For any z ∈ C×, H(t, W o Γ, k, r, \) admits a ”scaling by degree”
automorphism
(2.7) x 7→ znx if x ∈ H(t, W o Γ, k, r, \) has degree 2n.
Extend the sign representation to a character sgn of W o Γ, trivial on Γ. We have the Iwahori–Matsumoto and sign involutions
(2.8)
IM : H(t, W o Γ, k, r, \) → H(t, W o Γ, k, r, \) IM(Nw) = sgn(w)Nw, IM(r) = r, IM(ξ) = −ξ sgn : H(t, W o Γ, k, r, \) → H(t, W o Γ, k, r, \)
sgn(Nw) = sgn(w)Nw, sgn(r) = −r, sgn(ξ) = ξ w ∈ W o Γ, ξ ∈ t∨. Upon specializing r = 1, these induce isomorphisms
IM : H(t, W oΓ, k, \) → H(t, W oΓ, k, \), sgn : H(t, W oΓ, k, \) → H(t, W oΓ, −k, \) More generally, we can pick a sign (sα) for every simple reflection sα ∈ W , such that (sα) = (sβ) if sα and sβ are conjugate in W o Γ. Then extends uniquely to a character of W o Γ trivial on Γ (and every character of W o Γ which is trivial on Γ has this form). Define a new parameter function k by
k(α) = (sα)k(α).
Then there are algebra isomorphisms (2.9)
φ : H(t, W o Γ, k, r, \) → H(t, W o Γ, k, r, \), φ : H(t, W o Γ, k, \) → H(t, W o Γ, k, \),
φ(Nw) = (w)Nw, φ(r) = r, φ(ξ) = ξ, w ∈ W o Γ, ξ ∈ O(t).
Notice that for the sign character of W , φ agrees with sgn from (2.8) on H(t, W o Γ, k, \) but not on H(t, W o Γ, k, r, \).
For R irreducible of type Bn, Cn, F4or G2, there are two further nontrivial possible
’s. Consider the characters s, l of W with
s(sα) =
1 α long
−1 α short , l(sα) =
1 α short
−1 α long .
Since Γ acts isometrically on a, l and s are Γ-invariant. Thus we obtain algebra isomorphisms
φs : H(t, W oΓ, k, \) → H(t, W oΓ, sk, \), φl: H(t, W oΓ, k, \) → H(t, W oΓ, lk, \).
Lemma 2.2. Let H(t, W o Γ, k, \) be a twisted graded Hecke algebra with a real- valued parameter function k. Then it is isomorphic to a twisted graded Hecke algebra H(t, W o Γ, k, \) with k : R → R≥0, via an isomorphism φ that is the identity on O(t ⊕ C).
Proof. Define
(sα) =
1 k(α) ≥ 0
−1 k(α) < 0 .
Since k is Γ-invariant, this extends to a Γ-invariant quadratic character of W . Then
φ has the required properties.
With the above isomorphisms we will generalize the results of [Sol6, §6.2], from graded Hecke algebras with positive parameters to twisted graded Hecke algebras with real parameters.
For the moment, we let H stand for either H(t, W o Γ, k, r, \) or H(t, W o Γ, k, \).
Every finite dimensional H-module V is the direct sum of its generalized O(t)-weight spaces
Vλ:= {v ∈ V : (ξ − ξ(λ))dim Vv = 0 ∀ξ ∈ O(t)} λ ∈ t.
We denote the set of O(t)-weights of V by
Wt(V ) = {λ ∈ t : Vλ 6= 0}.
Let a− be the obtuse negative cone in RR ⊂ a determined by (R, ∆). We denote the interior of a− in RR by a−−. We recall that a finite dimensional H-module V is tempered if
Wt(V ) ⊂ a−⊕ ia
and that V is essentially discrete series if, with z as in (2.5):
Wt(V ) ⊂ a−−⊕ (z ∩ a) ⊕ ia.
For a subset U of t we let Modfl,U(H) be the category of finite dimensional H-modules V with Wt(V ) ⊂ U . For example, we have the category of H-modules with ”real”
weights Modfl,a(H). We indicate a subcategory/subset of tempered modules by a subscript ”temp”. In particular, we have the category of finite dimensional tempered H-modules Modfl(H)temp.
We want to compare the irreducible representations of
H(t, W o Γ, k, \) = H(t, W o Γ, k, r, \)/(r − 1) with those of
H(t, W o Γ, 0, \) = H(t, W o Γ, k, r, \)/(r).
The latter algebra has Irr(C[W o Γ, \]) as the set of irreducible representations on which O(t) acts via evaluation at 0 ∈ t. The correct analogue of this for H(t, W o Γ, k, \), at least with k real-valued, is
Irra(H(t, W o Γ, k, \))temp := Irr(H(t, W o Γ, k, \))temp∩ Modfl,a(H(t, W o Γ, k, \)).
As C[W o Γ, \] is a subalgebra of H(t, W o Γ, k, \), there is a natural restriction map ResW oΓ : Modfl(H(t, W o Γ, k, \)) → Modf(C[W o Γ, \]).
However, when k 6= 0 this map usually does not preserve irreducibility, not even on Irra(H(t, W o Γ, k, \))temp.
In the remainder of this section we assume that the parameter function k only takes real values. Let be as in Lemma 2.2. Since φ is the identity on O(t ⊕ C), it induces equivalences of categories
Modfl,U(H(t, W o Γ, k, \)) −→ Modfl,U(H(t, W o Γ, k, \)) U ⊂ t, Modfl(H(t, W o Γ, k, \))temp −→ Modfl(H(t, W o Γ, k, \))temp
and a bijection
Irra(H(t, W o Γ, k, \))temp −→ Irra(H(t, W o Γ, k, \))temp. Theorem 2.3. Let k : R → R be a Γ-invariant parameter function.
(a) The set ResW oΓ(IrraH(t, W o Γ, k, \)temp) is a Z-basis of Z Irr(C[W o Γ, \]).
Suppose that the restriction of k to any type F4 component of R has k(α) = 0 for a root α in that component or is the form k0 for a character : W (F4) → {±1} and a geometric k0 : F4 → R>0.
(b) There exist total orders on Irra(H(t, W o Γ, k, \)temp) and on Irr(C[W o Γ, \]), such that the matrix of the Z-linear map
ResW oΓ : Z Irra(H(t, W o Γ, k, \))temp→ Z Irr(C[W o Γ, \]) is upper triangular and unipotent.
(c) There exists a unique bijection
ζH(t,W oΓ,k,\)): Irra(H(t, W o Γ, k, \))temp→ Irr(C[W o Γ, \]) such that ζH(t,W oΓ,k,\)(π) always occurs in ResW oΓ(π).
Proof. (a) is known from [Sol2, Proposition 1.7]. The proof of that shows we can reduce the entire theorem to the case where \ is trivial. We assume that from now on, and omit \ from the notations.
Parts (b) and (c) were shown in [Sol6, Theorem 6.2], provided that k(α) ≥ 0 for all α ∈ R. Choose as in Lemma 2.2, so that k : R → R≥0. For V ∈ Modfl(H(t, W, k)) we have
ResW(φ∗V ) = ResW(V ) ⊗ , so we obtain a commutative diagram
(2.10)
Z Irra(H(t, W, k))temp ResW
−−−→ Z Irr(W )
↓ φ∗ ↓ ⊗
Z Irra(H(t, W, k))temp
ResW
−−−→ Z Irr(W )
All the maps in this diagram are bijective and the vertical maps preserve irreducibil- ity. Thus the theorem for H(t, W, k) implies it for H(t, W, k).
The commutative diagram (2.10) also allows us to extend [Sol6, Lemma 6.5] from H(t, W, k) to H(t, W, k). Then we can finish our proof for H(t, W o Γ, k) by applying
[Sol6, Lemma 6.6].
Remark. Geometric parameter functions will appear in Section 3. We make the allowed parameter functions for a type F4 root system explicit. Write k = (k(α), k(β)) where α is short root and β is a long root. The possibilities are
(0, 0), (c, 0), (0, c), (c, c), (2c, c), (c/2, c), (4c, c), (−c, c), (−2c, c), (−c/2, c), (−4c, c), where c ∈ R× is arbitrary. We expect that Theorem 2.3 also holds without extra conditions in type F4.
Theorem 2.4. Let H(t, W oΓ, k, \) be as in Theorem 2.3.b. There exists a canonical bijection
ζH(t,W oΓ,k,\)): Irr(H(t, W o Γ, k, \)) → Irr(H(t, W o Γ, 0, \)) which (as well as its inverse)
• respects temperedness,
• preserves the intersections with Modfl,a,
• generalizes Theorem 2.3.c, via the identification
Irra(H(t, W o Γ, 0, \))temp = Irr(C[W o Γ, \]).
Proof. As discussed in the proof of Theorem 2.3.a, we can easily reduce to the case where \ is trivial. In [Sol6, Proposition 6.8], that case is derived from [Sol6, Theorem 6.2] (under more strict conditions on the parameters k). Using Theorem 2.3 instead of [Sol6, Theorem 6.2], this works for all parameters allowed in Theorem 2.3.
Although [Sol6, Proposition 6.8] is only formulated for irreducible representations in Modfl,a(H(t, W o Γ, k)), the argument applies to all of Irr(H(t, W o Γ, k)).
3. Cuspidal local systems and equivariant cohomology
We follow the setup from [Lus1, Lus3, AMS1, AMS2]. In these references a graded Hecke algebra was associated to a cuspidal local system on a nilpotent orbit for a complex reductive group. For applications to Langlands parameters we deal not only with connected groups, but also with disconnected reductive groups G.
Recall from [AMS1] that a quasi-Levi subgroup of G is a group of the form M = ZG(Z(L)◦), where L is a Levi subgroup of G◦. Thus Z(M )◦ = Z(L)◦ and M ←→ L = M◦ is a bijection between the quasi-Levi subgroups of G and the Levi subgroups of G◦.
Definition 3.1. A cuspidal quasi-support for G is a triple (M, CvM, qE ) where:
• M is a quasi-Levi subgroup of G;
• CvM is the Ad(M )-orbit of a nilpotent element v ∈ m = Lie(M ).
• qE is a M -equivariant cuspidal local system on CvM, i.e. as M◦-equivariant local system it is a direct sum of cuspidal local systems.
We denote the G-conjugacy class of (M, CvM, qE ) by [M, CvM, qE ]G. With this cuspidal quasi-support we associate the groups
(3.1) NG(qE ) = StabNG(M )(qE ) and WqE = NG(qE )/M.
Such cuspidal quasi-supports are useful to partition the set of G-equivariant local systems on nilpotent orbits in g = Lie(G). Let E be an irreducible constituent of qE as M◦-equivariant local system on CvM (which by the cuspidality of E equals the Ad(M◦)-orbit of v). Then
WE◦ := NG◦(M◦)/M◦ ∼= NG◦(M◦)M/M
is a subgroup of WqE. It is normal because G◦ is normal in G. Write T = Z(L)◦ and t = Lie(T ). It is known from [Lus1, Proposition 2.2] that R(G◦, T ) ⊂ t∨ is a root system with Weyl goup WE◦.
Let P◦ be a parabolic subgroup of G◦ with Levi decomposition P◦ = M◦ n U . The definition of M entails that it normalizes U , so
P := M n U
is a again a group, a ”quasi-parabolic” subgroup of G. We put NG(P, qE ) = NG(P, M ) ∩ NG(qE ), ΓqE = NG(P, qE )/M.
The same proof as for [AMS2, Lemma 2.1.b] shows that
(3.2) WqE = WE◦o ΓqE.
The WqE-action on T gives rise to an action of WqE on O(t) = S(t∨).
We specify our parameters c(α). For α in the root system R(G◦, T ), let gα ⊂ g be the associated eigenspace for the T -action. Let ∆P be the set of roots in R(G◦, T ) which are simple with respect to P . For α ∈ ∆P we define c(α) ∈ Z≥2 by
(3.3) ad(v)c(α)−2: gα⊕ g2α→ gα⊕ g2α is nonzero, ad(v)c(α)−1: gα⊕ g2α→ gα⊕ g2α is zero.
Then (c(α))α∈∆P extends to a WqE-invariant function c : R(G◦, T )red → C, where the subscript ”red” indicates the set of indivisible roots. Let \ : (WqE/WE◦)2 → C× be a 2-cocycle (to be specified later). To these data we associate the twisted graded Hecke algebra H(t, WqE, c, r, \), as in Proposition 2.1.
To make the connection of the above twisted graded Hecke algebra with the cuspidal local system qE complete, we involve the geometry of G and g. We work in the G-equivariant bounded derived category DG(X), as in [BeLu], [Lus1, §1] and [Lus3, §1]. Despite the terminology, this is not exactly the bounded derived category of the category of G-equivariant constructible sheaves on a G-variety X. Write
treg= {x ∈ t : Zg(x) = l} and gRS = Ad(G)(CvM + treg+ u).
Consider the varieties
˙g = {(X, gP ) ∈ g × G/P : Ad(g−1)X ∈ CvM+ t + u},
˙g◦ = {(X, gP ) ∈ g × G◦/P◦ : Ad(g−1)X ∈ CvM + t + u},
˙gRS = ˙g ∩ (gRS × G/P ).
We let G × C× act on these varieties by
(g1, λ) · (x, gP ) = (λ−2Ad(g1)x, g1gP ).
Equivariant cohomology for elements of DG×C×(X) is defined via push-forward to a point. By [Lus1, Proposition 4.2] there is a natural isomorphism
(3.4) HG×C∗ ×( ˙g) ∼= O(t) ⊗CC[r].
Consider the maps
(3.5) CvM ←− {(x, g) ∈ g × G : Ad(gf1 −1)x ∈ CvM + t + u}−→ ˙g,f2 f1(x, g) = prCM
v (Ad(g−1)x), f2(x, g) = (x, gP ).
The group G × C×× P acts on {(x, g) ∈ g × G : Ad(g−1)x ∈ CvM+ t + u} by (g1, λ, p) · (x, g) = (λ−2Ad(g1)x, g1gp).
Notice that qE is M × C×-equivariant, because C× is connected and stabilizes nilpo- tent M -orbits. Further f1 is constant on G-orbits, so f1∗qE is naturally a G × C×- equivariant local system. Let ˙qE be the unique G × C×-equivariant local system on
˙g such that f2∗qE = f˙ 1∗qE , and let ˙qERS be its pullback to ˙gRS. Let pr1 : ˙g → g be the projection on the first coordinate. Its restriction
pr1,RS : ˙gRS → gRS
is a fibration with fibre NG(M )/M , so (pr1,RS)!qE˙ RS is a local system on gRS. Let K = ICG×C×(g, (pr1,RS)!qE˙ RS)
be the equivariant intersection cohomology complex on g defined by (pr1,RS)!qE˙ RS. It is shown in [Lus3, Proposition 7.12] that
(3.6) K = pr1,!qE˙ in DG×C×(g).
Let qE∨ be the dual equivariant local system on CvM. It is also cuspidal and (pr1,RS)!qE˙ ∨RS is another local system on gRS. It determines an equivariant in- tersection cohomology complex K∨ on g, which is isomorphic with pr1,!qE˙ ∨.
We can relate ˙g and K to their versions for G◦, as follows. Write
(3.7) G = G
γ∈NG(P,M )/M
G◦γM/M and G/P = G
γ∈NG(P,M )/M
G◦γP/P.
Then we can decompose (3.8) ˙g = G
γ∈NG(P,M )/M{(X, gγP ) ∈ ˙g : g ∈ G◦} = G
γ∈NG(P,M )/M
{(X, gγP γ−1) : X ∈ g, g ∈ G◦/γP◦γ−1, Ad(g−1)X ∈ Ad(γ)(CvM + t + u)}
= G
γ∈NG(P,M )/M ˙g◦γ. Here each term ˙g◦γ is a twisted version of ˙g◦. Consequently K is a direct sum of G◦× C×-equivariant subobjects, each of which is a twist of the K for (G◦M, CvM, qE ) by an element of NG(M )/M .
Considering (pr1,RS)!qE˙ RS as a local system on gRS, [AMS1, Lemma 5.4] and [Lus3, Proposition 7.14] say that
(3.9) C[WqE, \qE] ∼= End0D
G×C×(gRS) (pr1,RS)!qE˙ RS∼= End0D
G×C×(g)(K), where \qE : (WqE/WE◦)2 → C× is a suitable 2-cocycle. As in [AMS2, (8)], we record the subalgebra of endomorphisms that stabilize Lie(P ):
(3.10) End+G (pr1,!qE˙ ∼= C[ΓqE, \qE].
Now we associate to (M, CvM, qE ) the twisted graded Hecke algebra H(G, M, qE ) := H(t, WqE, c, r, \qE),
where the parameters c(α) come from (3.3). As in [AMS2, Lemma 2.8], we can consider it as
H(G, M, qE ) = H(t, WE◦, c, r) o End+G((pr1,!qE ),˙
and then it depends canonically on (G, M, qE ). We note that (3.2) implies (3.11) H(G◦NG(P, qE ), M, qE ) = H(G, M, qE).
There is also a purely geometric realization of this algebra. For Ad(G) × C×-stable subvarieties V of g, we define, as in [Lus1, §3],
(3.12)
V = {(X, gP ) ∈ ˙g : X ∈ V},˙
V = {(X, gP, g¨ 0P ) : (X, gP ) ∈ ˙V, (X, g0P ) ∈ ˙V}.
The two projections π12, π13: ¨V → ˙V give rise to a G × C×-equivariant local system qE = ˙¨ qE ˙qE∨ on ¨V, which carries a natural action of (3.4). As in [Lus1], the action of C[WqE, \−1qE] on K∨ leads to
(3.13) actions of C[WqE, \qE] ⊗ C[WqE, \−1qE] on ¨qE and on HjG◦×C×( ¨V, ¨qE ).
We will apply this with V = g and with V = gN, the variety of nilpotent elements in g. In [Lus1] and [AMS2, §2] a left action ∆ and a right action ∆0 of H(G, M, qE) on H∗G×C×( ¨gN, ¨qE ) are constructed. Let
pr1,N : ˙gN = ˙g ∩ (gN × G/P ) → gN
be the restriction of pr1. Let ˙qEN be the pullback of ˙qE to ˙gN and define KN = (pr1,N)!qE˙ N ∈ DG×C×(gN).
Theorem 3.2. (a) Let V = g or V = gN. The actions ∆ and ∆0 identify H∗G×C×( ¨V, ¨qE ) with the biregular representation of H(G, M, qE).
(b) There exists a convolution product on H∗G×C×(¨g, ¨qE ) which makes it isomorphic (as a graded algebra) to End∗D
G×C×(g)(K). Similarly H∗G×C×( ¨gN, ¨qE ) ∼= End∗D
G×C×(gN)(KN).
(c) Parts (a) and (b) induce canonical isomorphisms of graded algebras H(G, M, qE ) → End∗DG×C×(g)(K) → End∗D
G×C×(gN)(KN).
Proof. (a) When G is connected, this is shown for V = gN in [Lus1, Corollary 6.4]
and for V = g in the proof of [Lus3, Theorem 8.11]. In [AMS2, Corollary 2.9 and
§4] both are generalized to possibly disconnected G.
(b) In [Lus3, §2] it is shown that
(3.14) End∗D
G×C×(g)(K) ∼= HG×C∗ ×(¨g, i!( ˙qE D ˙qE )),
where i denotes the embedding ¨g→ ˙g × ˙g. The Verdier duality operator D satisfies D ˙qE = ˙qE∨ and i!= Di∗D, so the right hand side of (3.14) is
(3.15) HG×C∗ ×(¨g, Di∗D( ˙qE ˙qE∨)) = HG×C∗ ×(¨g, Di∗( ˙qE∨ ˙qE )).
By definition of equivariant homology [Lus3, §1.17], this equals (3.16) H∗G×C×(¨g, i∗( ˙qE ˙qE∨)) = H∗G×C×(¨g, ¨qE ).
The same arguments apply with (gN, KN) instead of (g, K).
(c) In [Lus3, Theorem 8.11] the first isomorphism is shown when G is connected.
Using part (a) the same argument applies when G is disconnected. The second algebra isomorphism is a consequence of parts (a,b) and functoriality. Let σ ∈ t, so that M = ZG(T ) ⊂ ZG(σ). We would like to compare Theorem 3.2 with its version for (ZG(σ), M, qE ). First we analyse the variety
(G/P )σ := {gP ∈ G/P : σ ∈ Lie(gP g−1)}.
This is also the fixed point set of exp(Cσ) in G/P .
Lemma 3.3. For any gP ∈ (G/P )σ, the subgroup gP◦g−1∩ ZG◦(σ) of ZG◦(σ) is parabolic.
