A vanishing theorem for Dirac cohomology of standard modules
aa r X i v : . [ m a t h . R T ] D ec A VANISHING THEOREM FOR DIRAC COHOMOLOGY OFSTANDARD MODULES
KEI YUEN CHAN
Abstract.
This paper studies the Dirac cohomology of standard modules in the set-ting of graded Hecke algebras with geometric parameters. We prove that the Diraccohomology of a standard module vanishes if and only if the module is not twisted-elliptic tempered. The proof makes use of two deep results. One is some structuralinformation from the generalized Springer correspondence obtained by S. Kato andLusztig. Another one is a computation of the Dirac cohomology of tempered modulesby Barbasch-Ciubotaru-Trapa and Ciubotaru.We apply our result to compute the Dirac cohomology of ladder representations fortype A n . For each of such representations with non-zero Dirac cohomology, we associateto a canonical Weyl group representation. We use the Dirac cohomology to concludethat such representations appear with multiplicity one. Introduction p -adicreductive groups. Motivated by some analogies between p -adic and real reductive groups,Barbasch-Ciubotaru-Trapa [BCT] define a Dirac operator and develop the Dirac coho-mology theory for graded Hecke algebras. The theory has been further developed in[CT, CM, COT] and there are applications on W -character formulas [CT, CH] and a clas-sification of discrete series [CO].This paper studies the Dirac cohomology of standard modules for graded Hecke algebras.Determining their Dirac cohomology requires understanding on some fine structure. Itturns out that part of the information can be obtained from results of generalized Springercorrespondence by Kato [Ka2] and Lusztig [Lu, Lu1, Lu3, Lu5] via the geometric realizationof standard modules. More details and applications of our result will be discussed in nexttwo sections. Main results.
For any complex reductive group H , let H ◦ be the identity componentof H .We first introduce some notations. Let G be a complex connected reductive group withits Lie algebra g . Let N G be the set of nilpotent elements in g . Let L be a Levi subgroup of G and let O be a L -orbit in N G . Let L be a cuspidal local system of O . The datum ( L, O , L ) forms a cuspidal triple. Let P be a parabolic subgroup with the Levi decomposition LU .Let T = Z ◦ L , where Z L is the center of L . Let h be the Lie algebra of T . Let h ∨ be the dualspace of h . All these data determine a Weyl group W , a root system R , a set Π of simpleroots in R and a parameter function c : Π → R (see Section 4.1 for the detailed notations).Lusztig [Lu1, Lu3] constructed geometrically a graded Hecke algebra H , which also has analgebraic description H ( V, W, Π , r , c ) (see Section 2.1).Before defining the Dirac cohomology, we introduce more notations. Let V ′ = C ⊗ Z R .Let C ( V ′ ) be the Clifford algebra for V ′ (Section 2.3). Let S be a fixed choice of simplemodule of C ( V ′ ) . Let f W be the spin double cover of W (Section 2.4). f W then definesa twisted group algebra C [ f W ] , which is a natural subalgebra of C ( V ′ ) . There is also adiagonal embedding, denoted ∆ : C [ f W ] → H ⊗ C ( V ′ ) , which plays an important role in theDirac cohomology.The Dirac element, denoted D , for H is an element in H ⊗ C ( V ′ ) which has a remarkableformula for D (2.1). Given an H -module X , define the Dirac cohomology of X (as in[BCT]): H D ( X ) = ker π ( D )ker π ( D ) ∩ im π ( D ) , where π : H ⊗ C ( V ′ ) → End C ( X ⊗ S ) is the map defining the H ⊗ C ( V ′ ) -action on X ⊗ S . Wealso have a natural action of f W on X ⊗ S via the diagonal embedding ∆ , and such actionturns out to commute with the action of D , up to a sign. Hence, H D ( X ) is equipped witha f W -representation structure. A crucial result in the Dirac cohomology theory, so-calledVogan’s conjecture [BCT, Theorem 4.4], states that if X is irreducible and H D ( X ) = 0 ,then the f W -isotypic component in H D ( X ) determines the central character of X .A goal of this paper is to compute the Dirac cohomology of standard modules. Let e ∈ N G and let r ∈ C × . Let s ∈ g be a semisimple element satisfying [ s, e ] = 2 re . Let A ( e, s ) be the component group for e and s , and let ζ ∈ Irr A ( e, s ) . For the datum ( s, e, r, ζ ) ,Lusztig geometrically constructs a module E s,e,r,ζ (see Section 4.3). We shall assume that E s,e,r,ζ is non-zero and call it a standard module . An interesting case is that when s = h e isthe middle element in the sl -triple for e . In that case, E rh e ,e,r,ζ is a tempered module (seeDefinition 4.5). When e has a solvable centralizer in g , it is shown in [Ci2] and [BCT] that H D ( E rh e ,e,r,ζ ) = 0 . We call E rh e ,e,r,ζ to be a twisted-elliptic tempered module , motivatedby the close connections to twisted elliptic pairings for Weyl groups and component groups[CH] (also see [Ch4]).The first main result in this paper is the following: Theorem 1.1. (Theorem 6.4) Let E be a standard module as above. Then H D ( E ) = 0 ifand only if E is not a twisted-elliptic tempered module. IRAC COHOMOLOGY OF STANDARD MODULES 3
We shall explain the strategy for proving Theorem 1.1 in the next subsection. We remarkthat in order to apply results of Kato and Lusztig, a main ingredient of our proof is to studycertain deformation behavior of the Dirac cohomology, which is carried out in Section 3.For unequal (not necessarily geometric) parameter case of type BC, we also deduce aversion of Theorem 1.1 from results and methods of [Ka2, CK] in Section 8.We also obtain the twisted Dirac index of a standard module consequently. Here thetwisted Dirac index of an H -module X (with a Z -graded decomposition X = X + ⊕ X − ,see Section 2.2) is defined as: I ( X ) = X + ⊗ S − X − ⊗ S as virtual f W -representations. Here S = S when dim V ′ is even and S is the direct sumof two simple C ( V ′ ) -modules when dim V ′ is odd. The terminology of the Dirac index issuggested by its relation to the Dirac cohomology in Section 6.4 (also see [CT, CH, BPT]). Corollary 1.2. (Corollary 6.6) Let E be a standard module. Define E ′ as in Lemma 5.6.Then I ( E ′ ) = 0 if and only if E is not twisted-elliptic tempered. Dirac index can descend to the Grothendieck group of H -modules, but Dirac cohomologycannot. Hence Dirac cohomology is an invariant encoding more information. When E istempered, the Dirac index I ( E ) is computed in [CH]. A twisted Dirac index of standardmodules for real groups has been recently computed in [BPT].The Dirac index of a simple module can be, in principle, computed by using a characterformula from a Kazhdan-Lusztig type polynomial. Perhaps a significance of Theorem 1.1 isthat the Dirac cohomology of a simple module may be computed if a suitable categorificationof the character formula is established. For example, such strategy can be done for theladder representations, where a BGG-type resolution exists.Ladder representations of GL ( n, F ) are introduced by Lapid-Mingueź [LM] for the studyof a determinantal formula of Tadić, and they coincide with the calibrated representations ofRam [Ra] in the graded Hecke algebra level [BC2]. Speh representations form a subclass ofladder representation. Hence our result for the Dirac cohomology of ladder representationscan be viewed as an extension of the Speh representation case by Barbasch-Ciubotaru [BC].Interestingly, our method does not need a lot of knowledge on the W -structure of a ladderrepresentation and hence we obtain some closed W -structure as a consequence. Theorem 1.3. (Theorems 7.8) Let H l be the graded Hecke algebra of type A l − (see Section7.1 for notations). Let X be a ladder representation whose central character is the same asthat of a twisted-elliptic tempered module E . Then H D ( X ) ∼ = H D ( E ) = 0 . Theorem 1.3 leads to the question of identifying the W -representation which contributesthe non-zero Dirac cohomology. We do that via the Arakawa-Suzuki functor. Such canon-ical W -type in terms of Young diagram parametrization can be constructed by a simpleprocedure starting from a Zelevinsky segment (Section 7.6). Using the Dirac cohomology,we show that that W -type appears with multiplicity one: KEI YUEN CHAN
Corollary 1.4. (Theorem 7.21) Let X be as in Theorem 1.3. Let W = S l be the permuta-tion group on l elements. Then there exists a unique irreducible W -representation σ suchthat Hom f W ( H D ( X ) , σ ⊗ S ) = 0 and Hom W ( σ, X | W ) ∼ = C . There are some other possible approaches for obtaining those W -structures. Our resultfor Dirac cohomology provides a feasible and simple argument for the proof.1.3. Outline of ’if ’ proof of Theorem 1.1.
Let E be a standard module of H which isnot tempered. The graded Hecke algebra H admits a natural Z -grading (Section 2.2). Forthe introduction purpose, we assume that E can be extended to a Z -graded H -module.Let A W = S ( V ) ⋊ W be the skew group ring (Section 2.5). We obtain an A W -modulefrom E as follows. Pick a remarkable W -subspace σ of E which comes from a generalizedSpringer correspondence and generates E . Define E i = (cid:8) px : x ∈ σ, p ∈ S ≤ i ( V ) (cid:9) , where S ≤ i ( V ) be the space containing polynomials of degree less than or equal to i . Nowset E σ := L i E i +1 /E i , which is equipped an A W -action with elements in v ∈ V ⊂ S ( V ) sending E i +1 /E i → E i +2 /E i +1 and elements in w ∈ W sending E i +1 /E i → E i +1 /E i .(The action coincides with the one in Section 3.5.) Regarding S ( V ) as a natural Z -gradedalgebra, we also have a Z -grading on A W . The module E σ can be extended to a Z -graded A W -module.To realize the A W -structure of E σ , we need to introduce a module of Kato. We define ≤ to be a partial ordering on Irr W from the closure ordering in the generalized Springercorrespondence. For τ ∈ Irr W , we can associate to an A W -module K τ defined in [Ka2](see (4.6) for the precise definition). Lemma 1.5.
There exists an A W -module filtration (cid:8) Y i (cid:9) ki =0 on E σ such that Y ⊂ Y ⊂ Y ⊂ . . . ⊂ Y k = E σ and for i = 0 , . . . , k − , Y i +1 /Y i is isomorphic to a module K σ i for some σ i ∈ Irr W . One can view A W as a specific case of the general graded Hecke algebras and hence wehave an analogue of the Dirac cohomology, which is denoted by H D A (see Section 2.5). Wewant to compute H D A ( K σ ) ( σ ∈ Irr W ).The generalized Springer correspondence of Lusztig [Lu] and results of Kato [Ka2] showthat for each τ ∈ Irr W , there is a tempered module, denote X τ such that its associated A W -structure X τ , satisfying X τ ∼ = K τ (see Corollary 4.8 for the precise statement).On the other hand, the unitarity on X τ implies that H D ( X τ ) ∼ = H D A ( X τ ) ∼ = H D A ( K τ ) IRAC COHOMOLOGY OF STANDARD MODULES 5 (see Proposition 3.13). The Dirac cohomology H D ( X τ ) has been known from the compu-tations in [Ci2] and [BCT], and Vogan’s conjecture [BCT]. Hence H D A ( K τ ) is (almost)determined.We now turn back to the standard module E . To transfer the information from H D A ( K τ ) to H D ( E ) , we need the following result, which is a version of Theorem 3.11: Theorem 1.6. H D ( E ) is a quotient of L i H D A ( K σ i ) . Now with Lemma 1.5 and the computation of H D A ( K τ ) , it suffices to see that the f W -representations in H D A ( K σ i ) cannot appear in H D ( E ) . To this end, we show that theoperator D acts by a non-zero scalar on any corresponding isotypic components of f W from H D A ( K σ i ) in E . This proves H D ( E ) = 0 .1.4. Acknowledgments.
The problem of computing the Dirac cohomology of standardmodules arose from discussions with Dan Ciubotaru. The author is grateful to him for sev-eral useful discussions and suggestions during this work. He also thanks Maarten Solleveldfor useful comments for a earlier version of this paper. The author was supported by theCroucher Postdoctoral Fellowship. 2.
Preliminaries
Graded Hecke algebra H . In this section, we work over a more general setting. Weshall specialize to the graded Hecke algebra of geometric parameters in Section 4.Let V be complex linear space and let V ∨ = Hom C ( V, C ) . Let W ⊂ O ( V ) be a groupgenerated by simple reflections. Let R be a (not necessarily reduced) root system associatedto W and let Π be a fixed set of simple roots. Let l : W → Z be the length function for W . For α ∈ R , let s α be the associated simple reflection for V . For α ∈ Π , let α ∨ be theassociated coroot i.e. s α ( v ) = v − α ∨ ( v ) α for any v ∈ V . Let w be the longest element in W .Recall that V ∨ ∼ = V as W -representations. This defines a symmetric bilinear form on V given by h v , v i V = v ( η ( v )) , where η : V → V ∨ is an isomorphism. We shall normalize h , i V such that h α, α i V = 2 for α ∈ R . We shall sometimes write h , i for h , i V .Let c : Π → R be a W -invariant parameter function. Write c α for c ( α ) . Let S ( V ) be thepolynomial algebra for V . Let C [ W ] be the group algebra of W . Definition 2.1. [Lu1] The graded Hecke algebra H = H ( V, W, Π , r , c ) is a complex asso-ciative algebra with generators { p ∈ S ( V ) } , { t w : w ∈ W } , r satisfying the relations:(1) the natural map S ( V ) to H given by p v is an algebra injection;(2) the natural map C [ W ] = ⊕ w ∈ W C f w to C [ W ] given by f w t w is an algebrainjection;(3) for α ∈ R , t s α v − s α ( v ) t s α = r c α α ∨ ( v ) ;(4) r is in the center of H .In particular, H ∼ = S ( V ) ⊗ C [ W ] ⊗ C [ r ] naturally as vectors spaces. Let θ : H → H bean algebra automorphism determined by θ ( v ) = − w ( v ) and θ ( t w ) = t w ww − . For an KEI YUEN CHAN H -module X , let θ ( X ) be an H -module determined by π θ ( X ) ( h ) x = π X ( θ ( h )) x, where π X (resp. π θ ( X ) ) is the map defining the action of H on X (resp. θ ( X ) ).2.2. Z -grading on H -modules. For v ∈ V ⊂ S ( V ) , define e v = ( v − v ∗ ) . Define aHermitian-linear anti-involution ∗ : H → H determined by e v ∗ = − e v ( v ∈ V ) , t ∗ w = t w − ( w ∈ W ) , r ∗ = r An H -module X is said to be unitary if there exists a non-degenerate positive-definiteHermitian form h ., . i ∗ on X satisfying the property h h.x , x i ∗ = h x , h ∗ .x i ∗ . The terminology of unitarity comes from p -adic groups [BM].The algebra H admits a natural Z -grading given by deg( e v ) = 1 (mod ) and deg( t w ) = 0 (mod ). It is straightforward to check from the defining relations that the Z -grading iswell-defined. An H -module X is said to be Z -graded if there exists a decomposition of X = X + ⊕ X − such that e v.X ± = X ∓ for any v ∈ V .We state two useful lemmas: Lemma 2.2.
Let X be an H -module. If X ∼ = θ ( X ) , then X can extend to a Z -gradedmodule. Moreover, if X is irreducible, then the Z -grading is unique.Proof. Note that t w θ has order . Let X ± be the eigenspace of the eigenvalue ± for t w θ .By using t w θ ( e v ) = − e v for any v ∈ V , one shows that the decomposition X = X + ⊕ X − gives the desired Z -grading. (cid:4) The Dirac operator D . Let V ′ = C ⊗ Z R ⊂ V . Let C ( V ′ ) be the Clifford algebraassociated to h ∨ over C given by V ′ ⊗ V ′ / h v ⊗ v + h v, v i : v ∈ V ′ i .Let n = dim V ′ . The space V ′ admits a natural real form, which is V ′ := R ⊗ Z R . Let ǫ , . . . , ǫ n be an orthogonal basis for V ′ with respect to h , i . There is a natural embedding V ′ ֒ → C ( V ′ ) , which will be denoted by v e g v . For α ∈ R + , define e s α = e g α | α | . Following [BCT], define the Dirac element D ∈ H ⊗ C ( V ′ ) D = X e ǫ i ⊗ e g ǫ i . The element D is independent of a choice of an orthogonal basis. The formula for D isgiven by [BCT, Theorem 3.5] D = − n X i =1 ǫ i ⊗ − r X α> ,β> ,s α ( β ) < c α c β | α || β | s α s β ⊗ e s α e s β (2.1)Let S be a fixed choice of simple module of C ( V ′ ) . (When dim V ′ is odd, C ( V ′ ) has twoisomorphic classes of simple modules. When dim V ′ is even, C ( V ′ ) has only one isomorphism IRAC COHOMOLOGY OF STANDARD MODULES 7 class of simple modules.) By the definition of D , we have D ( X + ⊗ S ) ⊂ X − ⊗ S, D ( X + ⊗ S ) ⊂ X − ⊗ S. The Dirac cohomology of an H -module X is defined as: H D ( X ) = ker π ( D )ker π ( D ) ∩ im π ( D ) , (2.2)where π : H ⊗ C ( V ′ ) → End C ( X ⊗ S ) is the map defining the H ⊗ C ( V ′ ) -action on X ⊗ S .Later on, we shall simply write ker D , im D for ker π ( D ) and im π ( D ) respectively.2.4. The spin cover f W . For more details of the spin cover, we refer the reader to [Ci2]and [BCT].Let f W be the group generated by the elements e g α ( α ∈ R + ) in C ( V ′ ) . The mapdetermined by e g α s α is a two-to-one map. We call f W to be the spin cover of W . Let C [ f W ] be the subalgebra generated by the elements e g α ( α ∈ R + ). We define a diagonalembedding ∆ : C [ f W ] → H ⊗ C ( V ′ ) given by e s α t s α ⊗ e s α . (2.3) Lemma 2.3. [Ci2] (also see [Ch, Lemma 4.1] ) The element X α> ,β> ,s α ( β ) < c α c β | α || β | s α s β ⊗ e s α e s β lies in the center of ∆( C [ f W ]) . Lemma 2.4. [BCT, Lemma 3.4]
For any α ∈ R + , ∆( e s α ) D = − D ∆( e s α ) . Lemma 2.4 implies that H D ( X ) is equipped with a natural f W -action via the map ∆ .2.5. The operator D A . Let A W := A W ( V ) = S ( V ) ⋊ W be the skew group ring, which isequipped with a Z -graded structure given by deg( v ) = 1 for v ∈ V ⊂ S ( V ) and deg( w ) = 0 for w ∈ W . Define D A ∈ A W ⊗ C ( V ) as: D A = n X i =1 ǫ i ⊗ c ǫ i . It is straightforward to compute that D A = − P ni =1 ǫ ⊗ . If X is a graded A W -module,then D A acts identically by zero on X . For an A W -module X , we define similarly H D A ( X ) = ker D A ker D A ∩ im D A , (2.4)where we realize D A to be the operator on X from its A W -action on X .We define ∆ A : C [ f W ] → A W ⊗ S given by e s α t s α ⊗ e s α . We still have a version ofLemma 2.4. KEI YUEN CHAN
Six-term exact sequence.
Let X be an H -module admitting a Z -grading. Let e X ± = X ± ⊗ S . Note that D ( X ± ⊗ S ) ⊂ X ∓ ⊗ S . Define D ± = D | X ± ⊗ S .We define H ± D ( X ) = ker D ± / (ker D ± ∩ im D ∓ ) . We remark that the notion here does not coincide with the one in [CT] since we use adifferent Z -grading on X ⊗ S . Definition 2.5. An H -module X is said to admit a central character if X is finite-dimensional and any element z in Z ( H ) acts by a scalar on X . By Schur’s lemma, any irre-ducible module always admits a central character. It is known that Z ( H ) = S ( V ) W × C [ r ] [Lu1, 6.5]. We shall identify naturally the set of central characters χ ( W γ ∨ ,r ) with ( W γ ∨ , r ) ∈ ( V ∨ /W ) × C such that χ ( W γ ∨ ,r ) ( z, r ′ ) = γ ∨ ( z ) rr ′ . Proposition 2.6.
Let X be a Z -graded H -module admitting a central character. Supposethere exists a short exact sequence of H -modules for X : → X → Y → Z → . Then we have the following six-term short exact sequence: H + D ( X ) / / H + D ( Y ) / / H + D ( Z ) (cid:15) (cid:15) H − D ( Z ) O O H − D ( Y ) o o H − D ( X ) o o . The statement still holds if we replace Z -graded H -modules by Z -graded A W -modules andreplace H D by H D A . For a proof of Proposition 2.6 and the definition of the connecting homomorphisms, onemay refer to a similar setting in [HP]. By definitions, we have H D ( A ) = H + D ( A ) ⊕ H − D ( A ) .In particular, we have: Corollary 2.7. (1) If H D ( X ) = H D ( Z ) = 0 , then H D ( Y ) = 0 . (2) If H D ( Y ) = 0 , then H D ( X ) ∼ = H D ( Z ) . Iwahori-Matsumoto involution.Definition 2.8.
The Iwahori-Matsumoto IM is a linear involution IM : H → H charac-terized by IM ( v ) = − v for v ∈ V , IM ( t w ) = ( − l ( w ) t w . For an H -module X , IM ( X ) is an H -module isomorphic to X as vector spaces with H -actiongiven by: π IM ( X ) ( h ) x = π X ( IM ( h )) x, where π IM ( X ) (resp. π X ) are the maps defining the H -action.We shall implicity use the following fact: Lemma 2.9.
Let X be an H -module. Then H D ( X ) = 0 if and only if H D ( IM ( X )) = 0 .Proof. Striaghtforward from definitions. (cid:4)
IRAC COHOMOLOGY OF STANDARD MODULES 9 Deformation for Dirac cohomology
Deformation for modules.
Let S ≤ i ( V ) be the space containing polynomials of de-gree less than or equal to i . Let H i = (cid:8) t w p : w ∈ W, p ∈ S ≤ i ( V ) (cid:9) for i ≥ and let H − = 0 .We have a natural linear isomorphism: A W ∼ = M i ≥ H i / H i − . (3.5)Let σ be a W -representation. Define a W -filtration on H ⊗ C [ W ] σ by F ( σ ) i := (cid:8) p ⊗ u : u ∈ λ and p ∈ S ≤ i ( V ) (cid:9) . Definition 3.1.
Let X = X + ⊕ X − be a Z -graded H -module. Let σ be a W -subspaceof X + or X − . Then we define a W -filtration, depending on σ , on X as follows. Define anatural map ψ σ : H ⊗ C [ W ] σ → X given by ⊗ x x . If ψ σ is surjective, we shall say that σ is a choice of deformation for X . If X is irreducible, the PBW basis for H implies thatany W -subspace σ of X ± is a choice of deformation for X .We now assume σ is a choice of deformation for X . Let X σ,i = ψ σ ( F ( σ ) i ) or simply X i ifthe choice of σ is clear. Define X σ = L i X i /X i − . There is a natural graded A W -structureon X σ which can be explicitly described as follows: Denote by pr σ,i : X i → X i /X i − ⊂ X σ a natural projection map. (We simply write pr i .) For pr i +1 ( x ) ∈ X i +1 /X i ,(1) for v ∈ V ⊂ A W , v. pr i +1 ( x ) = pr i +2 ( e v.x ) ∈ X i +2 /X i +1 , where e v acts by the H -action on x ;(2) for t w ∈ C [ W ] ⊂ A [ W ] , t w . pr i +1 ( x ) = pr i +1 ( t w .x ) ∈ X i +1 /X i , where t w .x is givenby the H -action on X . Lemma 3.2.
The above A W -structure on X σ is well-defined.Proof. By the definitions of X i , the action is independent of a choice of a representative for pr i +1 ( x ) ∈ X i +1 /X i . We also have to check the defining relations for A W . For v , v ∈ V ,a direct computation gives that e v e v − e v e v ∈ C [ W ] and hence ( e v e v − e v e v ) .x ∈ X i +1 .Thus pr i +3 (( e v e v − e v e v ) .x ) = 0 as an element in X i +3 /X i +2 .Now we consider w ∈ W and v ∈ V . It is again from direct computation that t w e v = ] w ( v ) t w . Thus pr i +2 (( t w e v − ] w ( v ) t w ) .x ) = 0 as desired. (cid:4) By switching X + and X − if necessary, we shall assume that σ ⊂ X + from now on.Inductively, we can fix a W -map ι σ,i : X i /X i − → X i ⊂ X such that(1) pr i ◦ ι σ,i (¯ y ) = ¯ y for any ¯ y ∈ X i +1 /X i ;(2) im ι σ,i ⊂ X + if i ≡ (mod ); and im ι σ,i ⊂ X − if i ≡ (mod ).There is no canonical choice for ι σ,i in general but it is not important in our study.Then we obtain a W -map ι σ = L i ι σ,i : X σ → X . We simply write ι for ι σ if there isno confusion. As a W -map, define κ σ = ι − σ : X → X σ . We simply write κ for κ σ .If we identify X = L i im ι σ,i , then we have κ = L i pr i | im ι σ,i . Example 3.3.
Let W be the Weyl group of type A . Let V = C ⊗ Z R . Let C x be a -dimensional H -module such that v.x = 0 for all v ∈ V . Then H ⊗ S ( V ) C x has copy of the trivial representation triv and copy of the sign representation sgn . Choose σ to be the triv -isotypic component. Let = u ∈ σ . Then X σ has a basis { u, v .u } for some non-trivial v ∈ V . The A w -structure is given by v ( v .u ) = 0 and t w .u = u and t w .u = ( − l ( w ) u for w ∈ S .We remark that we essentially use the Z -grading in the following lemma: Lemma 3.4.
For v ∈ V and x ∈ im ι σ,i , κ ( e v.x ) − v.κ ( x ) ∈ M j
For v ∈ V and e x ∈ im e ι σ,i , e κ ( D. e x ) − D A . e κ ( e x ) ∈ M j
This follows from Lemma 3.4 and the definitions of D and D A . (cid:4) Lemma 3.6. e κ (ker D ) ⊂ ker D A Proof.
This follows from Lemma 3.5 and the fact that D A . e κ ( e x ) ∈ X i +1 /X i ⊗ S . (cid:4) Lemma 3.7.
For α ∈ R + and e x ∈ X ⊗ S , ∆ A ( e s α ) . e κ ( e x ) = κ (∆( e s α ) . e x ) Proof.
This follows from the fact that κ is a W -map. (cid:4) Dirac cohomology. If X is a H -module admitting a central character, then ker D acts by a scalar on each f W -isotypic component of X by using the formula (2.1) and Lemma2.3. We shall implicitly use this fact below. Lemma 3.8.
Let X be a Z -graded H -module admitting a central character. Let e x ∈ X σ ⊗ S . Let e τ be an irreducible genuine f W -representation. Then D A e x lies in the e τ -isotypiccomponent of X σ if and only if e x lies in the sgn ⊗ e τ -isotypic component of X σ .Proof. This follows from an analogous version of Lemma 2.3. (cid:4)
Lemma 3.9.
Let X be a Z -graded H -module admitting a central character. Let e σ be anirreducible f W -representation. Then D acts by zero on e σ -isotypic component of X if andonly if D acts by zero on sgn ⊗ e σ -isotypic component for X .Proof. This follows from the formula of D (2.1) and Lemma 2.3. (cid:4) IRAC COHOMOLOGY OF STANDARD MODULES 11
Theorem 3.10.
Let X be a Z -graded H -module admitting a central character. Let σ bea choice of deformation for X . Suppose H D A ( X σ ) = 0 . Then H D ( X ) = 0 .Proof. Let K = ker D . We shall proceed by an inductive argument. By switching X + and X − if necessary, we assume σ ⊂ X + . We fix e ι and then obtain e κ as in Section 3.1.For simplicity, let e X i = ( X i ⊗ S ) ∩ K . Let k be the least integer such that e X k = 0 . Let e x ∈ e X k ∩ ker D . By Lemma 3.6, we have D A . e κ ( e x ) = 0 . Suppose e x = 0 and hence e κ ( e x ) = 0 .Then there exists ¯ y ∈ X k − /X k − such that D A . ¯ y = e κ ( e x ) by H D A ( X σ ) = 0 . However byLemmas 3.7, 3.8 and 3.9, we have D acts by zero on a non-zero element e ι (¯ y ) , contradictingthe minimality of our choice of k . Hence, e x = 0 and so e X k ∩ ker D = 0 .Let e Y = e X k ⊕ D ( e X k ) ⊂ K . From above discussion, we have ker D | e Y = ker D | e Y ∩ im D | e Y .We now proceed to consider the least k ′ ( ≥ k + 1) such that e X k ′ / e Y = 0 . Let e x ′ ∈ ( e X k ′ ∩ im e ι k ′ ) \ e Y . We claim that D. e x ′ / ∈ e Y . Otherwise D A . e κ ( e x ′ ) = 0 using a similar argumentin the previous paragraph and the proof of Lemma 3.6. Now H D A ( X σ ) = 0 implies thereexists ¯ y ′ ∈ L j Let X be a Z -graded finite-dimensional H -module admitting a centralcharacter. Set X ⊗ S = e X ⊕ e X as a decomposition of f W -representations such that D ( e X k ) ⊂ e X k and D A ( e κ ( e X k )) ⊂ e κ ( e X k ) ( k = 1 , ). Furthermore, assume that the decompositionsatisfies the property that X i ⊗ S = ( e X ∩ ( X i ⊗ S )) ⊕ ( e X ∩ ( X i ⊗ S )) for all i . Then if ker D A | e κ ( e X ) ker D A | e κ ( e X ) ∩ im D A | e κ ( e X ) = 0 then H D ( X ) is isomorphic to a quotient of e X as f W -representations.Proof. Using the assumptions, we have H D ( X ) ∼ = ker D | e X ker D | e X ∩ im D | e X ⊕ ker D | e X ker D | e X ∩ im D | e X . Using the argument in Theorem 3.10, the latter summand in the right hand side is zero.Then the corollary follows. (cid:4) Deformation for unitary modules. Certain deformation for unitary modules be-haves nicely since we have H D ( X ) = ker D = ker D .We recall a well-known property for the Dirac cohomology of unitary modules, whichcan be proven by linear algebra and the property that D ∗ = D . Proposition 3.12. Let X be a unitary H -module. Then H D ( X ) = ker D . Proposition 3.13. Let X be an irreducible unitary ( Z -graded) H -module. Let σ be achoice of deformation for X with σ ⊂ X + . Then H D A ( X σ ) ∼ = H D ( X ) .Proof. Since D acts as a diagonal matrix on X and ker D = ker D , we have a f W -decomposition X ⊗ S = ker D ⊕ L λ =0 e X λ , where e X λ = ker( D − λ ) . e X λ is compatiblewith the filtration { X i } on X i.e. X i ⊗ S = (ker D ∩ ( X i ⊗ S )) ⊕ L λ =0 ( e X λ ∩ ( X i ⊗ S )) since D ( X i ⊗ S ) ⊂ X i ⊗ S . Claim: ker D A | κ ( e X λ ) = im D A | κ ( e X λ ) ∩ ker D A | κ ( e X λ ) for each λ = 0 . Proof of claim: Let P r κ ( e x r ) ∈ ker D A | e X λ be an element satisfying e x r ∈ ι ( X r /X r − ) ⊗ S .By a grading consideration, we have e κ ( e x r ) ∈ ker D A for each r . Then D ( e x r ) ∈ X r − ⊗ S byLemma 3.6.Let y = D. e x r . We have D.y = λ e x r for some λ = 0 . By Lemma 3.5, D A .κ ( y ) = λκ ( e x r ) .Hence we obtain κ ( e x r ) ∈ im D A for each r . Hence we have ker D A | e κ ( e X λ ) = im D A | e κ ( e X λ ) ∩ ker D A | e κ ( e X λ ) . This proves the claim.Now with Lemma 3.6, we obtain the statement. (cid:4) We shall prove that a unitary H -module can extend to a Z -graded module in Lemma4.2. 4. Structure of tempered modules and generalized Springercorrespondence This section reviews results in [Lu1, Lu3, Lu5, Ka2] and our goal is to obtain Corollary4.8.4.1. Graded Hecke algebras of geometric parameters. Let G be a complex connectedreductive algebraic group with the Lie algebra g . Definition 4.1. [Lu3] A cuspidal triple for G is a triple ( L, O , L ) such that L is a Levisubgroup of G , O is a nilpotent L -orbit in the Lie algebra l of L , and L is a L -equivariantcuspidal local system for O in the sense of [Lu].Fix a cuspidal triple ( L, O , L ) . Let T = Z ◦ L , where Z L is the centralizer of L . Let h bethe Lie algebra of T . Let h ∨ be the dual space of h . Let W = N G ( T ) /L , where N G ( T ) isthe normalizer of T in G . The root decomposition of g by h defines a root system R ⊂ h ∨ .Let P = LU be a choice of parabolic subgroups with the Levi part L . We have aprojection map: µ : G × P ( O ⊕ u ) → g given by µ ( g, x ) = ad( g )( u ) . Denote the image of µ by N . Let j : O ֒ → O be the inclusionand let pr : O⊕ u → O be the projection map. By the cleanness property, we have j ∗ L ∼ = j L and hence pr ∗ j ! L defines a P × C ∗ -equivariant sheaf on O ⊕ u . Here P × C × acts on O ⊕ u by the action ( g, r ) .x = r − ad( g )( x ) . We then obtain a corresponding G × C × -equivariantsheaf ˙ L on G × P ( O ⊕ u ) . IRAC COHOMOLOGY OF STANDARD MODULES 13 The parabolic subgroup P determines a set Π of simple roots in R . We can obtain aparameter function c ′ : Π → R as in [Lu3, 8.7]. We set c = c ′ . Such normalization is forthe consistency of the computation in [BCT] later. Lusztig [Lu3, Sec. 8] proves that thereare canonical isomorphisms Hom D bG × C × ( N ) ( µ ∗ ˙ L , µ ∗ ˙ L ) ∼ = C [ W ] , Ext • D bG × C × ( N ) ( µ ∗ ˙ L , µ ∗ ˙ L ) ∼ = H , where H = H ( h ∨ , Π , W, r , c ) (Definition 2.1).We have the following commutative diagram (see [Lu3, 7.14, 8.10(a)]): Hom D bG × C × ( N ) ( µ ∗ ˙ L , µ ∗ ˙ L ) (cid:15) (cid:15) / / C [ W ] (cid:15) (cid:15) Ext • D bG × C × ( N ) ( µ ∗ ˙ L , µ ∗ ˙ L ) / / H , where the right vertical map is the injection given by w t w and the left vertical map isgiven [Lu3, 7.14, 8.10(a)]. Here D bG × C × ( N ) is the G × C × -equivariant derived category.The geometric parameters for graded Hecke algebras are described in [Lu3, Section 2.13].In particular, when G is simply connected and almost simple, L is the maximal torus and O is the zero orbit with a constant sheaf L , we have c ≡ . Lemma 4.2. Assume c α = 0 for all α ∈ Π . Let X be an irreducible unitary H -module with r acting by a real scalar and with all weights being real. Then X can extend to a Z -gradedmodule.Proof. By definitions, X ∼ = X ∗ . We also have X ∗ ∼ = θ ( X ) by [Ch2, Lemma 4.5] and [Ch5,Lemma 2.5]. Hence X ∼ = θ ( X ) . The lemma now follows from Lemma 2.2. (cid:4) Generalized Springer correspondence. Let e ∈ N . Let Z G × C × ( e ) (resp. Z G ( e ) )be the centralizer of e in G × C × (resp. in G ). Denote by O e the G -orbit of e in N . Let A ( e ) = Z G × C × ( e ) /Z G × C × ( e ) ◦ ∼ = Z G ( e ) /Z G ( e ) ◦ be the component group. Let B e = µ − ( e ) .Define H • ( B e , ˙ L ) = H • ( i ! e µ ∗ ˙ L [2dim N − B e ]) as in [Ka2, Theorem 3.1] (also see [Ka2,Remark 3.2]), where i e : { e } → N is the natural inclusion map. The space H • ( B e , ˙ L ) is equipped with a natural W -structure and A ( e ) -action which commutes with the W -structure [Lu1], [Ka2, Theorem 3.1(6)]. For e ∈ N and ζ ∈ Irr A ( e ) , let K e,ζ = H • ( B e , ˙ L ) ζ := Hom A ( e ) ( ζ, H • ( B e , ˙ L )) , regarded as a W -representation. Let P be the collection all pairs of ( O , ζ ) such that O isa nilpotent G -orbit in N , ζ ∈ Irr A ( e ) ( e ∈ O ) and K e,ζ = 0 . We have the following uppertriangulation property: Theorem 4.3. ( [Lu] , [Ka2, Theorem 3.1] ) There is a bijective map, denoted Φ : P → Irr W ,such that for any ( O , ζ ) ∈ P Hom W (Φ( O , ζ ) , K e ′ ,ζ ′ ) = 0 if O e ′ 6⊆ O or ζ = ζ ′ (for O = O e ′ ); and when O e ′ = O and ζ = ζ ′ , dim Hom W (Φ( O , ζ ) , K e ′ ,ζ ) =1 . Let A W = C [ h ∨ ] ⋊C [ W ] . It is shown in [Ka2, Theorem 3.1(1)] (also see [Lu3, Sec. 7]) that A W can be identified with Ext • D bG ( N ) ( µ ∗ ˙ L , µ ∗ ˙ L ) , where D bG ( N ) is the G -equivariant derivedcategory for N . One can extend the W -structure on K e,ζ to an A W -module structure [Ka2,Theorem 3.1].For σ ∈ Irr W , define K σ = A W ⊗ C [ W ] σ as an A W -module with elements in ⊗ σ having degree . Define the partial ordering ≤ on P given by the closure ordering i.e. ( O , ζ ) ≤ ( O ′ , ζ ′ ) if and only if O ⊆ O ′ . For σ ∈ Irr W , define modules in [Ka2]:(4.6) K σ = K σ / X f ∈ gHom A W ( K τ ,K σ ) > ,τ ≤ σ im f Here gHom A W ( K τ , K σ ) > is defined as follows. Let X = ⊕ i X i and Y = ⊕ i Y i be graded A W -modules. A linear map from X to Y is graded if f ( X ) ⊂ Y i for some i . Let gHom A W ( X, Y ) > be the set of all graded maps f from X to Y satisfying that f ( X ) Y . Theorem 4.4. [Ka2, Theorem 3.3] Let ( O , ζ ) ∈ P . Let e ∈ O . As A W -modules, K e,ζ ∼ = K Φ( O e ,ζ ) . Deformed structure of a tempered module. For r ∈ C × , denote by Re( r ) thereal part of r . Recall that C ⊗ Z R admits a natural real form and hence h ∨ has a real formextending the one on C ⊗ Z R . We then similarly have a notion of Re( s ) for s ∈ h .Let ( O , ζ ) ∈ P . The space H Z G × C × ( e ) ◦ • ( B e , ˙ L ) admits an H -action as well as a A ( e ) -action ([Lu3, 10.11]). Let m ⊂ g ⊕ C be the Lie algebra of Z G × C × ( e ) ◦ .Define E e = H Z G × C × ( e ) ◦ • ( B e , ˙ L ) .The space H • G × C × := H • G × C × ( pt, C ) is the center of H via the geometric construction inSection 4.1 [Lu1, 8.13]. It can be canonically identified with the polynomial functions on g ⊕ C which are invariant under the action of G [Lu1, 1.11(a)]. Let ( s, r ) be a semisimpleelement in g ⊕ C . Define χ ( s,r ) : H • G × C × → C given by χ ( s,r ) ( f ) = f ( s, r ) . Let I ( s,r ) be themaximal ideal in H • G × C × containing all functions vanishing at ( s, r ) . Let Mod ( s,r ) H be thecategory of finite-dimensional H -modules annihilated by some powers of I ( s,r ) .We now further assume that [ s, e ] = 2 re . Then ( s, r ) ∈ m ⊕ C , where m ⊕ C is the Liealgebra of Z G × C × ( e ) ◦ . Hence χ ( s,r ) descends to a map from H • Z G × C × ( e ) ◦ to C . This definesa -dimensional H • Z G × C × ( e ) ◦ -module, denoted C ( s,r ) .We shall assume r ∈ R × in the remainder of this paper. (By rescaling the action on S ( V ) , one indeed has a natural equivalene of categories Mod ( r ′ s,r ′ ) H ∼ = Mod ( rs,r ) H .) Wefollow the terminology of temperedness of Lusztig [Lu5]: Definition 4.5. An H -module X in Mod ( s,r ) H is said to be tempered if any S ( h ∨ ) -weight s ∈ h of X satisfies Re( ω α ( s )) / Re( r ) ≥ for all α ∈ Π . Here ω α is the fundamental weightassociated to α satisfying β ∨ ( ω α ) = δ α,β .We remark that most of other references use the condition that Re( ω α ( s )) / Re( r ) ≤ instead of Re( ω α ( s )) / Re( r ) ≥ for the notion of temperedness. Those two notions arerelated by the Iwahori-Matsumoto involution. IRAC COHOMOLOGY OF STANDARD MODULES 15 Definition 4.6. Let Z G × C × ( e, s ) = Z G × C × ( e ) ∩ ( Z G ( s ) × C × ) . Define A ( e, s ) = Z G × C × ( e, s ) /Z G × C × ( e, s ) ◦ to be the component group. Define E s,e,r = C ( s,r ) ⊗ H • ZG × C × ( e ) ◦ H Z G × C × ( e ) ◦ • ( B e , ˙ L ) , which again admits H -module structure and admits A ( e, s ) -module structure. By regarding A ( e, s ) as a subgroup of A ( e ) , the W × A ( e, s ) -structure can be identified with the one of H • ( B e , ˙ L ) [Lu3, 10.13].For ζ ∈ Irr A ( e, s ) , define, as an H -module, E s,e,r,ζ = Hom A ( e,s ) ( ζ, E s,e,r ) . We shall call E s,e,r,ζ is a standard module . Now according to [Lu1, 10.8, 10.12], we have E s,e,r,ζ as an object in Mod ( s,r ) H .For each e ∈ N , let { e, h e , f } be a sl -triple. Then E rh e ,e,r,ζ is a tempered moduleby [Lu5, Theorem 1.21]. We remark that h e is not unique up to conjugation if G is notsemisimple. Lemma 4.7. Any tempered module of the form E rh e ,e,r,ζ admits a Z -grading.Proof. By doing some rescaling on the action of V ⊂ S ( V ) on E rh e ,e,r,ζ , the statement canbe reduced to the case that r = 1 . Thus it suffices to show that E h e ,e, ,ζ is unitary and hasreal weights, by using Lemma 2.2. It is equivalent to show that the Iwahori-Matsumotodual of E h e ,e, ,ζ is unitary. To show the last statement, one can see the arguments in [Ci,Pg 463, 464] or [So, Sec. 7], using [Op, 2.22]. For the assertion that E h ,e, ζ has realweights, it follows from [Lu5, Corollary 1.18 and Theorem 1.22]. (cid:4) Since Φ( O e , ζ ) appears with multiplicity one in E rh e ,e,r,ζ (Theorem 4.3), Φ( O e , ζ ) is achoice of deformation for E rh e ,e,r,ζ . We state our main conclusion in this section: Corollary 4.8. Let ( O e , ζ ) ∈ P . Set E = E rh e ,e,r,ζ . Recall that E Φ( O e ,ζ ) is an A W -moduledefined in Definition 3.1. Then E Φ( O e ,ζ ) ∼ = K Φ( O e ,ζ ) . Proof. By the definition of E Φ( O e ,ζ ) , there is a surjective map from A ⊗ C [ W ] Φ( O e , ζ ) to E Φ( O e ,ζ ) . By considering the W -structure from Theorem 4.3, we have a surjective mapfrom K Φ( O e ,ζ ) to E Φ( O e ,ζ ) . The statement now follows from Theorem 4.4. (cid:4) Definition 4.9. We keep using above notations. We shall say that Φ( O e , ζ ) is the lowest W -type of E rh e ,e,r,ζ . 5. Structure of standard modules This section is a continuation of Section 4. We shall keep using notations in Section 4.In particular, we fix a graded Hecke algebra H of geometric parameters. Standard modules and central characters. Recall that a standard module isdefined in Definition 4.6. Using [Lu5, Corollary 1.18], one can write a standard module interms of a module parabolically induced from an irreducible one. Moreover, the center of H (see [Lu3, 8.13]) also lies in the parabolic subalgebra of H . This gives the following: Lemma 5.1. The standard module E s,e,r,ζ admits a central character (Definition 2.5). Lowest W -types.Notation 5.2. Let e ∈ N . Let Irr e W be the set of irreducible W -representation σ suchthat σ is a lowest W -type of E rh e ,e,r,ζ for some ζ ∈ Irr A ( e ) (see Definition 4.9). Definition 5.3. Let E s,e,r,ζ be a standard module. By [Lu5, Theorem 1.15], E s,e,r,ζ has aunique simple quotient. Denote such unique simple quotient by L s,e,r,ζ . Corollary 5.4. Let E := E s,e,r,ζ be a standard module as before. Set E T = Hom A ( e,s ) ( ζ, E rh e ,e,r ) .Here A ( e, s ) is regarded as a natural subgroup of A ( e, h e ) . Write E T = E ⊕ . . . ⊕ E r as thedirect sum of irreducible tempered modules. Let σ i ∈ Irr e W be the lowest W -type of each E i . Then as a W -representation, L s,e,r,ζ contains copies of σ i appearing with the samemultiplicities.Proof. We shall prove by an inductive argument. We fix a semisimple element s ∈ h and r ∈ R × . Let { X , . . . , X k } be a collection of mutually non-isomorphic simple modules in mod s,r H . By [Lu5, Theorem 1.15(b)], X i = L s,e i ,r,ζ i for some e i ∈ N satisfying [ s, e i ] = 2 re i and ζ i ∈ Irr A ( e, s ) . By suitable renaming, we shall assume O e i 6⊇ O e j if i < j . Denoteby p ( e i ,ζ i ) , ( e j ,ζ j ) the multiplicity of L s,e j ,r,ζ j in the composition series of E s,e i ,r,ζ i . By [Lu3,10.8], [ p ( e i ,ζ i ) , ( e j ,ζ j ) ] i,j forms an upper triangular matrix with in diagonal. Now with the W -structure of E s,e i ,r,ζ i from Theorem 4.3, we obtain the statement. (cid:4) Definition 5.5. In the notation of Corollary 5.4, each σ i is called a lowest W -type of E s,e,r,ζ . Lemma 5.6. Let E be a standard module. Let E ′ = E ⊕ θ ( E ) if E = θ ( E ) and let E ′ = E if E = θ ( E ) . Let σ be a lowest W -type of E . Let U σ be a subspace of E ′ such that U σ ∼ = σ and U σ ⊂ E ′± . Then U σ is a choice of deformation for E ′ .Proof. If E is a standard module, then θ ( E ) is also a standard module. Let L be the uniquesimple quotient of E . Then θ ( L ) is the unique simple quotient of θ ( E ) . By the uniquenessof a simple quotient, we have E ∼ = θ ( E ) if and only if L ∼ = θ ( L ) . In the case that E = θ ( E ) ,the condition U σ ⊂ E ′± implies that U σ E and U σ θ ( E ) . With Corollary 5.4 and thefact that the unique simple quotient appears with multiplicity one in the standard module,we have the lemma. (cid:4) Filtrations on deformed standard modules.Lemma 5.7. Let E = E s,e,r,ζ be a standard module. Set E ′ as in Lemma 5.6. Set E T = Hom A ( e,s ) ( ζ, E rh e ,e,r ) and write E T = E ⊕ . . . ⊕ E l as the direct sum of irreducible IRAC COHOMOLOGY OF STANDARD MODULES 17 tempered modules. Let σ i be the lowest W -type of each E i . Set σ = σ . Then there existsan A W -module filtration (cid:8) Y i (cid:9) pi =0 on E ′ σ such that Y ⊂ Y ⊂ Y ⊂ . . . ⊂ Y k = E ′ σ and for i = 0 , . . . , p − , Y i +1 /Y i is isomorphic to a module K σ j for some σ j .Proof. By Lemma 2.2, E ′ = E ′ + ⊕ E ′− admits a Z -grading. Let U σ be an irreducible W -subspace of E ′ with U σ ∼ = σ as W -representations and U σ ⊂ E ′± . By Lemma 5.6, U σ is a choice for deformation. Identify σ with U σ . Recall that the A W -module E ′ σ has a Z -grading, and hence we obtain a Z -graded decomposition E ′ σ = L mk =1 E ′ k . Since deg( t w ) = 0 , W acts on each E ′ k Let Z be the sum of all σ i -isotypic components in E ′ σ , where i runs for , . . . , p . Wedecompose Z = L pl =1 P l into irreducible W -spaces compatible with the A W -module gradingi.e. for each P l , there exists a unique index j ( P l ) such that P l ⊂ E ′ j ( P l ) . By suitablerelabeling, we shall assume that if l < l , then j ( P l ) ≥ j ( P l ) .Now we define M l to be the A W -submodule of E ′ σ generated by Z , . . . Z l . Denote by σ ( Z l ) to be one of the W -representations σ , . . . , σ p to which P l is isomorphic. Now byconsidering the natural map from A W ⊗ C [ W ] Z i to M i /M i − , we obtain a surjective mapfrom K σ ( Z i ) to M i /M i − . Now we have dim E ′ σ ( Z i ) = P i dim( M i /M i − ) ≤ dim K σ ( Z i ) .On the other hand, we have dim E ′ σ = P i dim E i = P i dim K σ i . By using Corollary 4.8and Theorem 4.3, we have P i dim K σ i = P k dim K σ ( Z k ) . This shows that the inequalityconcerning dimensions above has to be an equality. Thus dim ( M l /M l − ) = dim K σ ( Z l ) and so M l /M l − ∼ = K σ ( Z l ) . This completes the proof. (cid:4) Vanishing Theorem We keep using notations in Section 4. In particular, H is a graded Hecke algebra ofgeometric parameters.6.1. Twisted-elliptic tempered case.Definition 6.1. Recall that G is a complex connected reductive group. The Dynkindiagram automorphism arising from − w induces an automorphism, still denoted θ , on G . Let G = G ⋊ h θ i . For g ∈ G , let Z G ( g ) be the set of elements in G centralizing g . Following [CH], a nilpotent element e is θ -quasidistinguished if there is no semisimpleelement t ∈ G such that Z G ( tθ ) is semisimple and e is distinguished in Z G ( tθ ) . Heredistinguishedness is defined as [CM]. A tempered module of the form E rh e ,e,r,ζ (Section4.3) is said to be twisted-elliptic if e is θ -quasidistinguished.The terminology is suggested by the study of [Ci2, CH], which establishes connectionsto twisted elliptic pairings (also see [Ch4]). In [CH, Proposition 3.3], it is shown that any θ -quasidistinguished nilpotent element has a solvable centralizer in g and vice versa. Let Irr gen f W be the set of all genuine f W -representations. For each e σ ∈ Irr gen f W , define a ( e σ ) = − X α> ,β> ,s α ( β ) < c α c β | α || β | tr e σ ( e s α e s β )dim e σ . Recall that V ′ = C ⊗ Z R . Define V ′⊥ = { v ∈ V : h v, v ′ i = 0 for all v ′ ∈ V ′ } . We havea W -invariant orthogonal decomposition V = V ′ ⊕ V ′⊥ and let pr V ′ : V → V ′ be thecorresponding projection map. For v , v ∈ V , define h v , v i V ′ = h pr V ′ ( v ) , pr V ′ ( v ) i . Theorem 6.2. [Ci2, BCT] Let E be a twisted-elliptic tempered module and let W ( rh ) × r be the central character of E . Let Irr E f W be the collection of irreducible f W -representation e σ such that a ( e σ ) = h h, h i V ′ . Then (1) H D ( E ) = 0 ; (2) As f W -representations, any irreducible summand of H D ( E ) lies in Irr E f W .Proof. We remark that the notion of temperedness in [BCT] is different. In order to applythe result, we need the fact that for the Iwahori-Matsumoto dual of a tempered module is’tempered’ in the sense of [BCT].By suitably rescaling on V , we first reduce our statement to the case that E = E h e ,e, ,ζ for some θ -quasidistingusihed e . Now using the argument in the proof of Lemma 4.7,we have that IM ( E h e ,e, ,ζ ) is unitary and so the hypothesis of [BCT, Proposition 5.7] issatisfied. We now consider (1). For each nilpotent element e such that e has a solvablecentralizer, using [BCT, Theorem 5.1] (or [Ci2, Theorem 1.0.1]) and [Ci2, Section 3.10], wehave a f W -type e σ such that Hom f W ( e σ, IM ( E rh e ,e,r,ζ ) ⊗ S ) = 0 . (6.7)Now by [CH, Proposition 3.3], we can replace the nilpotent e in (6.7) with θ -quasidistinguished e . Then one can apply [BCT, Proposition 5.7] (or its proof) and Lemma 2.9 to obtain (1).For (2), it follows from the formula of D [BCT, Theorem 3.5]. (cid:4) Non-twisted-elliptic tempered case. The non-twisted-elliptic tempered case fol-lows from Vogan’s conjecture which is proved in [BCT, Theorems 4.4 and 5.8] and saysthat the Dirac cohomology of an H -module determines its central character if it is nonzero.Hence, we have the following: Theorem 6.3. [BCT, Theorem 5.8] Let E rh e ,e,r,ζ be a tempered H -module as above. If e isnot θ -quasidistinguished (equivalently E rh e ,e,r,ζ is not twisted-elliptic), then H D ( E rh e ,e,r,ζ ) =0 .Proof. It is equivalent to show that if H D ( IM ( E rh e ,e,r,ζ )) = 0 , then e is θ -quasidistinguished.This follows from [BCT, Theorem 5.8] and [CH, Proposition 3.3]. Notice that the argu-ment in [BCT, Theorem 5.8] still applies to general geometric parameter case by using asurjectivity property in [Ci2, Section 3.10]. (cid:4) IRAC COHOMOLOGY OF STANDARD MODULES 19 General case. We now prove the main result in this paper. Theorem 6.4. Let H be a graded Hecke algebra of geometric parameters (Section 4.1).Let E be a standard module (Definition 4.6). Then H D ( E ) = 0 if and only if E is nottwisted-elliptic-tempered .Proof. We have shown the statement for the case that E is tempered in Theorem 6.2 andTheorem 6.3. From now on, we assume that E is not tempered. Write E = E s,e,r,ζ (Definition 4.6). Define E ′ = E ⊕ θ ( E ) if E = θ ( E ) and let E ′ = E if E = θ ( E ) as inLemma 5.6. Now E possess a Z -grading.Let σ be a lowest W -type for E . Let U σ be a subspace of E such that U σ ⊂ E ± and U σ ∼ = σ as W -representations. Identify U σ and σ . By Lemma 5.7, there exists a filtration (cid:8) Y i (cid:9) ki =0 on E ′ σ such that Y ⊂ Y ⊂ Y ⊂ . . . ⊂ Y k = E ′ σ and for each i = 0 , . . . , k − , Y i +1 /Y i is isomorphic to a module K σ i for some lowest W -type σ i of E ′ .Let R ( f W ) h e (possibly zero) be the set containing f W -representations whose summandsare in Irr h e f W , where Irr h e f W = ne σ ∈ Irr f W : a ( e σ ) = h h e , h e i V ′ o . Claim: H D A ( Y i +1 /Y i ) ∈ R ( f W ) h e . Proof: Set E T = E rh e ,e,r to be the tempered modulein Section 4.5. Since E T is unitary, the formula of D and Lemma 5.1 imply that a f W -representation e σ is a summand of ker π E T ( D ) = ker π E T ( D ) only if h h e , h e i V ′ = a ( e σ ) (seethe paragraph before [BCT, Section 4.1]). Here π E T defines the action of H on E T . Hencewe have H D ( E T ) = ker D ∈ R ( f W ) h e . Now Proposition 3.13 and Corollary 4.8 imply that H D ( Y i +1 /Y i ) ∈ R ( f W ) h e . Claim: H D A ( E σ ) ∈ R ( f W ) h e . Proof: This follows from a version of the six-term shortexact sequence (Proposition 2.6) for A W -modules and the previous claim.We are now ready to show that H D ( E ) = 0 . Decompose, as f W -representations, E σ ⊗ S = e X ⊕ e X such that e X ∈ R ( f W ) h e and e X has no summands in R ( f W ) h e . Then we can applyCorollary 3.11. With the previous claim, we have H D ( E ) ∈ R ( f W ) h e . Now take x ∈ H D ( E ) .The formula of D implies that ( − h s, s i V ′ + r a ( e σ )) x = 0 for some e σ ∈ Irr f W ∩ R ( f W ) h e .Hence we have ( −h s, s i V ′ + r h h e , h e i V ′ ) x = 0 . Since we are considering E is not tempered,we have h s, s i V ′ = r h h e , h e i V ′ (note that h s − rh e , rh e i V ′ = 0 ). Thus x = 0 and so H D ( E ) = 0 . This completes the proof. (cid:4) Twisted Dirac index. Our formulation of Dirac index is a slight variation of [CH].Let X = X + ⊕ X − be a Z -graded H -module admitting a central character.Define S = S when dim V ′ is even. Define S = S + ⊕ S − , where S + and S − are twoinequivalent simple C ( V ′ ) -modules. Define I ( X ) = ( X + − X − ) ⊗ S . Here we regard I ( X ) as a virtual f W -representation with the action given from the diagonalembedding ∆ (see 2.3).We similarly define as in Section 2.6 that π S ( D ) ± = π S ( D ) | X ± ⊗S , where π S : H ⊗ C ( V ′ ) → End C ( X ⊗ S ) is the map defining the action of H ⊗ C ( V ′ ) on X ⊗ S . Define H ± D ( X ) = ker π S ( D ) ± / (ker π S ( D ) ± ∩ im π S ( D ) ∓ ) . Lemma 6.5. As virtual f W -representations, I ( X ) = H + D ( X ) − H − D ( X ) . Proof. Decompose X ⊗ S into λ -eigenspaces e X λ of π S ( D ) . By linear algebra considerations, I ( X ) is reduced to the expression e X +0 − e X − , where e X ± = ( X ± ⊗ S ) ∩ e X . Now the expressionin right hand side follows from definitions. (cid:4) As a consequence, we have: Corollary 6.6. Let E be a standard module. Define E ′ as in Lemma 5.6. Then I ( E ′ ) = 0 if and only if E is not twisted-elliptic tempered. An alternate approach to the above corollary is to use a twisted Euler-Poincaré pairingin [Ch4] (also see [CT]).7. Dirac cohomology of ladder representations Ladder representations for p -adic groups are introduced by Lapid-Mínguez [LM] in termsof the Zelevinsky classification. We shall work on the graded Hecke algebra setting (see[BC2]). Fix integers l and n throughout the whole section.7.1. Zelevinsky segments and ladder representations. Let h l be the set of diagonalmatrices in gl ( l, C ) . Let O be the zero orbit and let L be the trivial local system on O .The datum ( h , O , L ) is a cuspidal triple. This gives a graded Hecke algebra of type A l − .We shall do some normalizations below.Let S l be the symmetric group permuting on l numbers. We also set W = S l . Let s i,i +1 ∈ S l be the transposition switching i and i + 1 . Let ǫ , . . . , ǫ l be a standard basisfor h l . Let R + l = { ǫ i − ǫ j : i < j } be a fixed set of positive roots. Let Π l be the set ofsimple roots in R + l . Set c ≡ . Define bilinear form h , i on h l satisfying h ǫ i , ǫ j i = δ i,j . Let H l = H ( h l , S l , Π l , c, r ) (Definition 2.1). We remark that our definition of H l differs fromthe one [AS] by a sign in a commutation relation. Definition 7.1. A pair ∆ = [ a, b ] of complex numbers is called a segment if b ≥ a . Let l (∆) = b − a + 1 , which will be called the length of ∆ . A multisegment m = { ∆ , . . . , ∆ n } is a collection of ordered (possibly repeated) segments ∆ satisfying P ni =1 l (∆ i ) = l . Theinteresting cases, especially for non-zero Dirac cohomology, are for a, b being integers ina segment ∆ = [ a, b ] . From now on, we shall also assume that a and b for a segment [ a, b ] are integers. For e ∈ Z and a multisegment m = { [ a , b ] , [ a , b ] , . . . , [ a n , b n ] } , define m ( m , e ) = card { i : a i ≤ e ≤ b i } . For example, if m = { [4 , , [2 , , [1 , } then m ( m , 1) = 1 , m ( m , 2) = 2 , m ( m , 3) = 2 , m ( m , 4) = 2 , m ( m , 5) = 1 . IRAC COHOMOLOGY OF STANDARD MODULES 21 Let m = { ∆ , . . . , ∆ n } be a multisegment. Let l i = l (∆ i ) for each i . Identify the group S l × . . . × S l n to be a natural subgroup of S l via the map ⊗ . . . ⊗ ⊗ s k,k +1 ⊗ ⊗ . . . ⊗ s A i + k,A i + k +1 , where s k,k +1 is in the k -th factor of LHS and A i = l + . . . + l i − . Let H l ⊗ . . . H l n be the subalgebra generated by v ∈ V and t w for w ∈ S l × . . . × S l r . Definea -dimensional H l ⊗ . . . H l n -module m = C x determined by ǫ k .x = ( a i + k − A i − x for A i < k ≤ A i +1 t w .x = ( − l ( w ) x Define E ( m ) = H l ⊗ H l ⊗ ... ⊗ H ln m . Let Z l be the collection of all Zelevinsky segments m = { [ a , b ] , . . . , [ a n , b n ] } with theproperty that b > b > . . . > b n . For m ∈ Z l , E ( m ) has a unique simple quotientdenoted by L ( m ) , which is a special case of Langlands-Zelevinsky classification. (This canbe checked, for example, by results of [Ev].) This coincides with a standard module inSection 4.3 (see [Lu5, Section 3]), up to the Iwahori-Matsumoto dual. In view of Lemma2.9, this will not play a significant role in our arguments and will be implicitly used in therest of this section. Proposition 7.2. An H l -module X is twisted-elliptic tempered if and only if X = E ( m ) for some m ∈ Z l satisfying that m = { [ − b , b ] , . . . , [ − b n , b n ] } . Let Z ellip l be a subset of Z l consisting of m satisfying the property: there exists m ′ ofthe form { [ − b , b ] , . . . , [ − b n , b n ] } such that m ( m , e ) = m ( m ′ , e ) for all e ∈ Z .Note that for each m ∈ Z ellip l , there is exactly one element, denoted m temp , in Z ellip l suchthat E ( m temp ) = L ( m temp ) is the twisted-elliptic tempered module supported at the samecentral character of L ( m ) . From Proposition 7.2, we have: Corollary 7.3. For any m ∈ Z l , L ( m ) has a central character of a twisted-elliptic temperedmodule if and only if m ∈ Z ellip l . Definition 7.4. Define Z ladd l to be the collection of elements m = { [ a , b ] , . . . , [ a n , b n ] } in Z l such that a > a > . . . > a n . For any m ∈ Z ladd l , we call L ( m ) to be the ladderrepresentation for H l .7.2. Arakawa-Suzuki functor. We keep using notations in the previous subsection. Weshall consider gl n -modules in the BGG category O . Let n − be the Lie subalgebra of g containing all strictly lower triangular matrices. For X ∈ O , define H ( n − , X ) = X/ n − X, the -th n − -homology on X .Let V = C n be a gl ( n, C ) -representation by the matrix multiplication. Let t n be theset of diagonal matrices in gl ( n, C ) . We naturally identify t n with C n . Let ǫ ′ , . . . , ǫ ′ n be astandard basis for t n . Using the standard bilinear form on C n , we also identify t ∨ n with C n .Similar to h l in the previous section, we have a set R + n of positive roots and a set Π n ofsimple roots. Let ρ be the half sum of all the positive roots in R + n . For a t n -weight µ = ( a , . . . , a n ) , define L ( µ ) to be the irreducible module sl ( n, C ) withthe highest weight module ν in the BGG category O . Definition 7.5. [AS] Define the Arakawa-Suzuki functor F γ : O → H − mod as follows: F γ ( X ) = H ( n − , X ⊗ V ⊗ l ) γ . The S l -action on F γ ( X ) is defined by w. ( x ⊗ v ⊗ . . . ⊗ v l ) = ( − l ( w ) x ⊗ v w (1) ⊗ . . . ⊗ v w ( l ) . One extends the S l -structure to H l -module by using a Casimir element, see [AS]. We shallnot need the explicit description of the action for S ( h l ) .Let P ( V ⊗ l ) be the set of t n -weights of V ⊗ l . We suppose that γ − µ ∈ P ( V ⊗ l ) . Then γ − µ = P ni =1 l i ǫ i for non-negative integers l , . . . , l n satisfying l + l + . . . + l n = l . Let ρ = P α ∈ R + l α . Define m γ,µ := { [ µ ′ , µ ′ + l − , . . . , [ µ ′ n , µ ′ n + l n − } , where µ ′ i = h µ + ρ, ǫ i i = h µ, ǫ i i + n − i +12 .Let D on = { γ : h γ, α i ≥ for all α ∈ R + } . We recall a result of Suzuki. Theorem 7.6. [Su, Theorem 3.2.2] Let h γ + ρ, α i > and let µ ∈ V such that λ − µ ∈ P ( V ⊗ l ) . Then F γ ( L ( µ )) ∼ = L ( m γ,µ ) For each m = { [ a , b ] , . . . , [ a n , b n ] } ∈ Z ladd l , let γ = ( b , . . . , b n ) − ρ = ( b − n − , . . . , b n + n − ) . Let µ = ( a , . . . , a ) − ρ . Then µ satisfy the condition in Theorem 7.6 and m γ,µ = m .The Arakawa-Suzuki functor gives the following BGG resolution [Su]: Theorem 7.7. [Su, Theorem 5.1.1] Let m ∈ Z ladd l with γ and µ as above. There exists aBGG-type resolution for the H l -module L ( m γ,µ ) : → M w ∈ S n : l ( w )= l ( w ) E ( m wγ,µ ) → . . . → M w ∈ S n : l ( w )=1 E ( m wγ,µ ) → E ( m γ,µ ) → L ( m γ,µ ) → , where w is the longest element in S l , and m wγ,µ = (cid:8) [ a w (1) , b ] , . . . , [ a w ( n ) , b n ] (cid:9) . (If a w ( k ) >b k + 1 , then set E ( m wγ,µ ) = 0 as convention. ) Dirac cohomology. We need some notations for representations of S l . A partition λ of l is a sequence ( l , l , . . . , l n ) of integers such that l + . . . + l n = l . For a partition λ of l , we define l ( λ ) to be the number of parts in λ . A partition λ is said to be evenor odd if l − l ( λ ) is even or odd respectively [St, Section 2]. We denote σ λ to be theirreducible S l -representation associated to λ . In particular, if λ = ( l ) , then σ λ is the trivialrepresentation.Denote by e S l the spin double cover of S l . We now describe the irreducible genuinerepresentations of e S l [Sc] (see [St, Section 7]). Denote by DP l the set of partitions of l with distinct parts. Denote by DP + l (resp. DP − l ) the subset of DP l consisting of evenpartitions (resp. odd partitions). Following [St], one can associate one irreducible projectiverepresentations, denoted e σ λ , when λ ∈ DP + l and can associate to two irreducible projective IRAC COHOMOLOGY OF STANDARD MODULES 23 representations, denoted e σ ± λ when λ ∈ DP − l . Those e σ λ ( λ ∈ DP + l ) and e σ ± λ ( λ ∈ DP − l ) aremutually non-isomorphic to each other and form a complete list of irreducible projective e S l -representations. In particular, S = e σ ( l ) if l is odd and S = e σ ± l if l is even,For each m ∈ Z l , denote by λ ( m ) the associated partition i.e. the parts in λ ( m ) are givenby the numbers b i − a i + 1 . For example, if m = { [3 , , [2 , , [1 , } , λ ( m ) = (5 , , . Weshall sometimes write m for λ ( m ) if the context is clear. Theorem 7.8. Let m ∈ Z ellip l ∩ Z ladd l . Let λ = λ ( m temp ) . Set e σ = e σ λ . (a) If λ ∈ DP + l and set e σ = e σ + λ ⊕ e σ − λ . Then if λ = ( n ) , H D ( L ( m )) = k M i =1 e σ, where k = 1 ε λ ε ( n ) ( l ( λ ) − / Here ε λ ′ = √ if λ ′ ∈ DP − l and ε λ ′ = 1 if λ ′ ∈ DP + l . (b) If λ = ( n ) , H D ( L ( m )) = S .Proof. There is only one w ∈ S n such that m w = m temp . For any m ∈ { m w = 0 : w ∈ S n } \{ m temp } , E ( m ) is not a twisted-elliptic tempered and hence H D ( E ( m )) = 0 by Theorem6.4. Thus by applying the six-term exact sequence (Proposition 2.6) repeatedly, we have H D ( L ( m )) ∼ = H D ( E ( m temp )) .H D ( E ( m temp )) is computed in [BC, Lemma 3.6.2] from [St, Theorem 9.3]. (cid:4) [ a, b ] and [ a ′ , b ′ ] are said to be linked if b + 1 = a ′ or b ′ + 1 = a . Wesay that [ a, b ] is left (resp. right) linked to [ a ′ , b ′ ] if b + 1 = a ′ (resp. b ′ + 1 = a ).The following lemma immediately follows from the definition of Z ladd l . Lemma 7.9. Let m ∈ Z ladd l . For each segment [ a, b ] in m , there is at most one segmentin m which is left linked to [ a, b ] . The same is also true by replacing left with right. Definition 7.10. We say that a multisegment m has up-and-then-down property if thereexists an integer N such that for any i < j ≤ N , m ( m , i ) ≤ m ( m , j ) and for any N ≤ i < j , m ( m , i ) ≥ m ( m , j ) .Following from definitions, we immediately have the following: Lemma 7.11. For any m ∈ Z ellip l ∩ Z ladd l , m has up-and-then-down property. Algorithm for w ( m ) .Definition 7.12. Let m = { [ a , b ] , . . . , [ a n , b n ] } ∈ Z ladd l . Define ∼ as follows: [ a i , b i ] ∼ [ a j , b j ] if and only if [ a i , b i ] and [ a j , b j ] are linked. Let F ( m ) be the set of equivalence classesof the relation generated by ∼ .For each f ∈ F ( m ) , we write all the elements in the form [ a , b ] , . . . , [ a p , b p ] , where p = card( f ) and after suitable relabeling, we may assume that [ a i , a i +1 ] are right linked to [ a i +1 , b i +1 ] for each i . We let J ( f ) = [ a p , b ] . For such f , define a ( f ) = a p and b ( f ) = b . Algorithm: Let m ∈ Z ladd l . Suppose card F ( m ) = s .Step 1 We order the set F ( m ) in two ways: { f i , f i , . . . , f i s } and { f j , . . . , f j s } such that b ( f i ) > b ( f i ) > . . . > b ( f i r ) and a ( f j ) < a ( f j ) < . . . < a ( f j r ) .Step 2 For each f k , we write the segments in f k : f k = { [ a ( k, , b ( k, , [ a ( k, , b ( k, , . . . , [ a ( k, p k ) , b ( k, p k )] } , where p k is the number of segments in f k and those a and b satisfy the relationthat b ( k, e ) + 1 = a ( k, e − for e = 2 , . . . , p k .Step 3 Now define a function G : { a , . . . , a n } → { a , . . . , a n } such that G ( a ( i e , d )) = (cid:26) a ( j e , p j e ) if d = 1 a ( j e , d − if d > Step 4 We now define an element w ( m ) ∈ S n such that G ( a i ) = a w ( m )( i ) . Example 7.13. m = { [7 , , [4 , , [3 , } , we have w ( m ) = (1 , ∈ S . Example 7.14. Let m = { [5 , , [3 , , [2 , , [1 , } . Then J ( f i ) = [2 , , J ( f i ) = [3 , , J ( f i ) = [1 , , J ( f j ) = [1 , , J ( f j ) = [2 , , J ( f j ) = [3 , . Then we have w ( m ) =(1 , , , .7.6. Partition α ( m ) . Let m ∈ Z ladd l ∩ Z ellip l . We repeat Steps 1 and 2 in Section 7.5 toobtain the indices f i k , f j k , a ( k, e ) and b ( k, e ) . Definition 7.15. For a partition α = ( l , . . . , l n ) of l , let α T = ( t , . . . , t r ) , where α T isthe transpose of α .We define the length of the e -th hook of α to be the number l e + t e − e + 1 . Hence thelength of the e -th hook is to count the number of boxes starting from the rightmost of the e -th row to the diagonal box, and then from the diagonal box down to the bottom of the e -th row. For example, for a hook partition α = (5 , , , . The length of the first hook is8. We define the height of the e -th hook of α to be the number t e − e + 1 .For e = 1 , . . . , s , define hk( m , e ) = b ( i e , − a ( j e , p j e ) + 1 = b ( f i e ) − a ( f j e ) + 1 . Define ht( m , e ) = card { k : a ( j e , p e ) ≤ b k ≤ b ( i e , } . Define a partition α ′ ( m ) associated to m such that hk( m , e ) gives the length of the e -thhook and ht( m , e ) gives the heights of hooks. The definition of Z ladd l assures that α ′ ( m ) iswell-defined i.e. a sequence of non-increasing positive integers. Define α ( m ) = α ′ ( m ) T (thetranspose of α ′ ( m ) ). Lemma 7.16. Let e = 1 , . . . , s . Let w = w ( m ) defined in Section 7.5. Let N be the integersuch that a N = a ( i e , . Then ht( m , e ) = w ( N ) − N + e. IRAC COHOMOLOGY OF STANDARD MODULES 25 Proof. In the notation of Section 7.5, we have a corresponding function G which permutes a , . . . , a l . Then G ( a N ) = G ( a ( i e , a ( j e , p e ) . Now we have | w ( N ) − N | + 1 counts thenumber of elements in { i : b ∗ ≤ b i ≤ b ′ } where b ∗ = b w ( N ) and b ′ = b N if w ( N ) < N ; b ∗ = b w ( N ) and b ′ = b w ( N ) otherwise. Notethat we have card (cid:8) i : a w ( N ) ≤ b i < b N (cid:9) is e − (by the up-and-then-down property). Nowby considering two separate cases, we have the statement. (cid:4) Lemma 7.17. Let A ( e ) = { a s : a ( j e , p e ) ≤ b s ≤ b ( i e , , a s = a ( t, for some t } . Forany a i ∈ A ( e ) \ A ( e + 1) , there is a N such that a N = b i + 1 and furthermore, a i − k = a N .Proof. Let b i = a ( i k , q k ) = a ( j k ′ , q k ′ ) ∈ A ( e ) for some q k and q k ′ . We now look at f i k and f j k ′ in F ( m ) . The definitions of A ( k ) (which does not contain the rightmost end a ( j k , )guarantees that there are a N = b i . Now the property of Z ladd l and up-and-then-downproperty guarantee that there are exactly k − a t ’s such that a i < a t < a N = b i . (cid:4) Lemma 7.18. Keep using the notation in the previous lemma. Let |F ( m ) | = s . For ≤ k ≤ s , we have |A ( e ) \ A ( e + 1) | = ht( m , e ) − ht( m , e + 1) − (Here A ( s + 1) = ∅ .)Proof. By definitions, we have A ( e ) = ht( m , e ) − ( s − e + 1) . Here the term s − e + 1 comes from the strict equality for a ( j e , > b i in the definition of A ( e ) . Hence we have |A ( e ) \ A ( e + 1) | = ht( m , e ) − ht( m , e + 1) − . (cid:4) Existence of a W -representation.Proposition 7.19. Let m ∈ Z ladd l ∩ Z ellip l . Recall that α ( m ) is defined in Section 7.6. Set α = α ( m ) . Then Hom W ( σ α , L ( m ) | W ) = 0 . Proof. Construct α ( m ) as in Section 7.6. Write α ( m ) = ( m , . . . , m n ) as a partition of n .Let ω = ( m , . . . , m n ) as a t n -weight. By the classical Schur-Weyl duality, there is a subrep-resentation in V ⊗ l , which is isomorphic to α ( m ) ⊠ L ( ω ) as C [ S l ] ⊗ gl ( n, C ) -representation.(Here the S l -action on V ⊗ l is defined as in Definition 7.5.) Recall from Section 7.2 thatthere exist t n -weights γ, µ such that m = m γ,µ , and we have L ( m ) ∼ = F γ ( L ( µ )) in Theorem7.6. Thus to prove the proposition, it suffices to show that ( L ( µ ) ⊗ L ( ω )) γ is non-zero.This shall follow from the following claim and the PRV conjecture, proved by Kumar [Ku]. Claim: There exists w ∈ S n and w ∈ S n such that w µ + w ω = γ . Proof of the claim: We choose w = w ( m ) and set w := w for simplicity. Let m = { [ a , b ] , . . . , [ a n , b n ] } with a > a > . . . > a n . Let µ ′ = ( a , . . . , a n ) and let γ ′ =( b , . . . , b n ) . Recall that from the previous construction we have µ = µ ′ − ρ and γ = γ ′ − ρ + 1 . We also write ( a ′ , . . . , a ′ n ) = µ and ( b ′ , . . . , b ′ n ) = γ . We also construct the set F ( m ) = { f , . . . , f s } as in Definition 7.12.To prove the claim, it is equivalent to show that γ − w µ ∈ S n ω . To this end, it sufficesto show for each m i , there exists k such that b ′ k − a ′ w ( k ) = m i (with the same multiplicity).We first compute those m t . For ≤ t ≤ s , we have m t = hk( m , t ) − ht( m , t ) + e. We now consider t > s . For ht( s ) + s − ≥ t > s , we have m t = s . In general, for ht( m , s − x ) + ( s − x ) − ≥ t > ht( m , s − x + 1) + ( s − x ) , we have m t = s − x .We consider the tableaux obtained from γ − e µ w . For the first row, since w (1) = w (1) = n , we have b ′ − a ′ w (1) = b − n − 12 +1 − ( a w (1) − n − w (1) + 12 ) = b − a n +1 − ( n − 1) = hk( m , − ht( m , For ≤ t ≤ s , we choose N such that b N = b ( i t , 1) = b ( f i t ) . By definitions, a w ( N ) = a ( j t , p j t ) = a ( f j t ) . (Note that when t = 1 , i t = 1 and j t = n and the that case has beentreated.) b ′ N − a ′ w ( N ) = b N + 1 − n − N + 12 − ( a w ( N ) − n − w ( N ) + 12 )= b ( f t j ) + 1 − a ( j t , p j t ) − w ( N ) − N m , t ) − ht( m , t ) + e by Lemma 7.16We now consider t > s . We consider the case that ht( m , s ) + s − ≥ t > s . By Lemma 7.18 card { t : ht( m , s ) + s − ≥ t > s } = card A ( s ) . For any ht( m , s ) + s − ≥ t > s , we shall assign any N with a N ∈ A ( s ) , and make theassignment to be bijective. We now compute that b ′ N − a ′ w ( N ) = b N + 1 − n − N + 12 − ( a w ( N ) − n − w ( N ) + 12 )= b N + 1 − n − N + 12 − ( b N + 1 − n − N − s ) + 12= s The second equality in the above equations follows from Lemma 7.17. For the case that ht( m , s − x ) + ( s − x ) − ≥ t > ht( m , s − x + 1) + ( s − x ) , the proof is analogous to thecase that ht( m , s ) + s − ≥ t > s with the use of Lemmas 7.17 and 7.18. We omit thedetails. (cid:4) W -representations in twisted-elliptic ladder representations. Fix a choice S of basic spin representation of e S l (as in Section 2.3). Lemma 7.20. Let m ∈ Z ladd l ∩ Z ellip l . Let α = α ( m ) . Let λ = λ ( m temp ) . Then if α is nota hook or l is odd, dimHom f W ( e σ λ , σ α ⊗ S ) ≥ ε λ ε ( l ) ( l ( λ ) − / . IRAC COHOMOLOGY OF STANDARD MODULES 27 If α is a hook and n is odd, then dimHom f W ( e σ λ , σ α ⊗ S ) is or , depending on the choiceof the basic spin representation S .Proof. Note that if α is a hook, then λ = ( n ) . The lemma follows from [St, Theorem 9.3].In the notation of [St, Theorem 9.3], we have g λ,α ≥ since there is an obvious way tosatisfy the criteria. We only give an example. We consider the partition α = (4 , , , , and the corresponding λ = (8 , , . Now we can fill the numbers , ′ , , ′ , , ′ in theYoung tableaux for α as 1’ 1 1 11’ 2’ 2 21’ 2’ 31’ 2’1 2Then the word for α (terminology in [St, p. 125]) is ′ ′ ′ ′ ′ ′ ′ . Then we canwork out the multiplicity m i ( j ) [St, (8.4)]. To check the lattice property [St, Theorem9.3(b)(1)], the interesting case is that m (14) = m (14) = 3 . In that case, w = 1 = 2 , ′ and hence the lattice property is satisfied. (cid:4) Theorem 7.21. Let m ∈ Z ladd l ∩ Z ellip l . Let β = λ ( m ) T and let α = α ( m ) (see Section7.6). Then we have dim Hom W ( σ α , L ( m ) | W ) = dim Hom W ( σ β , L ( m ) | W ) = 1 . Proof. We first consider (1). It is well-known that dim Hom W ( σ β , L ( m ) | W ) = 1 (see Corol-lary 5.4). Let p = dim Hom W ( σ α , L ( m ) | W ) . By Proposition 7.19, we have p ≥ . By [BC,Proposition 4.4.2] and a property of D [BC4, (6.2.3)], we have H D ( L ( m )) = ker D = ker D .We have that D acts as a zero operator on the e σ λ -isotypic component by the formula(2.1). Thus with Lemma 7.20, we have dim Hom f W ( e σ λ , H D ( X )) = dim Hom f W ( e σ λ , ker D ) ≥ ε λ ε ( l ) ( l ( λ ) − / p Now Theorem 7.8 forces p = 1 . This proves (1). (cid:4) Unequal parameter case for type BC Affine Hecke algebras for unequal parameters are constructed geometrically by Kato[Ka]. The representation theory of those algebras is further studied in [CK] and [CKK]and others. For the harmonic analysis approach, see the study of Slooten [Sl] and Opdam-Solleveld [OS].8.1. Graded Hecke algebra for type C . Let V n = C n . Let ǫ , . . . , ǫ n be a standardbasis for V n equipped with a bilinear form given by h ǫ i , ǫ j i = δ ij . We also identify the dualspace V ∨ n with C n such that f ∈ V ∨ n ∼ = C n , v ∈ V n , f ( v ) = h f, v i . Let R n = { ǫ i − ǫ j : 1 ≤ i = j ≤ n } ∪ { ǫ i : i = 1 , . . . , n } be a root system of type C n . Let Π n = { ǫ i − ǫ i +1 : i = 1 , . . . , n − } ∪ { ǫ k } be a fixed setof simple roots in R n . Let W n be the Weyl group of type C n .For m > , the graded Hecke algebra H n,m (depending on c ) of type C n is generated bythe symbols { t w : w ∈ W } and { v ∈ V n } subject to the following relations:(1) v v = v v for v , v ∈ V n ;(2) t w t w = t w w for w , w ∈ W n ;(3) t s α v − s α ( v ) t s α = c α α ∨ ( v ) .(4) c α = 1 if α is a short root and c α = m if α is a long root.We drop the parameter r from our notation in Section 2.1 and it is not hard to recoverstatements with r . Any H n,m -module can extend to a Z -graded module since θ is theidentity in this case.We shall work on algebraic definitions: Definition 8.1. (1) Let J ⊂ Π . Let H J be the subalgebra of H generated by t s α ( α ∈ J ) and v ∈ V ⊂ S ( V ) .(2) An H n,m -module is said to be E a standard module if there exists an irreducible H J -module U satisfying the relation that E ∼ = H ⊗ H J U and for any S ( V ) -weight γ , γ ( ω α ) ≥ for any α ∈ J and γ ( α ) < for any α ∈ Π \ J .(3) An H n,m -module E is said to be tempered if E is standard with J = Π in (1).By [Lu5, Section 3], the above definition of standard modules coincide the one in Defi-nition 4.6 for geometric parameters.Notice that the definition of temperedness differs from [CK] by a sign. Results from [CK]can be recovered to our setting by applying the Iwahori-Matsumoto involution in Section2.7. We shall implicitly use this.8.2. A W -module structure on tempered modules. We follow the arguments in theproof of [CK, Corollary 4.23] (also see the proof of [Ka2, Pg 1062, Claim C]). For k ≥ ,let m be in the open interval of ( k, k + 1) with k ≥ . Let M P m be the set of markedpartitions (see [Ka2], [CK, Definition 1.28] for the definition) parametrizing irreducibletempered modules of H n,m with real central characters. We denote by temp m ( τ ) thetempered module parametrized by the marked partition τ ∈ M P m at m . Let m = k +12 .For any τ ∈ M P m = M P m , we have temp m ( τ ) ∼ = temp m ( τ ) as W n -modules. The W n -structures of temp m ( τ ) is then abstractly known from the gen-eralized Springer correspondence (see the geometric parameters in [Lu1, Sec. 2.13(e)]). Let P n,m be the set of all irreducible tempered modules with real central characters, equippedwith the partial ordering ≤ n,m from the corresponding generalized Springer correspondence.We then obtain the map Φ n,m : P n,m → Irr W n satisfying the upper triangular property (c.f. Theorem 4.3). IRAC COHOMOLOGY OF STANDARD MODULES 29 Let A W n = S ( V n ) ⋊ W n be the skew-group ring. For σ ∈ Irr W n , we can define an A W n -module K n,mσ as the one in (4.6) with respect to the ordering ≤ n,m .Following the argument in Corollary 4.8, we have: Proposition 8.2. Let E ∈ P . Let σ = Φ n,m ( E ) . Then E σ ∼ = K n,mσ . Here E σ is an A W -module defined similarly as the one in Definition 3.1 (i.e. ignoring theterm r ). A W -module structure on standard modules. We keep using notations in Section8.2. We also need some information for the central characters of temp( τ ) . In particular,let m be in an open interval. Following the parametrization of the marked partition,for τ ∈ M P ( m ) , the associated central character is given by the value a and s in thenotation of [CK] (also see [CK, Convention 1.32]). For τ , τ ∈ M P ( m ) , temp m ( τ ) and temp m ( τ ) have the same central character if and only if temp m ( τ ) and temp m ( τ ) havethe same central character. Recall that temp m ( τ i ) ( i = 1 , ) comes from the generalizedSpringer correspondence, and we can write into the form temp m ( τ i ) = E h ei ,e i , ,ζ i for somenilpotent e i ∈ N in the notation of Section 4.3. Since h e i determines the central characterof E h ei ,e i , ,ζ i , we must have that e and e lie in the same nilpotent orbit.Let E be a standard module with a real central character. Then there exists J ⊂ Π suchthat E = H ⊗ H J U with some H J -module U described in Definition 8.1. Let V ⊥ J = { v ∈ V n : h v, α i = 0 for any α ∈ J } . Now let U ′ be an H J -module isomorphic to U as vector space and the H J -module structureis determined by: π U ′ ( t w ) u = π U ( t w ) u ( w ∈ W ) ,π U ′ ( α ) u = π U ( α ) u ( α ∈ J ) ,π U ′ ( v ) u = 0 , where π U ′ (resp. π U ) is the map defining the action of H J on U ′ (resp. U ).Now set E T = H ⊗ H J U ′ . One checks from Definition 8.1(2) that E T is tempered. Wenow let E , . . . , E r be the collection of all the irreducible composition factors for E T . By theweight considerations, we deduce that E , . . . , E r have the same central character. (Hencethey have to correspond to the same nilpotent orbit in the sense of the first paragraph ofthis subsection.) Write the central character for E i as W n h and we shall call h h, h i to bethe length of E T .Now with the weight considerations as in the Langlands classification (see [Ev, KR]) andsuitable ordering using the generalized Springer correspondence, we can deduce a versionof Corollary 5.4 in this setting. From there, we can deduce a version of Lemma 5.7: Lemma 8.3. Let E be a standard module of H n,m . Let E T , E , . . . , E r be tempered modulesas above. Let σ i be the lowest type for E i for i = 1 , . . . , r . Set σ = σ . Then there exists a A W n -module filtration (cid:8) Y i (cid:9) pi =0 on E σ such that Y ⊂ Y ⊂ Y ⊂ . . . ⊂ Y k = E σ and for i = 0 , . . . , p − , Y i +1 /Y i is isomorphic to a module K n,mσ j for some σ j . Dirac cohomology of tempered modules. This section is essentially in the workof [Ci2]. We provide some explanations since the statement we look for is not stated in[Ci2].Let k ≥ . Following Section 3.1 in [CK], we consider m / ∈ Z and m ∈ ( k, k + 1) . Wefirstly consider a partition λ of n . According to [CK, Theorem 3.5] (also see [CK, Theorem3.16]), λ defines a tempered module, denoted E λ , with the central character W s λ . Here s λ ∈ C n is given as in [CK, Section 3.1], and can be computed from [CK, Definition 1.28]. Example 8.4. We consider H ,m . Identify C with h with h , i and the standard basis. Let σ = (3 , , . Then W s λ = W ( m, m + 1 , m + 2 , m − , m, m − , m − .We now study the Dirac cohomology of E λ . One can apply the formulas in [Ci2]. Fora partition λ m of n , we view it as a Young tableaux. For the box in the i -th row and j -thcolumn, we define c ( i, j ) = m + i − j , called a c -content. For a partition λ of n , we define p l ( λ + m ) to be the sum of l -th power of all the c -contents in the boxes of (the Youngtableaux of) λ . We have p ( λ m ) = p ( λ ) + 2 mp ( λ ) + nm . Using [Ci2, Section 3.7], weobtain a version of [Ci2, Theorem 1.1]. With [BCT] and the W -structure of the generalizedSpringer correspondence in the previous subsection, we have: Proposition 8.5. Let k ≥ . Let m ∈ ( k, k + 1) with m = Z . Let P n be the cardinalityof the set of partition of n . There is exactly P n isomorphism classes of tempered modulessuch that H D ( E ) = 0 . The central characters of those tempered modules are given by W s λ described above, where λ runs for all partitions of n . Dirac cohomology of standard modules.Theorem 8.6. Let m ≥ . Let H n,m be defined in Section 8.1. Let E be a standard moduleof H n,m (Definition 8.1). Then H D ( E ) = 0 if and only if E is not a tempered module inProposition 8.5.Proof. When m ≥ and m ∈ Z , H n,m is isomorphic to a geometric graded Hecke algebraof Lusztig [Lu1]. That case has been dealt in Theorem 6.4. We only have to consider m / ∈ Z . With Lemma 8.3, Propositions 8.2 and 8.5, we can modify the arguments inTheorem 6.4 to obtain the statement. We omit the details. (cid:4) A remark on noncrystallographic types. Other than unequal parameter case, apossible generalization is for R of noncrystallographic types. Results in Section 3 still apply.We also have some other partial information in the literature. Some structural informationfor tempered modules can be found in [KR] and [Kr]. For the Dirac cohomology, somecomputations are done in [Ch]. IRAC COHOMOLOGY OF STANDARD MODULES 31 References [AS] T. Arakawa and T. 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IRAC COHOMOLOGY OF STANDARD MODULES 33 Department of Mathematics, University of Georgia, Athens GA E-mail address ::