Cocenters and representations of pro- p Hecke algebras
aa r X i v : . [ m a t h . R T ] M a y COCENTERS AND REPRESENTATIONS OF PRO- p HECKE ALGEBRAS
XUHUA HE AND SIAN NIE
Abstract.
In this paper, we study the relation between the co-center ˜ H and the representations of an affine pro- p Hecke algebra˜ H = ˜ H (0 , − ). As a consequence, we obtain a new criterion onthe supersingular representation: a (virtual) representation of ˜ H is supersingular if and only if its character vanishes on the non-supersingular part of the cocenter ˜ H . Introduction
Let G be a p -adic group and ˜ W be its Iwahori-Weyl group. TheIwahori-Hecke algebra ˜ H q is a deformation of the group algebra of ˜ W .It plays an important role in the study of the ordinary representationsof G .For representations of G in characteristic p (the defining character-istic), one expects that there is a close relation between the mod- p representations of G and of the pro- p Hecke algebra ˜ H of G . The pro- p Hecke algebra is a deformation of the group algebra ˜ W (1), with param-eter q = 0. Here ˜ W (1) is the pro- p Iwahori-Weyl group, an extensionof ˜ W by a finite torus. For a pro- p Hecke algebra of a p -adic group (i.e. the associatedgroup ˜ W (1) is the pro- p Iwahori-Weyl group of a p -adic group), therepresentations are studied by Abe [1] and Vign´eras [20], based on theBernstein presentation and Satake-type isomorphism.In this paper, we study the representations via a different approach,the “cocenter program”.Let us first provide some background on the “cocenter program”.For a group algebra of a finite group, the cocenter is very simple. Theelements in the same conjugacy class of the group have the same imagein the cocenter of the group algebra and the cocenter has a standardbasis given by the conjugacy classes. There is a perfect pairing betweenthe cocenter of the group algebra and the Grothendieck group of finitedimensional (complex) representations, via the trace map. This is a“toy model” for the “cocenter program”. Mathematics Subject Classification.
Key words and phrases. affine Coxeter groups, Hecke algebras, p -adic groups.X. H. was partially supported by NSF DMS-1463852. For finite or affine Hecke algebras (with nonzero parameters), thecocenter is more complicated. The elements in the same conjugacyclass may not have the same image in the cocenter. However, basedon some remarkable properties on the minimal length elements in theWeyl group, one may show that the elements of minimal length in aconjugacy class of the Weyl group still have the same image in thecocenter of the Hecke algebra and the cocenter is still indexed by theconjugacy classes of the group.For finite Hecke algebras (with generic parameters), the relation be-tween the cocenter and the representations is fairly simple. The dimen-sion of the cocenter equals the number of irreducible representations,and the trace map gives a perfect pairing between the cocenter andrepresentations. For affine Hecke algebras, both the dimension of thecocenter and the number of irreducible representations are infinite andthe counting-number method does not simply apply. In [3], we in-troduced the rigid cocenter and rigid quotient of Grothendieck groupof representations, and shows that they form a perfect pairing underthe trace map, and the whole cocenter and all the finite dimensionalrepresentations can be understood via the rigid part of the parabolicsubalgebras.
For finite and affine pro- p Hecke algebra (with parameter q =0), the trace map from the cocenter to the linear functions on theGrothendieck group of finite dimensional representations, is surjective,but not injective. However, the knowledge of the structure of the co-center still does a big help in the understanding of representations.We first show in Theorem 2.4 that Theorem 0.1.
For finite and affine pro- p Hecke algebra ˜ H , the co-center is spanned by the image of T w , where w runs over the minimallength elements in its conjugacy class in ˜ W (1) . For finite pro- p Hecke algebras, the irreducible representations arejust characters. For affine pro- p Hecke algebras, it is easy to constructa family { π J, Γ , Ξ ,V } of representations by taking the parabolic inductionfrom characters of the parabolic algebras of ˜ H . See § Theorem 0.2.
Let ˜ H be an affine pro- p Hecke algebra. The set { π J, Γ , Ξ ,V } is a basis of the Grothendieck group R ( ˜ H ) . For pro- p Hecke algebras of p -adic groups, the above result is ob-tained by Abe [1] and Vign´eras [20]. Our strategy is different fromAbe and Vign´eras. We use the structure of the cocenter (see Theorem2.4) and the character formula established in § RO- p HECKE ALGEBRAS 3
Among all the representations of the affine pro- p Hecke algebra˜ H , the supersingular representations are the most important ones. Abe[1] showed that for pro- p Hecke algebras of p -adic groups, any repre-sentation can be obtained from supersingular ones by the parabolic in-ductions. The classification of supersingular representations for affinepro- p Hecke algebras is obtained by Vign´eras [20]. In Theorem7.8, wegive a new proof of the classification (in the more general setting) andwe also give a new criterion of supersingular representations.
Theorem 0.3.
A virtual representation π of ˜ H is supersingular if andonly if T r ( ˜ H nss , π ) = 0 , where ˜ H nss is the non-supersingular part ofthe cocenter defined in § The paper is organized as follows.In section 1, we recall the definition of pro- p Hecke algebras andtrace maps. In section 2, we study the cocenters of finite and affinepro- p Hecke algebras. In section 3, we discuss the cocenter and repre-sentations of finite pro- p Hecke algebras. In section 4, we define theparabolic subalgebras of affine pro- p Hecke algebras. In section 5, wediscuss the standard representatives associated to minimal length el-ements and study their power. Such results is used in section 6, inwhich we study the character formula for affine pro- p Hecke algebras.Finally, in section 7, we give a basis of the Grothendieck group of finitedimensional modules of affine pro- p Hecke algebras and gives a newcriterion of supersingular representations.
Acknowledgement.
After the paper was finished, we learned fromVign´eras that Theorem 4.2 is also proved in her paper [22] and herpreprint [21]. We thank her for sending us [21] and for many usefulcomments. Preliminary
We start with a sextuple (
W, S, Ω , ˜ W , Z, ˜ W (1)), where ( W, S ) isa Coxeter system, Ω is a group acting on W and stabilizing S , ˜ W = W ⋊ Ω, Z is a finite commutative group, and we have a short exactsequence 1 → Z → ˜ W (1) π −→ ˜ W → . Let ℓ be the length function on W . It extends to a length functionon ˜ W by requiring that ℓ ( τ ) = 0 for τ ∈ Ω, and inflates to a lengthfunction ℓ on ˜ W (1).Since Z is commutative, the conjugation action of ˜ W (1) on Z inducesan action of ˜ W on Z , which we denote by • .For any subset D of ˜ W , we denote by D (1) the inverse image of D in ˜ W (1). XUHUA HE AND SIAN NIE
Let w ∈ W . The support of w is defined to be the set of simplereflections that appear in some (or equivalently, any) reduced expres-sion of w and is denoted by supp( w ). For w ∈ W and τ ∈ Ω, wedefine supp( wτ ) to be ∪ i ∈ N τ i (supp( w )). For ˜ w ∈ ˜ W (1), we definesupp( ˜ w ) = supp( π ( ˜ w )). Now we recall the definition of generic pro- p Hecke algebra intro-duced by Vign´eras in [19].Let T = ∪ w ∈ W wSw − ⊆ W be the set of reflections in W . Let k bean algebraically closed field. We choose ( q t , c t ) ∈ k × k [ Z ] for t ∈ T (1)such that • q wtw − = q t for w ∈ ˜ W (1) and q tz = q t for z ∈ Z . • c wtw − = w • c t for w ∈ ˜ W (1) and c tz = c t z for z ∈ Z .Let ˜ H ( q, c ) be the associative k -algebra with basis ( T w ) w ∈ ˜ W (1) subjectto the following relations T w T w ′ = T ww ′ , for w, w ′ ∈ ˜ W (1) with ℓ ( ww ′ ) = ℓ ( w ) + ℓ ( w ′ ); T s = q s T s + c s T s , for s ∈ S (1) . We denote by H ( q, c ) the subalgebra of ˜ H ( q, c ) spanned by T w for w ∈ W (1).In this paper, we are mainly interested in the case where q t ≡
0. Wesimply write ˜ H for ˜ H (0 , c ) and write H for H (0 , c ). In this case, thesecond relation becomes T s = c s T s for s ∈ S (1). The algebra ˜ H playsan important role in the study of mod- p representations of reductivegroups over finite fields of characteristic p and over p -adic fields.In the case where W is a finite Coxeter group, we call ˜ H a finitepro- p Hecke algebra . In the case where W is an affine Weyl group, wecall ˜ H an affine pro- p Hecke algebra . Let [ ˜ H , ˜ H ] be the commutator of ˜ H , the subspace of ˜ H spannedby [ T w , T w ′ ] := T w T w ′ − T w ′ T w for w, w ′ ∈ ˜ W (1). Let ˜ H = ˜ H / [ ˜ H , ˜ H ] bethe cocenter of ˜ H . Denote by R ( ˜ H ) k the ( k -span of the) Grothendieckgroup of finite dimensional representations of ˜ H over k , i.e., the k -vector space with basis given by the isomorphism classes of irreduciblerepresentations of ˜ H . Consider the trace map T r : ˜ H → R ( ˜ H ) ∗ k , h ( V T r ( h, V )) . Similar maps for affine Hecke algebras with generic nonzero param-eters are studied in the joint work of Ciubotaru and the first-namedauthor [3]. It is proved in [3] that the trace map is injective and there isa “perfect pairing” between the rigid-cocenter and rigid-representationsof ˜ H q . RO- p HECKE ALGEBRAS 5 Cocenter of ˜ H For w, w ′ ∈ ˜ W (1) and s ∈ S (1), we write w s −→ w ′ if w ′ = sws − and ℓ ( w ′ ) ℓ ( w ). We write w → w ′ if there exists a sequence w = w , w , · · · , w n = w ′ of elements in ˜ W (1) such that for any k , w k − s k −→ w k for some s k ∈ S (1). We write w ≈ w ′ if w → τ w ′ τ − and τ w ′ τ − → w for some τ ∈ Ω(1). In this case, we say that w and w ′ are in thesame cyclic-shift class. Lemma 2.1.
Let w ∈ ˜ W (1) and s ∈ S (1) .(1) If ℓ ( sws − ) = ℓ ( w ) , then T w ≡ T sws − mod [ ˜ H , ˜ H ] .(2) If ℓ ( sws − ) < ℓ ( w ) , then T w ≡ c s − T ws mod [ ˜ H , ˜ H ] .Proof. (1) Without loss of generality, we may assume that ℓ ( sw ) = ℓ ( w ) −
1. Then T w = T s − T sw ≡ T sw T s − = T sws − mod [ ˜ H , ˜ H ] . Here the last equality follows from the fact that ℓ ( sw ) = ℓ ( sws − ) − ℓ ( sws ) = ℓ ( w ) −
2. So T w = T s − T sws T s − ≡ T s − T sws = c s − T s − T sws = c s − T ws mod [ ˜ H , ˜ H ] . Here the last equality follows from the fact that ℓ ( sws ) = ℓ ( sw ) − (cid:3) The following consequence follows easily from Lemma 2.1 (1).
Corollary 2.2.
Let w, w ′ ∈ ˜ W (1) with w ≈ w ′ . Then T w ≡ T w ′ mod [ ˜ H , ˜ H ] . Proof.
By definition, there exists a sequence w = w , w , · · · , w n = w ′ such that for any 1 < k < n , ℓ ( w k ) = ℓ ( w k − ) and w k = sw k − s − forsome s ∈ S (1) and w ′ = τ w n − τ − for some τ ∈ Ω(1). By Lemma2.1(1), T w ≡ T w n − mod [ ˜ H , ˜ H ]. By definition, T w ′ = T τ T w n − T τ − ≡ T w n − T τ − T τ = T w n − mod [ ˜ H , ˜ H ] . The corollary is proved. (cid:3)
Let ˜ W (1) min be the set of elements in ˜ W (1) that are of minimallength in their conjugacy classes. We have the following result. Theorem 2.3.
Assume that W is a finite Coxeter group or an affineWeyl group. Then for any w ∈ ˜ W (1) , there exists w ′ ∈ ˜ W (1) min suchthat ˜ w → ˜ w ′ .Proof. Since the length function on ˜ W (1) is induced from the lengthfunction on ˜ W through π , the statement follows directly from [5, The-orem 1.1] and [4, Theorem 2.6] (see also [10]) if W is a finite Coxetergroup, and from [11, Theorem 2.9] if W is an affine Weyl group. (cid:3) Now we prove the main result of this section.
XUHUA HE AND SIAN NIE
Theorem 2.4.
Let ˜ H be a finite or an affine pro- p Hecke algebra.Then the cocenter ˜ H is spanned by the image of T w for w ∈ ˜ W (1) min .Proof. Let x ∈ ˜ W (1). We prove by induction that the image of T x in˜ H is spanned by T w for w ∈ ˜ W (1) min .If x ∈ ˜ W (1) min , then the statement is obvious. If x / ∈ ˜ W (1) min , thenthere exists x ′ ∈ ˜ W (1) and s ∈ S (1) with x ≈ x ′ and ℓ ( sx ′ s − ) <ℓ ( x ′ ) = ℓ ( x ). By Corollary 2.2 and Lemma 2.1 (2), we have T x ≡ T x ′ ≡ c s T x ′ s − mod [ ˜ H , ˜ H ] . Note that ℓ ( x ′ s − ) < ℓ ( x ′ ) = ℓ ( x ). By inductive hypothesis, theimage of T x ′ s − in ˜ H is spanned by T w for w ∈ ˜ W (1) min . Hence theimage of T x in ˜ H is spanned by T w for w ∈ ˜ W (1) min . (cid:3) Let Cyc( ˜ W (1) min ) be the set of cyclic-shift classes in ˜ W (1) min .For Σ ∈ Cyc( ˜ W (1) min ), we denote by T Σ the image of T w in ˜ H forany w ∈ Σ. By Corollary 2.2, T Σ is well-defined. By Theorem 2.4,for a finite or an affine pro- p Hecke algebra, its cocenter is spanned by( T Σ ) Σ ∈ Cyc( ˜ W (1) min ) . It is interesting to see if the spanning set is in facta basis. It is known to be true for finite 0-Hecke algebras [8, Theorem4.4] and for affine 0-Hecke algebras [12, Theorem 0.1]. Finite pro- p Hecke algebras
In this section, we assume that ˜ H is a finite pro- p Hecke algebra andwe discuss the relation between the cocenter and representations of ˜ H . Recall that H is the subalgebra of ˜ H spanned by T w for w ∈ W (1).It is proved by Vign´eras [20, Proposition 2.1 & Proposition 2.2], everyirreducible representation of H is a character and is of the form Ξ χ, Γ ,where χ is a character of Z and Γ ⊆ { s ∈ S ; χ ( c s ) = 0 } . Here thecharacter Ξ χ, Γ is defined to beΞ χ, Γ ( T s ) = ( χ ( c s ) , if s ∈ Γ(1);0 , otherwise . We set Γ Ξ = Γ for Ξ = Ξ χ, Γ . Let Ω(Ξ)(1) be the stabilizer of Ξ inΩ(1). Let V be an irreducible representation of Ω(Ξ)(1). We say thepair (Ξ , V ) is permissible with respect to ( W (1) , Ω(1)) if Z ⊆ Ω(Ξ)(1)acts on V via Ξ. Set I (Ξ , V ) = Ind ˜ HH ⊗ k [ Z ] k [Ω(Ξ)(1)] (Ξ ⊗ V ) . We say that two permissible pairs (Ξ , V ) and (Ξ ′ , V ′ ) are equivalent ifthere exists γ ∈ Ω(1) such that ( γ Ξ , γ V ) = (Ξ ′ , V ′ ). Here γ Ξ (resp. γ V )denotes the twisted module of H (resp. Ω( γ Ξ)(1) = γ Ω(Ξ)(1) γ − ) by γ . In this case, we write (Ξ , V ) ∼ (Ξ ′ , V ′ ). It is proved by Vign´eras [20, RO- p HECKE ALGEBRAS 7
Proposition 6.17] that every irreducible representation of ˜ H is of theform I (Ξ , V ) and I (Ξ , V ) ∼ = I (Ξ ′ , V ′ ) if and only if (Ξ , V ) ∼ (Ξ ′ , V ′ ).The following formula follows easily from the definition of inducedmodules. Lemma 3.1.
Let (Ξ , V ) be a permissible pair. For w ∈ W (1) and τ ∈ Ω(1) we have
T r ( T wτ , I (Ξ , V )) = X γ ∈ Ω(1) / Ω(Ξ)(1) γ Ξ( T w ) T r ( τ, γ V ) . Here we set
T r ( τ, γ V ) = 0 if τ / ∈ Ω( γ Ξ)(1) . We denote by ˜ H = S ⊆ ˜ H the k -linear space generated by T wτ ,where w ∈ W (1) with supp( w ) = S and τ ∈ Ω(1). Denote by R ( ˜ H ) = S the k -linear space spanned by the simple ˜ H -modules I (Ξ , V ), where(Ξ , V ) is a permissible pair such that Γ Ξ = S .By Dedekind Theorem, the trace map T r : A → R ( A ) ∗ k is surjectivefor any k -algebra A . For finite pro- p Hecke algebra ˜ H , we have thefollowing refinement. Proposition 3.2.
Let Γ ⊆ S . Then the trace map T r : ˜ H = S → R ( ˜ H ) = S ∗ is surjective.Proof. Let M ∈ R ( ˜ H ) = S such that T r ( ˜ H = S , M ) = 0. We show that M = 0.Assume M = P [(Ξ ,V )] a [(Ξ ,V )] I (Ξ , V ), where a [(Ξ ,V )] ∈ k and [(Ξ , V )]ranges over the ∼ -equivalence classes of permissible pairs with Γ Ξ = S .Now we show that each coefficient a [( χ,V )] vanishes. Fix w ∈ W (1)with supp( w ) = S . For any w ∈ W (1) and τ ∈ Ω(1), T w T w is alinear combination of T w ′ with w ′ ∈ W (1) such that supp( w ′ ) = S . So T r ( T w T w T τ , M ) = 0 by assumption. Using Lemma 3.1, we have0 = T r ( T w T w T τ , M ) = X [(Ξ ,V )] X γ ∈ Ω(1) / Ω(Ξ)(1) a [(Ξ ,V )] γ Ξ( T w ) γ Ξ( T w ) T r ( τ, γ V )= X (Ξ ′ ,V ′ ) a (Ξ ′ ,V ′ ) Ξ ′ ( T w ) T r ( τ, V ′ )Ξ ′ ( T w ) , where in the last expression, (Ξ ′ , V ′ ) ranges over permissible pairs suchthat Γ Ξ ′ = S , and a (Ξ ′ ,V ′ ) = a [(Ξ ,V )] for (Ξ ′ , V ′ ) ∈ [(Ξ , V )].Now we regard P (Ξ ′ ,V ′ ) a (Ξ ′ ,V ′ ) Ξ ′ ( T w ) T r ( τ, V ′ )Ξ ′ ( T w ) as the virtualcharacter P (Ξ ′ ,V ′ ) a (Ξ ′ ,V ′ ) Ξ ′ ( T w ) T r ( τ, V ′ )Ξ ′ evaluated at T w . Since w runs over all the elements in W (1), we have X (Ξ ′ ,V ′ ) a (Ξ ′ ,V ′ ) Ξ ′ ( T w ) T r ( τ, V ′ )Ξ ′ = 0 . XUHUA HE AND SIAN NIE
Therefore, for each Ξ ′ , we have X V ′ ; (Ξ ′ , V ′ ) is permissible a (Ξ ′ ,V ′ ) Ξ ′ ( T w ) T r ( τ, V ′ ) = 0 . We regard P V ′ ; (Ξ ′ , V ′ ) is permissible a (Ξ ′ ,V ′ ) Ξ ′ ( T w ) T r ( τ, V ′ ) as the virtualcharacters P V ′ ; (Ξ ′ , V ′ ) is permissible a (Ξ ′ ,V ′ ) Ξ ′ ( T w ) T r ( − , V ′ ) evaluated at τ . Since τ runs over all the elements in Ω(Ξ ′ )(1), we have a (Ξ ′ ,V ′ ) Ξ ′ ( T w ) =0 for any V ′ . In particular, a (Ξ ,V ) Ξ( T w ) = 0. Since Ξ( T w ) = 0, wehave a [(Ξ ,V )] = a (Ξ ,V ) = 0 as desried. (cid:3) Affine pro- p Hecke algebras and parabolic algebras
Let R = ( X, R, Y, R ∨ , F ) be a based root datum, where X and Y are free abelian groups of finite rank together with a perfect pairing h , i : X × Y → Z , R ⊆ X is the set of roots, R ∨ ⊆ Y is the set ofcoroots and F ⊆ R is the set of simple roots. Let α α ∨ be thenatural bijection from R to R ∨ such that h α, α ∨ i = 2. For α ∈ R ,we denote by s α : X → X the corresponding reflections stabilizing R . Let S = { s α ; α ∈ F } be the set of simple reflections of theassociated finite Weyl group W . Let R + ⊆ R be the set of positiveroots determined by F . Let X + = { λ ∈ X ; h λ, α ∨ i > , ∀ α ∈ R + } .For any v ∈ X Q , we set J v = { s α ∈ S ; h v, α ∨ i = 0 } . For any J ⊆ S ,we set X + ( J ) = { λ ∈ X + ; J λ = J } . Let W a ff = Z R ⋊ W be the affine Weyl group and S a ff ⊃ S be the set of simple reflections in W . Then ( W a ff , S a ff ) is a Coxetergroup. Let ˜ W = X ⋊ W be the extended affine Weyl group. Then W a ff is a subgroup of ˜ W . For λ ∈ X , we denote by ǫ λ ∈ ˜ W thecorresponding translation element.Let V = X ⊗ Z R . For α ∈ R and k ∈ Z , set H α,k = { v ∈ V ; h v, α ∨ i = k } . Let H = { H α,k ; α ∈ R, k ∈ Z } . Connected components of V − ∪ H ∈ H H are called alcoves. Let C = { v ∈ V ; 0 < h v, α ∨ i < , ∀ α ∈ R + } be the fundamental alcove. We may regard W a ff and ˜ W as subgroupsof affine transformations of V , where t λ acts by translation v v + λ on V . The actions of W a ff and ˜ W on V preserve the set of alcoves.For any ˜ w ∈ ˜ W , we denote by ℓ ( ˜ w ) the number of hyperplanes in H separating C from ˜ wC . Then ˜ W = W a ff ⋊ Ω, where Ω = { ˜ w ∈ ˜ W ; ℓ ( ˜ w ) = 0 } is the subgroup of ˜ W stabilizing fundamental alcove C .The conjugation action of Ω on ˜ W preserves the set S a ff of simplereflections in W a ff . RO- p HECKE ALGEBRAS 9
Let Z be a finite commutative group and ˜ W (1) be a group con-taining Z as a normal subgroup and ˜ W (1) /Z ∼ = ˜ W . As in § p Hecke algebra ˜ H ( q, c ) and the affinepro- p Hecke algebra ˜ H . The parameters q s for s ∈ S (1) gives a multi-plicative function w q ( w ) on ˜ W (1) such that q ( ω ) = 1 if ω ∈ Ω(1)and q ( s ) = q s if s ∈ S (1).Examples of such algebras include the pro- p Iwahori-Hecke algebrasof reductive p -adic groups.By [18, Corollary 2], the map T w ι T w := ( − ℓ ( w ) q ( w ) T − w − givesan involution ι of ˜ H q . We still denoted by ι the induced involution of˜ H = ˜ H (0 , c ). For any J ⊆ S , we denote by R J the set of roots spanned by J and set R ∨ J = { a ∨ ; α ∈ R J } . Let R J = ( X, R J , Y, R ∨ J , J ) be the basedroot datum corresponding to J . Let W J ⊆ W and ˜ W J = X ⋊ W J bethe Weyl group and the extended affine Weyl group of R J respectively.We say ˜ w ∈ ˜ W J is J -positive if ˜ w ∈ t λ W J for some λ ∈ X such that h λ, α i > α ∈ R + r R J . Denote by ˜ W + J the set of J -positiveelements, which is a submonoid of ˜ W J , see [2, Section 6] and [17, II.4].We set H J = { H α,k ∈ H ; α ∈ R J , k ∈ Z } and C J = { v ∈ V ; 0 < h v, α ∨ i < , α ∈ R + J } . For any ˜ w ∈ ˜ W J , we denote by ℓ J ( ˜ w ) thenumber of hyperplanes in H J separating C J from ˜ wC J .Let ( W J ) a ff = Z R J ⋊ W J and let J a ff ⊇ J be the set of simplereflections of ( W J ) a ff . Then ˜ W J = ( W J ) a ff ⋊ Ω J , where Ω J = { ˜ w ∈ ˜ W J ; ℓ J ( ˜ w ) = 0 } . We denote by J the Bruhat order on ˜ W J . Note that J differs from the restriction to ˜ W J of the Bruhat order on ˜ W .We denote by ˜ W J (resp. J ˜ W ) the set of minimal coset representativesin ˜ W /W J (resp. W J \ ˜ W ). For J, K ⊆ S , we simply write ˜ W J ∩ K ˜ W as K ˜ W J . We define J W , W J and J W K in a similar way.The following result is proved in [1, Lemma 4.1]. Lemma 4.1.
Let x, y ∈ ˜ W J (1) . If x ∈ ˜ W + J (1) and y J x , then y ∈ ˜ W + J (1) . Let λ ∈ X (1). Then λ = λ λ − for some λ , λ ∈ X + (1). We set θ λ = T λ T − λ . It is easy to see that θ λ does not depend on the choicesof λ and λ . For λ, λ ′ ∈ X (1) we have θ λ θ λ ′ = θ λλ ′ . For u ∈ ˜ W J (1),there exist u ′ ∈ ˜ W + J (1) and λ ∈ X + ( J )(1) such that u = u ′ λ − . We set T Ju = T u ′ T − λ . It is easy to see that T Ju does not depend on the choicesof u ′ and λ .Let ˜ H J ( q, c ) be the k -linear subspace of ˜ H ( q, c ) spanned by T Ju for u ∈ ˜ W J (1). Let ˜ H + J ( q, c ) be the k -linear subspace of ˜ H ( q, c ) spannedby T u = T Ju for u ∈ ˜ W + J (1). We write ˜ H J for ˜ H J ( q, c ) specialized at q = 0 and ˜ H + J for ˜ H + J ( q, c ) specialized at q = 0. Theorem 4.2.
Let J ⊆ S . Then(1) The multiplication map on ˜ H ( q, c ) gives ˜ H J ( q, c ) a generic pro- p Hecke algebra structure.(2) The multiplication map on ˜ H = ˜ H (0 , c ) gives ˜ H + J a pro- p Heckealgebra structure.Remark.
We have a natural embedding˜ H + J ֒ → ˜ H , T J ˜ w T ˜ w . Notice that this embedding does not extend to an algebra homomor-phism ˜ H J → ˜ H since T Jλ for λ ∈ X + ( J )(1) is invertible in ˜ H J , but notinvertible in ˜ H .If ˜ H ( q, c ) is the pro- p Hecke algebra of a p -adic group, then ˜ H J isthe pro- p Hecke algebra of the corresponding Levi subgroup. In thiscase, the statements are obvious. The general situation requires morework and will be proved in the rest of this section.We have discuss some relations on θ , which essentially follows from[14, Lemma 2.5 & 2.7]. Lemma 4.3.
Let s ∈ S (1) and χ ∈ X (1) . Denote by α s the simpleroot corresponding to s .(1) If h χ, α ∨ s i = 0 , then T s θ χ = θ sχs − T s .(2) If h χ, α ∨ s i = 1 , then T − s − θ χ T − s − = θ sχs .(3) If h χ, α ∨ s i = 2 , χ ∈ X + (1) and α ∨ s ∈ Y , then (i) T w ′ T w ′′ θ χ − = T s θ s − χs ; (ii) T w ′ T ˜ s T w ′′ θ χ − = θ w ′ ˜ sw ′′ χ − ; (iii) T w ′ T ˜ s − T ˜ s T w ′′ θ χ − = θ χ T − s − .Here w ′ = λ s sω and w ′′ = ω − s − λ − s χsχ , λ s ∈ X + (1) such that π ( λ s ) is the fundamental weight corresponding to s , ω ∈ Ω(1) r Z and ˜ s = ω − sω ∈ S a ff (1) . Lemma 4.4.
For t ∈ J a ff (1) , we have the quadratic relation ( T Jt ) = c t T Jt + q t T t . Proof. If t ∈ J (1), then T Jt = T t and the statement is trivial. Nowwe assume t ∈ J a ff (1) r J (1). Since Z is finite and ˜ W J (1) is finitelygenerated, there exists a central element µ of X + J (1) such that tµ ∈ ˜ W + ( J )(1). Then T Jt = T tµ T − µ = T − µ T µt . It remains to show T µt = c t T µ t + q t t T µ .Assume π ( t ) = ǫ α s α ∈ W J a ff for some maximal short root α ∈ R + J . Let w ∈ W J (1) and s ∈ J (1) such that s α = wsw − and ℓ ( s α ) = 2 ℓ ( w ) + 1. Let λ s ∈ X + (1) such that π ( λ s ) is the fundamentalweight corresponding to s . Since the quadratic relation of T t is equiva-lent to that of T tz for any z ∈ Z , we can assume t = wλ s sλ − s w − .Set λ = λ s sλ − s s − ∈ X (1). We have T Jµt = θ wµλw − T − ws − w − = RO- p HECKE ALGEBRAS 11 θ wµλw − T − w − T − s − T − w . Let w = s · · · s n be a reduced expression of w with each s i ∈ S (1). Then h s k − · · · σ ( α ) , α ∨ k i = 1 for k = 1 , . . . , n ,where α i is the simple root corresponding to s i . Applying Lemma 4.3(2), for λ ′ ∈ wλw − Z we have(a) T − w θ wµλw − T − w − = T − s ··· s n θ s − wµλw − s T − s ··· s n ) − = · · · = θ µλ . Similarly, we have(b) T − w − θ sµλs T − w = θ wµλw − . Case(1): α ∨ / ∈ Y . Then λ s ∈ X (1) and q t = q s . We have( T Jt ) = θ wµλw − T − w − T − s − T − w θ wµλw − T − w − T − s − T − w = θ wµλw − T − w − T − s − θ µλ T − s − T − w = θ wµλw − T − w − T − s − θ µλ s T − s T s θ sλ − s s − T − s − T − w = θ wµλw − T − w − θ sµλ s s − ( c s + q s s T − s ) θ sλ − s s − T − s − T − w = c ( wλ s ) • s θ wµ λw − T − w − T − s − T − w + q s θ wµ λw − T − w − θ sµλ s s − s T − s θ sλ − s s − T − s − T − w = c t θ wµ λw − T − w − T − s − T − w + q s θ wµλw − T − w − θ µλsµλ s sλ − s T − w = c t θ wµ λw − T − w − T − s − T − w + q s θ wµλsµλ s sλ − s w − = c t T µ t + q t t T µ , where the second equality follows from (a); the sixth one follows fromLemma 4.3 (2); the seventh follows from Lemma 4.3 (1).Case(2): α ∨ ∈ Y . Then λ s / ∈ X (1). Let χ, w ′ , w ′′ , ˜ s be as in Lemma4.3 (3). One computes that( T Jt ) = θ wµλw − T − w − T − s − θ µλ T − s − T − w = θ wµλw − T − w − T − s − θ χ θ χ − µλ T − s − T − w = θ wµλw − T − w − T − s − θ χ T − s − θ s − χ − µλs T − w = θ wµλw − T − w − T − s − T w ′ T ˜ s − T ˜ s T w ′′ θ χ − s − χ − µλs T − w = θ wµλw − T − w − T − s − T w ′ ( c ˜ s T ˜ s − + q ˜ s ) T w ′′ θ χ − s − χ − µλs T − w = θ wµλw − c ( wsw ′ ) • ˜ s T − w − T − s − θ w ′ ˜ s − w ′′ χ − s − χ − µλs T − w + q ˜ s θ wµλw − T − w − θ sµλs T − w = θ wµλw − c ( wsw ′ ) • ˜ s T − w − T − s − θ µ T − w + q ˜ s θ wµλsµλsw − = θ wµ λw − c ( wsw ′ ) • ˜ s T − w − T − s − T − w + q ˜ s t T µ , where the first equality follows from (a); the third one follows from 4.3(1); the fourth and the sixth follow from 4.3 (3); the seventh followsfrom (b). Note that q ˜ s = q t since π (˜ s ) and π ( t ) are conjugate under˜ W . It remains to check wµ λsw ′ c ˜ s w ′− s − w − = c t wµ λw − , that is λsw ′ c ˜ s w ′− s − = c λ s • s λ s sλ − s s − = λ s sλ − s c λ s • s s − , which follows by observing that λs = λ s sλ − s and w ′ c ˜ s w ′− = c λ s • s . (cid:3) Lemma 4.5.
Let x, x ′ ∈ ˜ W + J (1) such that ℓ J ( x ′ x ′ ) = ℓ J ( x ) + ℓ J ( x ′ ) .Then ℓ ( xx ′ ) = ℓ ( x ) + ℓ ( x ′ ) . As a consequence, we have T Jy T Jy ′ = T Jyy ′ for y, y ′ ∈ ˜ W J (1) such that ℓ J ( yy ′ ) = ℓ J ( y ) + ℓ J ( y ′ ) .Proof. Write x = λu and x ′ = λ ′ u ′ , where λ, λ ′ ∈ X (1) and u, u ′ ∈ W (1). Since x, x ′ , xx ′ ∈ ˜ W + J (1), one computes that ℓ ( xx ′ ) = ℓ J ( xx ′ ) + X α ∈ R + r R J |h λ + π ( u )( λ ′ ) , α ∨ i| = ℓ J ( x ) + ℓ J ( x ′ ) + X a ∈ R + r R J h λ, α ∨ i + X a ∈ R + r R J h π ( u )( λ ′ ) , α ∨ i = ℓ J ( x ) + X a ∈ R + r R J h λ, α ∨ i + ℓ J ( x ′ ) + X a ∈ R + r R J h λ ′ , α ∨ i = ℓ ( x ) + ℓ ( x ′ )as desired. (cid:3) (1) Let ˜ H J ( q J , c J ) be the generic pro- p Hecke algebra associated to ˜ W J (1), where q Jt = q t and c Jt = c t for t ∈ T (1) ∩ ˜ W J (1). Denote by ( T w,J ) w ∈ ˜ W J (1) its Iwahori-Matsumotobasis. Combining Lemma 4.4 and Lemma 4.5, we see that there existsa surjective algebra homomorphism from ˜ H J ( q J , c J ) to ˜ H J sending T w,J to T Jw for w ∈ ˜ W J (1). It is easy to see ( T Jw ) w ∈ ˜ W J (1) is linear independent.Hence the homomorphism is an isomorphism.(2) Let x, x ′ ∈ ˜ W + J (1). We have to show T Jx T Jx ′ ∈ ˜ H + J . We argue byinduction on ℓ ( x ′ ). If ℓ ( x ′ ) = 0, T x T x ′ = T xx ′ ∈ ˜ H + J by Lemma 4.5.Assume T x T x ′′ ∈ ˜ H + J for any x, x ′′ ∈ ˜ W + J (1) with ℓ ( x ′′ ) < ℓ ( x ′ ). Againby Lemma 4.5, it remains to consider the case ℓ ( xx ′ ) < ℓ ( x ) + ℓ ( x ′ ).Let x ′ = ωs · · · s n be a reduced expression with respect to ℓ J , where ω ∈ Ω J (1) and s i ∈ J a ff (1) for 1 i n . By assumption, there exists1 m n such that ℓ J ( x ′ ωs · · · s m − ) = ℓ J ( x ) + ℓ J ( ωs · · · s m − )and xωs · · · s m < J xωs · · · s m − . By the exchange condition for theCoxeter group W J a ff , xωs · · · s m = yωs · · · s m − for some y < J x suchthat ℓ J ( yωs · · · s m − ) = ℓ J ( y ) + ℓ J ( ωs · · · s m − ). So y ∈ ˜ W + J (1) byLemma 4.1.By part (1) and Lemma 4.4, one computes that T Jx T Jx ′ = T Jx T Jωs ··· s m − T Js m T Js m +1 ...s n = T Jxωs ··· s m − T Js m T Js m +1 ··· s n = T Jx T Jωs ··· s m − c s m T Js m +1 ...s n + q s m T Jy T Jωs ··· s m − T Js m +1 ...s n . RO- p HECKE ALGEBRAS 13
Note that T Jωs ··· s m − T Js m +1 ...s n is a linear combination of T Jx ′′ such that x ′′ < J x ′ . Again by Lemma 4.1, we have x ′′ ∈ ˜ W + J (1). Now thestatement follows by induction hypothesis. Standard representatives
By Theorem 2.4, the cocenter ˜ H of an affine pro- p Hecke algebra ˜ H is spanned by the image of T ˜ w for ˜ w ∈ ˜ W (1) min . In this section, we willcompute the trace of T ˜ w for ˜ w ∈ ˜ W (1) min using certain elements in theparabolic subalgebra ˜ H + J of ˜ H . Let n = ♯W . For any ˜ w ∈ ˜ W (1), ˜ w n = λ for some λ ∈ X (1).Let ν ˜ w = λ/n ∈ X Q and ¯ ν ˜ w ∈ X + Q be the unique dominant element inthe W -orbit of ν w . It is easy to see that the map ˜ W → V, ˜ w ¯ ν ˜ w isconstant on each conjugacy class of ˜ W .We say that an element ˜ w ∈ ˜ W (1) is straight if ℓ ( ˜ w n ) = nℓ ( ˜ w ) for any n ∈ N . By [7, Lemma 1.1], ˜ w is straight if and only if ℓ ( ˜ w ) = h ¯ ν ˜ w , ρ ∨ i ,where ρ is the half sum of positive coroots. A conjugacy class thatcontains a straight element is called a straight conjugacy class.It is proved in [11, Proposition 2.8] that for each cyclic-shift class in˜ W (1) min , we have some nice representatives. Proposition 5.1.
For any ˜ w ∈ ˜ W (1) min , there exists a subset K ′ ⊆ S a ff with W K ′ finite, a straight element y ∈ K ′ ˜ W K ′ (1) with yK ′ (1) y − = K ′ (1) , and an element w ∈ W K ′ (1) such that ˜ w ˜ ≈ wy . Here W K ′ ⊆ W a ff denotes the subgroup generated by reflections of K ′ . In the situation of Proposition 5.1, we call wy a standard repre-sentative of the cyclic-shift class of ˜ w . By [7, Proposition 2.2], ¯ ν ˜ w =¯ ν wy = ¯ ν y . The expression of standard representative relates each conju-gacy class of ˜ W with a straight conjugacy class. It plays an importantrole in the study of combinatorial properties of conjugacy classes ofaffine Weyl groups [11], σ -conjugacy classes of p -adic groups [6] andrepresentations of affine Hecke algebras with nonzero parameters [3].However, for a given cyclic-shift class in ˜ W min (1), the standard rep-resentatives are in general, not unique. This leads to some difficulty inunderstanding the cyclic-shift classes in ˜ W min (1) and their relations tothe representations of ˜ H .To overcome the difficulty, we use the notion of standard quadruples.We say that ( J, x, Γ , C ) is a standard quadruple of ˜ W if • J ⊆ S ; • x ∈ Ω J and h ν x , α ∨ i > α ∈ R + r R J ; • Γ ⊆ J a ff such that W Γ finite and the conjugation action of x stabilizes Γ. • C is an elliptic Ad( x )-twisted conjugacy class of W Γ . We say that (
J, x, Γ , C ) and ( J ′ , x ′ , Γ ′ , C ′ ) are strongly equivalent if J = J ′ and there exists ω ∈ Ω J such that ( x ′ , Γ ′ , C ′ ) is obtained from( x, Γ , C ) by conjugation by ω . It is proved in [9, Proposition 3.23] that Proposition 5.2.
The map ( J, x, Γ , C ) [ C min x ] induces a bijec-tion between the strongly equivalence classes of standard quadruplesand cyclic-shift classes in ˜ W min . Let wy be a standard representative as in Proposition 5.1. Let J = J ¯ ν y ⊆ S and K = ∪ i π ( y ) i supp( w ) π ( y ) − i ⊆ S a ff . By loc. cit., W E is finite. Let h ∈ J W (1) such that h ( ν y ) = ¯ ν y . Set x = hyh − and Γ = πhKπh − ⊆ J a ff . Denote by C the Ad( π ( x ))-twisted conjugacy classof W Γ . By construction, C is elliptic. One checks that ( J, π ( x ) , Γ , C ) isthe standard quadruple corresponding to the cyclic-shift class of π ( wy )in Proposition 5.2. In this paper, we are mainly interested in the pair( J, Γ) associated to wy . We call it the associated standard pair . Notethat Γ ⊆ J a ff with W Γ finite.Now we state the main result of this section. Proposition 5.3.
Let w, y, h, J, K be as in § n ≫ , T nwy ≡ ( T Jhwyh − ) n mod [ ˜ H , ˜ H ] . Remark.
Note that ( T Jhwyh − ) n ∈ ˜ H + J ⊆ ˜ H .The following is a variation of the length formula in [13]. Lemma 5.4.
For w ∈ W (1) and α ∈ R , set δ w ( α ) = ( , if wα ∈ R + ;1 , if wα ∈ R − . Then for any x, y ∈ W (1) and µ ∈ X (1) , we have that ℓ ( xt µ y ) = X α ∈ R + |h µ, α ∨ i + δ x ( α ) − δ y − ( α ) | . Proposition 5.5.
Let ( J, Γ) be a standard pair. Let x ∈ Ω J (1) suchthat ν x ∈ X + Q ( J ) . Then for any u ∈ W Γ (1) we have(1) for n ≫ and h ∈ J W (1) , ℓ ( h − ux n h ) = ℓ ( ux n ) .(2) for n ≫ , ℓ ( ux n + n ) = ℓ ( ux n ) + ℓ ( x n ) , where n = ♯W .Proof. We have ux n = λw for some λ ∈ X (1) and w ∈ W J (1). Since h ν x , α ∨ i > α ∈ R + r R J , we have h λ, α ∨ i > α ∈ R + r R + J as n ≫ RO- p HECKE ALGEBRAS 15
Notice that for α ∈ R J , δ h − ( α ) = δ α . Now ℓ ( h − ux n h ) = X α ∈ R + |h λ, α ∨ i + δ h − ( α ) − δ h − w − ( α ) | = X α ∈ R + J |h λ, α ∨ i − δ w − ( α ) | + X α ∈ R + r R + J |h λ, α ∨ i + δ h − ( α ) − δ h − w − ( α ) | = X α ∈ R + J |h λ, α ∨ i − δ w − ( α ) | + X α ∈ R + r R + J (cid:0) h λ, α ∨ i + δ h − ( α ) − δ h − w − ( α ) (cid:1) = X α ∈ R + J |h λ, α ∨ i − δ w − ( α ) | + X α ∈ R + r R + J h λ, α ∨ i + ♯ { α ∈ R + r R + J , h − ( α ) ∈ R − }− ♯ { α ∈ R + r R + J , h − w − ( α ) ∈ R − } = X α ∈ R + J |h λ, α ∨ i − δ w − ( α ) | + X α ∈ R + r R + J h λ, α ∨ i + ℓ ( h ) − ℓ ( h )= X α ∈ R + J |h λ, α ∨ i − δ w − ( α ) | + X α ∈ R + r R + J h λ, α ∨ i . This proves part (1).For part (2), ℓ ( ux n + n ) = X α ∈ R + |h λ + n ν x , α ∨ i − δ w − ( α ) | = X α ∈ R + J |h λ + n ν x , α ∨ i − δ w − ( α ) | + X α ∈ R + r R + J h λ + n ν x , α ∨ i = X α ∈ R + J |h λ, α ∨ i − δ w − ( α ) | + X α ∈ R + r R + J h λ, α ∨ i + X α ∈ R + r R + J h n ν x , α ∨ i = ℓ ( ux n ) + ℓ ( x n ) . (cid:3) As a consequence, we have
Corollary 5.6.
Let w, y, h, J be as in § n ≫ we have wy n ˜ ≈ hwy n h − .Proof. Let x = hyh − and u ∈ hwh − . Suppose that h = s · · · s k for s , · · · , s k ∈ S (1). Set h i = s · · · s i for 1 i k . Then h i ∈ J W (1).By Proposition 5.5 (1), ℓ ( h − i ux n h i ) = ℓ ( h − i +1 ux n h i +1 ) for 0 i k −
1. Hence h − i ux n h i ˜ ≈ h − i +1 ux n h i +1 for 0 i k −
1. Therefore ux n ˜ ≈ h − ux n h = wy n . (cid:3) Assume T w T ywy − · · · T y n − wy − n T y n = X w ′ ∈ W K (1) a w ′ T w ′ with a w ′ ∈ k . Let H K ⊆ ˜ H (resp. H J, Γ ⊆ ˜ H J ) be the subalgebra gen-erated by T w for w ∈ W K (1) (resp. by T Jw for w ∈ W Γ (1)). By Lemma T w ′ T Jhw ′ h − gives an algebra isomorphism between H K and H J, Γ . Thus T Jhwh − T Jhywy − h − · · · T Jhy n − wy − n h − = X w ′ ∈ W K (1) a w ′ T Jhw ′ h − . Now one computes that T nwy = T w T ywy − · · · T y n − wy − n T y n = ( X w ′ ∈ W K (1) a w ′ T w ′ ) T y n = X w ′ ∈ W K (1) a w ′ T w ′ y n ≡ X w ′ ∈ W K (1) a w ′ T hw ′ y n h − mod [ ˜ H , ˜ H ] . Moreover X w ′ ∈ W K (1) a w ′ T hw ′ y n h − = X w ′ ∈ W K (1) a w ′ T Jhw ′ y n h − = ( X w ′ ∈ W K (1) a w ′ T Jhw ′ h − ) T Jhy n h − = T Jhwh − T Jhywy − h − · · · T Jhy n − wy − n h − T Jhy n h − = ( T Jhwyh − ) n . Some character formulas
Let M ∈ R ( ˜ H ) k . For any J ⊆ S , we set M J = ∩ λ ∈ X + ( J )(1) T λ M .Since M is a finite dimensional, there exists µ ∈ X + ( J )(1) such that M J = T µ M . Moreover, since the action of T λ on M J is invertible forany λ ∈ X + ( J )(1), we may regard M J as an ˜ H J -module. For Γ ⊆ J a ff ,let Ω J (Γ) = { τ ∈ Ω J (1); π ( τ )Γ π ( τ ) − = Γ } and M J, Γ = T Jw Γ M J , where w Γ ∈ ˜ W J (1) such that π ( w Γ ) is the longestelement of W Γ . Then M J, Γ is an Ω J (Γ)(1)-module.Let H J, Γ ⊆ H J be the subalgebra spanned by T Ju with u ∈ W Γ (1).By § H J, Γ ⋊ Ω J (Γ)(1) is of the form I (Ξ , V ) forsome permissible pair (Ξ , V ) with respect to ( W Γ (1) , Ω J (Γ)(1)). Let u ∈ ˜ W J (1) such that supp J ( u ) = supp J ( w Γ ) = Γ. One checks directlythat T Ju I (Ξ , V ) = T Jw Γ I (Ξ , V ). In particular, we have M J, Γ = T Ju M J ∈ R ( H J, Γ ⋊ Ω J (Γ)(1)). Let ℵ = { ( J, Γ); J ⊆ S , Γ ⊆ J a ff } and let ℵ ∗ = { ( J, Γ) ∈ℵ ; ♯W Γ < + ∞} be the set of standard pairs. We define an equivalencerelation ∼ and a partial order < on ℵ as follows. Let ( J, Γ) , ( J ′ , Γ ′ ) ∈ ℵ .We say ( J, Γ) ∼ ( J ′ , Γ ′ ) if J = J ′ and Γ ′ = π ( τ )Γ π ( τ ) − for some τ ∈ Ω J (1). We say that ( J, Γ) < ( J ′ , Γ ′ ) if either J ( J ′ or J = J ′ andΓ ) π ( τ )Γ ′ π ( τ ) − for some τ ∈ Ω J (1).For ( J, Γ) ∈ ℵ , denote by P ( J, Γ) the set of permissible pairs (Ξ , V )with respect to ( W Γ (1) , Ω J (Γ)(1)) such that Ξ( T Jw ) = 0 for each w ∈ RO- p HECKE ALGEBRAS 17 W Γ (1). Let (Ξ , V ) ∈ P ( J, Γ) and let I (Ξ , V ) be the H J, Γ ⋊ (Ω J (Γ)(1))-module constructed as in Section 3. Set ˜ H J (Γ) = H J ⋊ (Ω J (Γ)(1)).We denote by I (Ξ , V ) the extension of I (Ξ , V ) by zero as a module of˜ H J (Γ) by requiring that T w I (Ξ , V ) = 0 if π ( w ) / ∈ W Γ ⋊ Ω J (Γ). Define π J, Γ , Ξ ,V = ˜ H ⊗ ˜ H + J ( ˜ H J ⊗ ˜ H J (Γ) I (Ξ , V ) ) . It is easy to see that π J, Γ , Ξ ,V and π J, γ Γ , γ Ξ , γ V are isomorphic as ˜ H -modules for γ ∈ Ω J (1). Theorem 6.1.
Let ( J, Γ) , ( J ′ , Γ ′ ) ∈ ℵ ∗ and (Ξ , V ) ∈ P ( J, Γ) . Then ( π J, Γ , Ξ ,V ) J ′ = ( ⊕ γ ∈ Ω J (1) / Ω J (Γ)(1) γ I (Ξ , V ) , if J = J ′ ;0 , if J * J ′ . Moreover, ( π J, Γ , Ξ ,V ) J ′ , Γ ′ = ( I (Ξ , V ) , if ( J, Γ) = ( J ′ , Γ ′ );0 , if ( J, Γ) (cid:10) ( J ′ , Γ ′ ) . Proof.
Let λ ∈ X + ( J ′ )(1) such that T λ π J, Γ , Ξ ,V = ( π J, Γ , Ξ ,V ) J ′ . We mayreplace λ with some appropriate power of itself so that T Jγ − λγ ∈ ˜ H J (Γ)for any γ ∈ Ω J (1). Let M = ⊕ γ ∈ Ω J (1) / Ω J (Γ)(1) T Jγ ⊗ I (Ξ , V ) . By [15, Proposition 5.2], we have π J, Γ , Ξ ,V = ⊕ d ∈ W J (1) ι T d ⊗ M. For s ∈ S (1), T λι T s = ( ι T s T s − λs , if s ∈ J ′ (1);0 , otherwise.Thus for d ∈ W J (1),(a) T λι T d = ( ι T d T d − λd , if d ∈ W J ′ (1);0 , otherwise,where d − λd ∈ λZ since λ ∈ X ( J ′ )(1).Assume J * J ′ . We show that ( π J, Γ , Ξ ,V ) J ′ = T λ π J, Γ , Ξ ,V = 0. By (a),it suffices to show T λ M = 0. One checks that T λ ( T Jγ ⊗ I (Ξ , V ) ) = T Jλ ( T Jγ ⊗ I (Ξ , V ) ) = T Jγ ⊗ ( T Jγ − λγ I (Ξ , V ) ) . Since J J ′ , there exists β ∈ R J such that h ν λ , β ∨ i 6 = 0. We have γ − λγ = wλw − for some w ∈ W J (1). Thus h ν γ − λτ , π ( w ) − ( β ∨ ) i 6 = 0.Hence γ − λγ / ∈ W Γ ⋊ Ω J (Γ)(1) and T Jγ − λγ I (Ξ , V ) = 0.Assume J = J ′ . By (a), we have( π J, Γ , Ξ ,V ) J ′ = T λ π J, Γ , Ξ ,V = M = ⊕ γ ∈ Ω J (1) / Ω J (Γ)(1) I ( γ Ξ , γ V ) . Let u ∈ W Γ ′ with supp J ( u ) = Γ ′ . If ( J, Γ) (cid:10) ( J, Γ ′ ), that is, Γ ′ * γ Γfor each γ ∈ Ω J (1). So γ Ξ( T u ) = 0 and hence( π J, Γ , Ξ ,V ) J, Γ = T u M = ⊕ γ ∈ Ω J (1) / Ω J (Γ)(1) T u I ( γ Ξ , γ V ) = 0 . If Γ = Γ ′ , once checks similarly T u I ( γ Ξ , γ V ) = 0 if and only if γ ∈ Ω J (Γ)(1). So ( π J, Γ , Ξ ,V ) J, Γ = I (Ξ , V ) and proof is finished. (cid:3) Now we state the main result of this section.
Theorem 6.2.
Let ( J, Γ) ∈ ℵ ∗ and (Ξ , V ) ∈ P ( J, Γ) . Let wy be astandard representative and ( J ′ , Γ ′ ) be the standard pair assoicated toit. Let h ∈ J W (1) with h ( ν y ) = ¯ ν y . Then T r ( T wy , π J, Γ , Ξ ,V )= (P γ ∈ Ω J (Γ)(1) / Ω J (Γ , Ξ)(1) γ Ξ( T hwh − ) T r ( hyh − , γ V ) , if ( J, Γ) = ( J ′ , Γ ′ );0 , if ( J, Γ) (cid:10) ( J ′ , Γ ′ ) . Here Ω J (Γ , Ξ)(1) is the stabilizer of Ξ in Ω J (Γ)(1) . Lemma 6.3.
Let ( J, Γ) be a standard pair. Let x, x ′ ∈ Ω J (Γ)(1) suchthat ν x ∈ X + ( J ) Q , and let u ∈ W Γ (1) with supp J ( u ) = Γ . Let M ∈ R ( ˜ H ) . Then for n ≫ , we have T r ( T ux ′ x n , M ) = T r ( T Jux ′ x n , M J, Γ ) . Proof.
Let µ ∈ X + ( J )(1) such that M J = T µ M . Notice that n ν x ∈ X + ( J ), where n = ♯W . There exists m ∈ N such that x mn µ − ∈ X + ( J ). By Proposition 5.5 (2), for n ≫ ℓ ( ux ′ x n + mn ) = ℓ ( ux ′ x n ) + ℓ ( x mn ) = ℓ ( ux ′ x n ) + ℓ ( x mn µ − ) + ℓ ( µ ) and T ux ′ x n + mn = T ux ′ x n T x mn µ − T µ . Moreover, for n ≫ ux ′ x n + mn ∈ ˜ W + J (1) and T ux ′ x n + mn = T Jux ′ x n + mn .Since 0 → ker( T µ : M → M ) → M → M J →
0, we have
T r ( T ux ′ x n + mn , M ) = T r ( T ux ′ x n T x mn µ − T µ , M ) = T r ( T ux ′ x n + mn , M J )= T r ( T Jux ′ x n + mn , M J ) . Notice that T Jux ′ x n + mn = T Jux ′ ( T Jx ) n + mn = T Jx ′ ( T Jx ) n + mn T Ju ′ for some u ′ with supp J ( u ′ ) = Γ. Since 0 → ker( T Ju ′ : M J → M J ) → M J → M J, Γ → , we have T r ( T Jux n + mn , M J ) = T r ( T Jx ′ ( T Jx ) n + mn T Ju ′ , M J ) = T r ( T Jux ′ x n + mn , M J, Γ )as desired. (cid:3) RO- p HECKE ALGEBRAS 19
Let x = hyh − ∈ Ω J ′ (1) and u = hwh − ∈ W Γ ′ (1). Let n ∈ N . Then ( T Jhwyh − ) n = ( T Jux ) n is a linearcombination of T u ′ x n with u ′ ∈ W Γ ′ (1). Thus, for M ∈ R ( ˜ H ) and n ≫ T r ( T nwy , M ) = T r (( T Jhwyh − ) n , M ) = T r (( T J ′ ux ) n , M J ′ , Γ ′ ) , where the first equality follows from Proposition 5.3, and the secondone follows from Lemma 6.3. Thus by Theorem 6.1, T r ( T nwy , π J, Γ , Ξ ,V ) ) = T r ( T J ′ ux , ( π J, Γ , Ξ ,V ) J ′ , Γ ′ )= ( T r ( T Jux , I (Ξ , V )) , if ( J, Γ) = ( J ′ , Γ ′ );0 , if ( J, Γ) (cid:10) ( J ′ , Γ ′ ) . Now the statement follows from Lemma 3.1. Representations of ˜ H We first give a basis of R ( ˜ H ) k . Theorem 7.1.
The set { π J, Γ , Ξ ,V ; ( J, Γ) ∈ ℵ ∗ / ∼ , (Ξ , V ) ∈ P ( J, Γ) / ∼} is a k -basis of R ( ˜ H ) k . Lemma 7.2.
Let A be a k -algebra. Let τ ∈ A and ζ ∈ R ( A ) . Assumethere exists an invertible central element µ ∈ A such that T r ( τ µ n , ζ ) =0 for n ≫ . Then T r ( τ, ζ ) = 0 .Proof. Assume ζ = P V a V V , where a V ∈ k and V ranges over simplemodules. Since µ is central, µ acts on V by a scalar χ V,µ ∈ k × . Byassumption, for n ≫ T r ( τ µ n , ζ ) = X V a V T r ( τ µ n , V ) = X V χ nV,µ a V T r ( τ, V ) . Due to the non-vanishing of Vandermonde determinant, for each f ∈ k × , we have X V,χ
V,µ = f a V T r ( τ, V ) = 0 . So T r ( τ, ζ ) = P V T r ( τ, V ) = 0. (cid:3) First we show that { π J, Γ , Ξ ,V ; ( J, Γ) ∈ ℵ ∗ / ∼ , (Ξ , V ) ∈ P ( J, Γ) / ∼} is linearly independent in R ( ˜ H ) k .Suppose P J, Γ , Ξ ,V a J, Γ , Ξ ,V π J, Γ , Ξ ,V = 0 with a J, Γ , Ξ ,V ∈ k .Let ( J , Γ ) ∈ ℵ ∗ / ∼ be a minimal element such that a J , Γ , Ξ ,V = 0for some (Ξ , V ). Since Z is finite and Ω is finitely generated, thereexists a central element µ of Ω J (Γ )(1) with µ ∈ X + ( J )(1). Let u ∈ W Γ (1) with supp J ( u ) = Γ and x ∈ Ω J (Γ)(1). Combining Lemma6.3 with Theorem 6.1, we deduce that for n ≫
00 = X J, Γ , Ξ ,V a J, Γ , Ξ ,V T r ( T uxµ n , π J, Γ , Ξ ,V )= X (Ξ ,V ) ∈ P ( J , Γ ) / ∼ a J , Γ , Ξ ,V T r ( T uxµ n , π J , Γ , Ξ ,V )= X (Ξ ,V ) ∈ P ( J , Γ ) / ∼ a J , Γ , Ξ ,V T r ( T Juxµ n , ( π J , Γ , Ξ ,V ) J , Γ )= X (Ξ ,V ) ∈ P ( J , Γ ) / ∼ a J , Γ , Ξ ,V T r ( T Juxµ n , I (Ξ , V ))By Lemma 7.2, we have that X (Ξ ,V ) ∈ P ( J , Γ ) / ∼ a J , Γ , Ξ ,V T r ( T Jux , I (Ξ , V )) = 0 . Thanks to Proposition 3.2 (where we take S = Γ and Ω = Ω J (Γ )), a J , Γ , Ξ ,V = 0 for every (Ξ , V ). That is a contradiction. Next we show that ( π J, Γ , Ξ ,V ) ( J, Γ) ∈ℵ ∗ , (Ξ ,V ) ∈ P ( J, Γ) spans R ( ˜ H ).For any M ∈ R ( ˜ H ), let ℵ ∗ ( M ) be the set of pairs ( J, Γ) in ℵ ∗ / ∼ which is associated to some standard representative ˜ w ∈ ˜ W (1) min suchthat T r ( T ˜ w , M ) = 0.Fix a total order on ℵ ∗ that is compatible with the partial order givenin § ℵ ∗ ( M ).If ℵ ∗ ( M ) = ∅ , then T r ( T w , M ) = 0 for all w ∈ ˜ W (1) min . By Theorem2.4, T r ( h, M ) = 0 for all h ∈ ˜ H . Hence M = 0.Now suppose that ℵ ∗ ( M ) = ∅ . Let ( J, Γ) be the minimal element in ℵ ∗ ( M ). We regard M J as a virtual H J, Γ ⋊ Ω J (Γ)(1)-module. Then M J is linear combination of I (Ξ , V ), where (Ξ , V ) ranges over permissiblepair with respect to ( W Γ (1) , Ω J (Γ)(1)). Therefore, M J, Γ = X (Ξ ,V ) ∈ P ( J, Γ) / ∼ a (Ξ ,V ) I (Ξ , V ) , where a (Ξ ,V ) ∈ k . We set U ( J, Γ) = X (Ξ ,V ) ∈ P ( J, Γ) / ∼ a (Ξ ,V ) π J, Γ , Ξ ,V . By Theorem 6.1, M J, Γ = U ( J, Γ) J, Γ . By Theorem 6.2, we deduce that T r ( T ˜ w , M ) = T r ( T ˜ w , U ( J, Γ)) for each standard representative ˜ w ∈ ˜ W (1) min whose associated standard pair is equivalent to ( J, Γ).Set M ′ = M − U ( J, Γ) . By Theorem 6.1, for any standard repre-sentative ˜ w ′ ∈ ˜ W (1) min with T r ( T w ′ , M ′ ) = 0, its associated standardpair ( J ′ , Γ ′ ) is larger than ( J, Γ) in the fixed linear order. By inductive
RO- p HECKE ALGEBRAS 21 hypothesis, M ′ is a linear combination of ( π J, Γ , Ξ ,V ) ( J, Γ) ∈ℵ ∗ , (Ξ ,V ) ∈ P ( J, Γ) .So does M , as desired. Motivated by [3], we introduce rigid part of the cocenter. Recallthat { T w } w ∈ ˜ W (1) min spans ˜ H .Let ˜ H rig be the subspace of ˜ H spanned by T w for w ∈ ˜ W (1) min with ν w central and let ˜ H nrig be the subspace of ˜ H spanned by T w for w ∈ ˜ W (1) min with ν w non-central. We call ˜ H rig the rigid part of thecocenter and ˜ H nrig the non-rigid part of the cocenter.Let ˜ H nss = ˜ H nrig + ι ( ˜ H nrig ). We call ˜ H nss the non-supersingularpart of the cocenter. Lemma 7.3.
Let w ∈ ˜ W (1) such that ♯W supp( w ) = + ∞ , then the imageof T w in ¯˜ H lies in ˜ H nrig .Proof. If w ∈ ˜ W (1) min , then ν x is central if and only if ♯W supp x < + ∞ and the statement follows. Assume the statement holds for any w ′′ ∈ ˜ W (1) with ℓ ( w ′′ ) < ℓ ( w ). Assume w / ∈ ˜ W (1) min . By Theorem 5.1,there exists w ∈ ˜ W (1) and s ∈ S aff (1) such that w ˜ ≈ w ′ and sw ′ s Let M ∈ R ( ˜ H ) k . Then M is rigid if and only if M is spanned by ( π S , Γ , Ξ ,V ) ( S , Γ) ∈ℵ ∗ , (Ξ ,V ) ∈ P ( S , Γ) .Proof. By Theorem 6.1, π S , Γ , Ξ ,V is rigid.On the other hand, assume that M = X ( J, Γ) ∈ℵ ∗ / ∼ with J ( S , (Ξ ,V ) ∈ P ( S , Γ) / ∼ a J, Γ , Ξ ,V π J, Γ , Ξ ,V with a J, Γ , Ξ ,V ∈ k . Let ( J , Γ ) be a minimal standard pair such that a J , Γ , Ξ ,V = 0 for some (Ξ , V ). Using Lemma 7.3 and the sameargument as in § a J , Γ , Ξ ,V = 0 for all (Ξ , V ).That is a contradiction. (cid:3) We also have the following result, which will be used in the study ofsupersingular represenations. Lemma 7.5. Let M = P ( S , Γ) ∈ℵ / ∼ , (Ξ ,V ) ∈ P ( S , Γ) a Γ , Ξ ,V π S , Γ , Ξ ,V for some a Γ , Ξ ,V ∈ k . Then M is rigid if and only if a Γ , Ξ ,V = 0 unless ♯W Γ < ∞ .Proof. We prove by contradiction. Let ( S , Γ ) be the minimal pairin ℵ / ∼ such that ♯W Γ = + ∞ and a Γ , Ξ ,V = 0 for some (Ξ , V ) ∈ P ( S , Γ ). Let τ ∈ Ω(1) and w , w ∈ W Γ (1) with supp( w ) = Γ .Then T w T w T τ is a linear combination of T x with supp( x ) ⊇ Γ . Since ♯W Γ = + ∞ , we have T w T w T τ ∈ ¯˜ H nrig by Lemma 7.3. Then0 = T r ( T w T w T τ , M ) = X (Ξ ,V ) ∈ P ( S , Γ ) / ∼ a Γ , Ξ ,V T r ( T w T w T τ , π S , Γ , Ξ ,V ) , where the second equality follows from the observation that T x π S , Γ , Ξ ,V =0 if ( S , Γ) (cid:10) ( S , supp( x )). By the same argument as in § a Γ , Ξ ,V vanishes. That is a contradiction. (cid:3) Now we describe the image of rigid representations of ˜ H under theinvolution ι . Proposition 7.6. Let ( S , Γ) ∈ ℵ and let (Ξ , V ) ∈ P ( S , Γ) . Let Γ ′ = { s ∈ S a ff r Γ; Ξ( c s ) = 0 } and Ξ ′ be the character of H Γ ′ definedby Ξ ′ | Z = Ξ | Z and Ξ ′ ( s ) = Ξ( c s ) for s ∈ Γ ′ . Then ι ( π S , Γ , Ξ ,V ) ∼ = π S , Γ ′ , Ξ ′ ,V . Lemma 7.7. Let Υ be a character on H . Then Ω(Υ)(1) = Ω(Υ | Z )(1) ∩ Ω(Γ Υ )(1) , where Γ Υ = { s ∈ S a ff ; Υ( s ) = 0 } .Proof. By definition, Ω(Υ)(1) ⊆ Ω(Υ | Z )(1) ∩ Ω(Γ Υ )(1). On the otherhand, let γ ∈ Ω(Υ | Z )(1) ∩ Ω(Γ Υ )(1). If s ∈ S a ff (1) r Γ Υ (1), then γsγ − ∈ S a ff r Γ Υ and Υ( T γsγ − ) = Υ( T s ) = 0. If s ∈ Γ Υ (1), then γsγ − ∈ Γ Υ (1) and Υ( T γsγ − ) = Υ( c γsγ − ) = Υ( γc s γ − ) = Υ( c s ) =Υ( T s ). Therefore, γ ∈ Ω(Υ)(1). (cid:3) We denote by Ξ the extension ofΞ on H defined by Ξ ( T w ) = 0 if w / ∈ W Γ (1). Note that for any s ∈ S a ff (1), ι ( T s ) = − T s + c s . By Lemma 7.7,Ω(Ξ )(1) = Ω(Ξ | Z )(1) ∩ Ω(Γ)(1) , Ω( ι (Ξ ))(1) = Ω(Ξ | Z )(1) ∩ Ω(Γ ′ )(1) . Let γ ∈ Ω(Ξ | Z )(1). Then γ preserves { s ∈ S a ff ; Ξ( c s ) = 0 } . Hence γ ∈ Ω(Γ)(1) if and only γ ∈ Ω(Γ ′ )(1). Thus Ω(Ξ )(1) = Ω( ι (Ξ ))(1).We have π S , Γ , Ξ ,V = ˜ H ⊗ ˜ H (Γ) I (Ξ , V ) ∼ = ˜ H ⊗ ˜ H (Γ) ( ˜ H (Γ) ⊗ H ⋊ Ω(Ξ )(1) (Ξ ⊗ V )) ∼ = ˜ H ⊗ H ⋊ Ω(Ξ )(1) (Ξ ⊗ V ) . RO- p HECKE ALGEBRAS 23 Note that ι (Ξ ) = Ξ ′ . Then ι ( π S , Γ , Ξ ,V ) ∼ = ι ( ˜ H ⊗ H ⋊ Ω(Ξ )(1) (Ξ ⊗ V )) ∼ = ˜ H ⊗ H ⋊ Ω( ι (Ξ ))(1) ( ι (Ξ ) ⊗ V ) ∼ = ˜ H ⊗ H ⋊ Ω(Ξ ′ )(1) (Ξ ′ ⊗ V ) ∼ = π S , Γ ′ , Ξ ′ ,V . Let q be an indeterminant. We denote by ˜ H q the generic pro- p Hecke algebra ˜ H ( q, c ) over k [ q ] such that q t = q for all t ∈ T (1).Notice that ˜ H = ˜ H q /q ˜ H q . Let ˜ w ∈ ˜ W (1). There exists w ∈ W (1), λ , λ ∈ X + (1) such that ˜ w = wλ λ − . Following Vign´eras, we define E ˜ w = q ( ℓ ( µ ) − ℓ ( µ ) − ℓ ( w )+ ℓ ( ˜ w )) T wλ T − λ ∈ ˜ H q . It is known that E ˜ w is independent of the choices of w , λ and λ . By[18], the set { E ˜ w ; ˜ w ∈ ˜ W } forms a basis of ˜ H q .We say that M ∈ R ( ˜ H ) k is supersingular if E w M = 0 for w ∈ ˜ W (1)with ℓ ( w ) ≫ 0. We have the following criterions on the supersingularrepresentations. Theorem 7.8. Let M ∈ R ( ˜ H ) k . The following conditions are equiva-lent:(1) M is supersingular.(2) T r ( ˜ H nss , M ) = 0 .(3) M is spanned by π S , Γ , Ξ ,V for ( S , Γ) ∈ ℵ ∗ and (Ξ , V ) ∈ P ( S , Γ) with ♯W Γ , ♯W Γ ′ < ∞ , where Γ ′ = { π ( s ); s ∈ S a ff (1) r Γ(1); Ξ( c s ) = 0 } .Remark. The equivalence between (1) and (3) is first obtained by Ol-livier in [16, Theorem 5.14] and Vign´era in [20, Theorem 6.18] if ˜ H isthe pro- p Iwahari-Hecke algebra of a p -adic group. Lemma 7.9. Let x, y ∈ ˜ W (1) with ℓ ( x ) ℓ ( y ) . Then q ( ℓ ( x ) − ℓ ( y )+ ℓ ( yx )) T y T − x − ∈ (cid:0) ⊕ z ∈ ˜ W (1) ,ℓ ( z ) > ( ℓ ( y ) − ℓ ( x )+ ℓ ( yx )) Z T z (cid:1) + q Z [ q ] ˜ H q ,q ( ℓ ( y ) − ℓ ( x )+ ℓ ( xy )) T x T − y − ∈ (cid:0) ⊕ z ∈ ˜ W (1) ,ℓ ( z ) > ( ℓ ( y ) − ℓ ( x )+ ℓ ( xy )) Z ι T z (cid:1) + q Z [ q ] ˜ H q . Proof. We prove the first statement. The second one can be proved inthe same way.We argue by induction on ℓ ( x ). If ℓ ( x ) = 0, then statement is ob-vious. Assume ℓ ( x ) > x ′ with ℓ ( x ′ ) < ℓ ( x ). Let s ∈ S a ff (1) such that sx < x .If ys < y , then q ( ℓ ( x ) − ℓ ( y )+ ℓ ( yx )) T y T − x − = q ( ℓ ( sx ) − ℓ ( ys )+ ℓ ( yssx )) T ys T − sx ) − and ℓ ( y ) − ℓ ( x )+ ℓ ( yx ) = ℓ ( ys ) − ℓ ( sx )+ ℓ ( yssx ). The statement followsfrom induction hypothesis. If ys > y , then q ( ℓ ( x ) − ℓ ( y )+ ℓ ( yx )) T y T − x − = q ( ℓ ( sx ) − ℓ ( ys )+ ℓ ( yx )) T ys T − sx ) − + q ( ℓ ( sx ) − ℓ ( y )+ ℓ ( yx ) − (1 − q ) T y T − sx ) − . By inductive hypothesis, q ( ℓ ( sx ) − ℓ ( ys )+ ℓ ( yx )) T ys T − sx ) − ∈ (cid:0) ⊕ z ∈ ˜ W (1) ,ℓ ( z ) > ( ℓ ( y ) − ℓ ( x )+ ℓ ( xy )) Z T z (cid:1) + q Z [ q ] ˜ H q . Let α be the simple root associated to s and β = x − ( α ). Then β < sx < x and yx ( β ) = y ( α ) > ys > y . Hence ysx = yxs β < yx . Therefore, ℓ ( ysx ) ℓ ( yx ) − ℓ ( sx ) − ℓ ( y ) + ℓ ( yx ) − > ℓ ( sx ) − ℓ ( y ) + ℓ ( ysx ).If ℓ ( ysx ) < ℓ ( yx ) − 1, then ℓ ( sx ) − ℓ ( y ) + ℓ ( yx ) − > ℓ ( sx ) − ℓ ( y ) + ℓ ( ysx ) and by inductive hypothesis, q ( ℓ ( sx ) − ℓ ( y )+ ℓ ( yx ) − (1 − q ) T y T − sx ) − ∈ q Z [ q ] ˜ H q and the statement holds in this case.If ℓ ( ysx ) = ℓ ( yx ) − 1, then ℓ ( y ) − ℓ ( x ) + ℓ ( yx ) = ℓ ( y ) − ℓ ( sx ) + ℓ ( ysx )and by inductive hypothesis, q ( ℓ ( sx ) − ℓ ( y )+ ℓ ( yx ) − (1 − q ) T y T − sx ) − ∈ (cid:0) ⊕ z ∈ ˜ W (1) ,ℓ ( z ) > ( ℓ ( y ) − ℓ ( sx )+ ℓ ( ysx )) Z T z (cid:1) + q Z [ q ] ˜ H q = (cid:0) ⊕ z ∈ ˜ W (1) ,ℓ ( z ) > ( ℓ ( y ) − ℓ ( x )+ ℓ ( yx )) Z T z (cid:1) + q Z [ q ] ˜ H q The statement also holds in this case. (cid:3) Corollary 7.10. Let ( S , Γ) ∈ ℵ ∗ . Let w ∈ ˜ W (1) with ℓ ( w ) > ♯W Γ .Then in ˜ H = ˜ H q /q ˜ H q , we have either E w ∈ ⊕ z ∈ ˜ W (1) , supp( z ) * Γ k T z or E ˜ w ∈ ⊕ z ∈ ˜ W (1) , supp( z ) * Γ k ι T z .Proof. By definition, E w = q ( ℓ ( x ) − ℓ ( y )+ ℓ ( yx )) T y T − x − for some x, y ∈ ˜ W (1) such that yx = w . Applying Lemma 7.9, we see that E w ∈⊕ z ∈ ˜ W (1) ,ℓ ( z ) >♯W Γ k T z if ℓ ( x ) ℓ ( y ) and E w ∈ ⊕ z ∈ ˜ W (1) ,ℓ ( z ) >♯W Γ k ι T z if ℓ ( y ) ℓ ( x ). The statement follows by noticing that supp( z ) * Γ if ℓ ( z ) > ♯W Γ . (cid:3) (1) ⇒ (2). Let w ∈ ˜ W (1) min withassociated standard pair ( J, Γ) such that J = J ¯ ν y ( S . It remains toshow T r ( T w , M ) = T r ( ι T w − , M ) = 0. By Proposition 5.3, there exists x ∈ Ω J (1) such that ν x = ¯ ν y and u ∈ W Γ (1) such that T nw ≡ ( T J ( ux ) ) n mod [ ˜ H , ˜ H ] for n ≫ 0. Note that ( T J ( ux ) ) n is a linear combinationof T Ju ′ x n , where u ′ ∈ W Γ (1). By definition, there exists a sufficientlylarge m such that x m ∈ X + (1) and T x m M = E x m M = 0. ByProposition 5.5, ℓ ( u ′ x n ) = ℓ ( u ′ x n − m ) + ℓ ( x m ) for u ′ ∈ W Γ (1) and n ≫ 0. Thus T Ju ′ x n M = T u ′ x n M = T u ′ x n − m T x m M = 0. Therefore, T r ( T nw , M ) = T r (( T Jux ) n , M ) = 0 for n ≫ 0. Hence T r ( T w , M ) = 0 RO- p HECKE ALGEBRAS 25 as desired. The equality T r ( ι T w − , M ) = 0 follows in a similar way bynoticing that ι T x − m = E x − m .(2) ⇒ (3). By Proposition 7.4, M and ι ( M ) lie in the k -span of( π S , Γ , Ξ ,V ) ( S , Γ) ∈ℵ ∗ , (Ξ ,V ) ∈ P ( S , Γ) . Write ι ( M ) = X ( S , Γ) ∈ℵ ∗ / ∼ , (Ξ ,V ) ∈ P ( S , Γ) a Γ , Ξ ,V π S , Γ , Ξ ,V for some a Γ , Ξ ,V ∈ k . By Proposition 7.6, M = X ( S , Γ) ∈ℵ ∗ / ∼ , (Ξ ,V ) ∈ P ( S , Γ) a Γ , Ξ ,V π S , Γ ′ Γ , Ξ , Ξ ′ Γ , Ξ ,V , where Γ ′ Γ , Ξ = { s ∈ S a ff r Γ; Ξ( c s ) = 0 } and Ξ ′ Γ , Ξ is the character of H Γ ′ Ξ , Γ defined by Ξ ′ Γ , Ξ | Z = Ξ | Z and Ξ ′ Γ , Ξ ( s ) = Ξ( c s ) for s ∈ Γ ′ Γ , Ξ . ByLemma 7.5, a Γ , Ξ ,V = 0 unless ♯W Γ ′ Γ , Ξ < ∞ . Part (2) is proved.(3) By definition, T x π S , Γ , Ξ ,V = ι T x π S , Γ ′ , Ξ ′ ,V ′ = 0 for any x ∈ ˜ W (1)such that supp( x ) * Γ and supp( x ) * Γ ′ . 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Department of Mathematics, University of Maryland, College Park,MD 20742, USA E-mail address : [email protected] Institute of Mathematics, Academy of Mathematics and SystemsScience, Chinese Academy of Sciences, 100190, Beijing, China E-mail address ::