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ANDREW FRANCIS. Abstract. ... ANDREW FRANCIS. Property II: For any ..... [9] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Uni-.
CENTRALIZERS OF IWAHORI-HECKE ALGEBRAS II: THE GENERAL CASE ANDREW FRANCIS Abstract. This note is a sequel to [4]. We establish the minimal basis theory for the centralizers of parabolic subalgebras of IwahoriHecke algebras associated to finite Coxeter groups of any type, generalizing the approach introduced in [3] from centres to the centralizer case. As a pre-requisite, we prove a reducibility property in the twisted J-conjugacy classes in finite Coxeter groups, which is a generalization of results in [7] and [4].

1. Introduction The minimal basis theory of [3] for centres of Iwahori-Hecke algebras was part of a minor flurry of work on the question, including work of Lenny Jones [10] and Meinolf Geck and Rapha¨el Rouquier [8], and provided a combinatorial approach to some kind of natural/canonical basis for the centres of these algebras. The current paper completes the generalization of this approach to centralizers of parabolic subalgebras, which was begun in [4] with a minimal basis theory for any centralizer of a parabolic subalgebra in types An or Bn . What this generalization to arbitrary finite Coxeter groups requires, by Theorem 1.1 below, is the verification of a pair of properties to be satisfied by the J-conjugacy classes in the Coxeter group. These properties are: Property I: All J-conjugacy classes are reducible. Date: September 11, 2001. 2000 Mathematics Subject Classification. 20C08 20F55. Key words and phrases. Hecke algebra, Coxeter group, conjugacy class, centralizer, minimal basis. 1

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Property II: For any J-conjugacy class C and w, w0 ∈ Cmin , we have w ∼GP w0 . The theorem depending on these is as follows: Theorem 1.1. ([4]: (4.4), (4.5)) If (W, S) is a finite Coxeter system, and J ⊆ S satisfies Properties I and II, then the set of J-class elements is the set of primitive minimal positive elements of ZH (HJ ). That is, ZH (HJ )+ min = {ΓC | C ∈ cclJ (W )}. Furthermore, ZH (HJ )+ min is an R-basis for ZH (HJ ), where R = Z[ξs ]s∈S is the base ring of the Iwahori-Hecke algebra H. Properties I and II are somewhat technical and combinatorial results in the Coxeter group, and proofs of particular cases (the fundamental one being [7]) have so far required case by case analyses and the use of the computer algebra packages GAP [11] and CHEVIE [5]. For the results in this paper we do not directly use any computer calculations, but nevertheless we rely on other theorems (in [7] and [6]) which do rely on case by case checks. In [4] it is also shown that Properties I and II depend on the reducibility of certain twisted J-conjugacy classes. While our interest in these is motivated by an application to bases of centralizers in Iwahori-Hecke algebras, recently at least two other papers have appeared studying twisted conjugacy classes for other reasons, and in different ways. In [6], Meinolf Geck, Sungsoon Kim, and G¨otz Pfeiffer study minimal elements of twisted conjugacy classes in finite Coxeter groups in an effort to extend the results of [7] by finding character tables of Iwahori-Hecke algebras with an associated automorphism fixing the set S of simple reflections (a “twist”). This work is motivated by work of Fran¸cois Digne and Jean Michel [2] on L-functions, and contains an analogue of a reducibility result from [7] for twisted conjugacy classes which we will use in the current paper.

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Weyl groups with automorphisms which fix the set of generators have also arisen in the study of the group of symmetries of an Artin system in the work of John Crisp [1]. 2. Background definitions Let W be a finite Coxeter group with generating set S of simple reflections, and length function l : W → N. Then for s, s0 ∈ S, W has relations (1)

s2 = 1

(2)

(ss0 )mss0 = 1

for some mss0 ∈ N. For general information about the structure of finite Coxeter groups, the reader is referred to [9]. For J ⊆ S, each finite Coxeter group is partitioned into J-conjugacy classes, corresponding to sets of elements conjugate by elements of WJ (the parabolic subgroup generated by J). Write DK,J for the set of distinguished representatives of the double cosets WK dWJ where J, K ⊆ S. Let σ be a bijection between subsets J and K of S, extended to an isomorphism σ : WJ → WK . Define a twisted J-conjugacy class Cσ (or Cσ,w when we want to stress a representative w of the class) to be a set of form {σ(u)wu−1 | u ∈ WJ }. Such twisted J-conjugacy classes partition the double coset σ(WJ )wWJ = WK dWJ . Note that when σ = 1 and J = K, this set is a straightforward J-conjugacy class, and when in addition J = S, we have a standard conjugacy class. Note also that when J = S then σ becomes an automorphism of W , which is the case studied in [6]. It is worth bearing these special cases in mind as we will make the following definitions in the generality of twisted J-conjugacy classes. For w, w0 ∈ Cσ for some twisted J-conjugacy class Cσ , we say w →J w0 if there exists a sequence r1 , r2 , . . . , rm of elements of J and a sequence w0 , . . . , wm of elements of Cσ such that if w0 = w, and

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wi = σ(ri )wi−1 ri (1 ≤ i ≤ m) then wm = w0 , and l(wi ) ≤ l(wi−1 ) with wi 6= wi−1 , for 1 ≤ i ≤ m. Let Cσ,min be the set of shortest elements in the twisted J-conjugacy class Cσ . We say a twisted J-conjugacy class Cσ is reducible if for all w ∈ Cσ there exists a v ∈ Cσ,min such that w →J v. If w, w0 ∈ Cσ have the same length, we say w ∼GP w0 if there exists a sequence of xi ∈ WJ and wi ∈ Cσ such that w = w0 , σ(xi )wi x−1 i = wi+1 , and wn+1 = w0 , with l(wi ) = l(w) and either l(σ(xi )wi ) = l(xi ) + l(wi ) or l(wi x−1 i ) = l(wi ) + l(xi ) for each i, 0 ≤ i ≤ n. The subscript GP is used as to the best of my knowledge this equivalence was first introduced in [7].

3. Reducibility in twisted J-conjugacy classes In this section we will provide proofs of Properties I and II. We will actually prove the marginally more general result of allowing our Jconjugacy classes to be twisted, as this will help smooth the inductive argument. But first we recall some results from [4]. Let d ∈ DK,J , and let σ be a bijection from J to K. Let Kl ⊆ K be the set {s ∈ K | sd = ds0 for some s0 ∈ J}, and let Jr be the subset of J consisting of those s0 for which sd = ds0 for some s ∈ Kl . Let σd be the bijection from Kl to Jr defined by d (setting σd (s) = s0 ∈ Jr if sd = ds0 ). Define the subset Jl of J to be Jl = σ −1 (Kl ). Lemma 3.1. ([4], (2.3), (2.4)) Let dw be an element of the twisted J-conjugacy class Cσ,dw , with d ∈ DK,J and w ∈ WJ . Then (i) dw is a shortest element of Cσ,dw if and only if w is a shortest element of its twisted Jl -conjugacy class Cσd σ,w . (ii) dw is reducible in its twisted J-conjugacy class Cσ,dw if and only if w is reducible in its twisted Jl -conjugacy class Cσd σ,w . We will also need the following result on full conjugacy classes.

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Theorem 3.2. ([6] Theorem (2.6)) If C is a twisted conjugacy class of W , then C is reducible, and w ∼GP w0 for any w, w0 ∈ Cmin . This gives Properties I and II in the case J = S. It remains to generalize these to arbitrary J ⊆ S. Theorem 3.3. All twisted J-conjugacy classes in a finite Coxeter group W are reducible. Proof. Let (W, S) be a finite Coxeter system, and let J ⊆ S. Since the result holds for J = S by [6], we may assume J ⊂ S. We will proceed by induction on |J|. If |J| = 1 the result can easily be seen, since any twisted J-conjugacy class has at most 2 elements. Suppose inductively that for |J| ≤ k, all twisted J-conjugacy classes are reducible. Let |J| = k + 1. It is clear that any element of WK dWJ can be reduced to an element of form dw, so it is sufficient to show that given an arbitrary dw for w ∈ WJ and d distinguished, we can reduce dw to a shortest element of its twisted J-conjugacy class. By Lemma 3.1, this is equivalent to showing that w is reducible in its twisted Jl -conjugacy class, where the twist is σd σ. Now if Jl ( J, we have the reducibility of w by induction. If on the other hand Jl = J, then since w ∈ WJ , we need w to be reducible in its twisted conjugacy class in W = WJ . This holds by 3.2, which completes the proof.

¤

Theorem 3.4. For any twisted J-conjugacy class Cσ and w, w0 ∈ Cσ,min , we have w ∼GP w0 . Proof. We need to show the existence of a sequence of twisted conjugations by elements of WJ , connecting w and w0 . Clearly w ∼GP du and w0 ∼GP du0 for d ∈ DK,J and some u, u0 ∈ WJ , so it suffices to show that any pair du, du0 ∈ Cσ,min satisfy du ∼GP du0 . To show this, in turn it is sufficient by Lemma 3.1 to show u ∼GP u0 in their σd σ twisted Jl -conjugacy class within WJ .

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We induce on |J|. If |J| = 1, the result is trivial since each twisted J-conjugacy class has at most two elements. Suppose then that it holds for |J| ≤ k, and fix a J with k + 1 elements. Take arbitrary elements in Cσ,w,min of form du and du0 . Then by Lemma 3.1 (i), we have that u and u0 are minimal elements of the twisted Jl -conjugacy class Cσd σ,u . If Jl

J, then by our induction hypothesis we have u ∼GP u0 . In

other words, we have a sequence of twisted conjugations by elements of Jl , where the twist is σd σ, connecting u and u0 . This same sequence then provides the required sequence connecting du with du0 in the twisted J-conjugacy class Cσ,w . If on the other hand Jl = J, then u and u0 are minimal elements in their σd σ twisted J-conjugacy class in WJ . That is, they are actually shortest elements in the full σd σ twisted conjugacy class within the finite Coxeter group WJ . Then by 3.2, u ∼GP u0 and so there exists a sequence of σd σ twisted conjugations by elements of WJl connecting them. Exactly the same sequence of conjugations, twisting instead by σ only, will connect du and du0 as required by the definition of ∼GP . ¤ 4. Centralizers in Iwahori-Hecke algebras Given a set {ξs | s ∈ S} of indeterminates, and R = Z[ξs ]s∈S , the Iwahori-Hecke algebra H over R is the associative algebra generated by the set {T˜s | s ∈ S} with relations T˜s2 = T˜1 + ξs T˜s and T˜w = T˜si1 . . . T˜sir when w = si1 . . . sir is a reduced expression for w. The relationship 1

1

between this definition of H and the more standard one over Z[q 2 , q − 2 ] can be found in [4]. We write H+ for the set of elements of H whose terms T˜w have coefficient in R+ := N[ξs ]s∈S . That is, H+ is the R+ -span of the set

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{T˜w | w ∈ W }. This positive part H+ of H is given a partial order by setting for a, b ∈ H+ , a ≤ b if b − a ∈ H+ . If h ∈ H+ , we say h is primitive if the monomial coefficients of terms T˜w in h have no common factors. If we set ZH (HJ )+ := ZH (HJ ) ∩ H+ , and define ZH (HJ )+ min to be the primitive minimal elements of ZH (HJ )+ with respect to the partial order ≤ on H+ , then the main theorem of [3] was that Z(H)+ min is an R-basis for the centre of H, called the minimal basis. The minimal ˜ C characterized by the properties that basis consists of class elements Γ ˜ C |ξ=0 = T˜C (the conjugacy class sum), and Γ ˜ C − T˜C contains no Γ shortest elements of any conjugacy class. To extend this structure to a centralizer of a parabolic subalgebra HJ for J ⊆ S, let C be a J-conjugacy class, and let ZH (HJ )+ min be the set of primitive minimal positive elements of the centralizer ZH (HJ ). Then ˜C the generalization requires proving the existence of J-class elements Γ which specialize to J-class sums, and that these J-class elements form the set ZH (HJ )+ min and an R-basis for ZH (HJ ). In [4] the above generalization to the centralizer setting was reduced to proving Properties I and II (Theorem 1.1). This was achieved only in types A and B. Now that we have the generalization of Properties I and II to any J ⊆ S, the generalization to centralizers of parabolic subalgebras of Iwahori-Hecke algebras of finite Coxeter groups is complete. Theorem 4.1. Suppose J ⊆ S. Then ˜ ZH (HJ )+ min = {ΓC | C ∈ cclJ (W )} and ZH (HJ )+ min is an R-basis for ZH (HJ ). References [1] John Crisp, Symmetrical subgroups of Artin groups, Adv. Math., 152, (2000), no. 1, 159–177. MR 2001c:20083

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[2] Fran¸cois Digne and Jean Michel, Fonctions L des vari´et´es de Deligne-Lusztig et descente de Shintani, M´em. Soc. Math. France (N.S.), Soci´et´e Math´ematique de France, Paris, no. 20, (1985), iv+144. MR 87h:20071 [3] Andrew Francis, The minimal basis for the centre of an Iwahori-Hecke algebra, J. Algebra, 221, (1999), no. 1, 1–28. MR 2000k:20005 [4]

, Centralizers of Iwahori-Hecke algebras, Trans. Amer. Math. Soc., 353, (2001), no. 7, 2725–2739. MR 1 828 470

[5] Meinolf Geck, Gerhard Hiß, Frank L¨ ubeck, Gunter Malle and G¨otz Pfeiffer, CHEVIE—a system for computing and processing generic character tables., Computational methods in Lie theory (Essen 1994), Appl. Algebra ENGRG. Comm. Comput., (1996), 7, no. 3, 175-210. [6] Meinolf Geck, Sungsoon Kim and G¨otz Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra, 229, (2000), no. 2, 570–600. MR 1 769 289 [7] Meinolf Geck and G¨otz Pfeiffer, On the irreducible characters of Hecke algebras, Adv. Math., 102, (1993), no. 1, 79–94. MR 94m:20018 [8] Meinolf Geck and Rapha¨el Rouquier, Centers and simple modules for IwahoriHecke algebras, Finite reductive groups (Luminy, 1994), Birkh¨auser Boston, Boston, MA, (1997), 251–272. MR 98c:20013 [9] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, (1990), xii+204. MR 92h:20002 [10] Lenny K. Jones, Centers of generic Hecke algebras, Trans. Amer. Math. Soc., 317, (1990), no. 1, 361–392. MR 90d:20030 [11] Martin Sch¨onert et al., GAP – Groups, Algorithms, and Programming, Lehrstuhl D f¨ ur Mathematik, Rheinisch Westf¨alische Technische Hochschule, Aachen, Germany, fifth ed., 1995. School of Quantitative Methods and Mathematical Sciences, University of Western Sydney E-mail address: [email protected] URL: http://www.uws.edu.au/vip/afrancis/