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On the centre of Iwahori-Hecke algebras

Timothée Marquis, Sven Raum

Abstract

We prove triviality of the centre of arbitrary Hecke algebras of irreducible non-finite non-affine type. This result is obtained as a consequence of the following structure result for conjugacy classes of the underlying Coxeter groups. If $W$ is any infinite irreducible Coxeter group and $w \in W$ is a nontrivial element that is assumed not be a translation in case $W$ is affine, then there is an infinite sequence of conjugates of $w$ by Coxeter generators whose length is non-decreasing and tends to infinity.

On the centre of Iwahori-Hecke algebras

Abstract

We prove triviality of the centre of arbitrary Hecke algebras of irreducible non-finite non-affine type. This result is obtained as a consequence of the following structure result for conjugacy classes of the underlying Coxeter groups. If is any infinite irreducible Coxeter group and is a nontrivial element that is assumed not be a translation in case is affine, then there is an infinite sequence of conjugates of by Coxeter generators whose length is non-decreasing and tends to infinity.
Paper Structure (10 sections, 18 theorems, 26 equations, 4 figures)

This paper contains 10 sections, 18 theorems, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Coxeter groups and the Davis complex
  3. Coxeter groups
  4. Coxeter complexes
  5. Residues
  6. Cosets of standard parabolic subgroups
  7. Davis complex
  8. Coxeter groups with an infinite proper parabolic subgroup
  9. The affine and compact hyperbolic cases
  10. The centre of Hecke algebras

Key Result

Theorem 1

Let $(W,S)$ be a Coxeter system of irreducible indefinite type, let $R$ be a ring and $(a_s,b_s)_{s \in S}$ a deformation parameter. Then the centre of the generic Hecke algebra $\mathcal{H}(W, S, (a_s, b_s)_{s \in S})$ is trivial.

Figures (4)

  • Figure 1: $W=\langle s,t,u\rangle$ of affine type $(3,3,3)$.
  • Figure 2: $W=\langle s,t,u\rangle$ of compact hyperbolic type $(4,4,4)$.
  • Figure 3: Lemma \ref{['lemma:sphericalcriterion']}
  • Figure 4: Proposition \ref{['prop:dvsdchvsdw']}

Theorems & Definitions (38)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Example 2.3
  • Lemma 2.4
  • proof
  • ...and 28 more