Centralizers in Hecke Algebras of Any Coxeter Group

TL;DR

This paper investigates the structure of centralizers in Hecke algebras associated with arbitrary Coxeter groups, extending known results to the indefinite case using algebraic and geometric methods.

Contribution

It provides a basis description for the centralizer in general Coxeter groups and introduces a classification of partial conjugacy classes and a new class polynomial.

Findings

Basis for the centralizer in arbitrary Coxeter groups

Classification of finite partial conjugacy classes

Introduction of a new class polynomial for the centralizer

Abstract

We study the centralizer of a parabolic subalgebra in the Hecke algebra associated with an arbitrary (possibly infinite) Coxeter group. While the center and cocenter have been extensively studied in the finite and affine cases, much less is known in the indefinite setting. We describe a basis for the centralizer, generalizing known results about the center. Our approach combines algebraic techniques with geometric tools from the Davis complex, a CAT(0)-space associated to the Coxeter group. As part of the construction, we classify finite partial conjugacy classes in infinite Coxeter groups and define a variant of the class polynomial adapted to the centralizer.