Hecke algebras with unequal parameters

TL;DR

This paper explores the structure and properties of Iwahori-Hecke algebras with unequal parameters, extending known results from the equal parameter case and incorporating recent advances in the theory of Soergel bimodules.

Contribution

It generalizes key results of Hecke algebras to the unequal parameter setting, integrating recent developments in Soergel bimodule theory.

Findings

Extended results from equal to unequal parameter cases

Connected Hecke algebra properties with Soergel bimodules

Updated the theory with recent Elias and Williamson results

Abstract

These are notes for the Aisenstadt lectures given in may/june 2002 at CRM, Montreal, enlarged and updated in 2014 by taking into account the recent results of Elias and Williamson on Soergel bimodules. The main object is the study of Iwahori-Hecke algebras arising from reductive groups over finite or p-adic fields. We try to extend various results known in the equal parameter case to the case of not necessarily equal parameters.