On Iwahori--Hecke algebras with unequal parameters and Lusztig's isomorphism theorem

TL;DR

This paper develops new methods to verify Lusztig's isomorphism theorem for Iwahori--Hecke algebras with unequal parameters, extending its applicability to all finite Coxeter groups without geometric assumptions.

Contribution

The authors introduce novel techniques to verify key properties of Kazhdan--Lusztig bases, enabling Lusztig's isomorphism construction for all types of Iwahori--Hecke algebras with arbitrary parameters.

Findings

Verified properties for two-parameter algebras of types I_2(m) and F_4

Extended Lusztig's isomorphism to all finite Coxeter groups

Removed restrictions on parameters in Lusztig's construction

Abstract

By Tits' deformation argument, a generic Iwahori--Hecke algebra $H$ associated to a finite Coxeter group $W$ is abstractly isomorphic to the group algebra of $W$. Lusztig has shown how one can construct an explicit isomorphism, provided that the Kazhdan--Lusztig basis of $H$ satisfies certain deep properties. If $W$ is crystallographic and $H$ is a one-parameter algebra, then these properties are known to hold thanks to a geometric interpretation. In this paper, we develop some new general methods for verifying these properties, and we do verify them for two-parameter algebras of type $I_2(m)$ and $F_4$ (where no geometric interpretation is available in general). Combined with previous work by Alvis, Bonnaf\'e, DuCloux, Iancu and the author, we can then extend Lusztig's construction of an explicit isomorphism to all types of $W$, without any restriction on the parameters of $H$.