Leading coefficients and cellular bases of Hecke algebras

TL;DR

This paper introduces a new approach to understanding the asymptotic ring of Hecke algebras, enabling cellular bases construction for broader classes of Coxeter groups and parameters, with potential extensions to complex reflection groups.

Contribution

It presents a novel method to derive Lusztig's asymptotic ring from generic representations, extending cellular basis constructions beyond Weyl groups and equal parameters.

Findings

The asymptotic ring J can be obtained without Kazhdan--Lusztig basis.

Cellular bases can be constructed for non-crystallographic Coxeter groups.

Existence of cellular structures in multi-parameter Hecke algebras.

Abstract

Let $\bH$ be the generic Iwahori--Hecke algebra associated with a finite Coxeter group $W$. Recently, we have shown that $\bH$ admits a natural cellular basis in the sense of Graham--Lehrer, provided that $W$ is a Weyl group and all parameters of $\bH$ are equal. The construction involves some data arising from the Kazhdan--Lusztig basis $\{\bC_w\}$ of $\bH$ and Lusztig's asymptotic ring $\bJ$. This article attemps to study $\bJ$ and its representation theory from a new point of view. We show that $\bJ$ can be obtained in an entirely different fashion from the generic representations of $\bH$, without any reference to $\{\bC_w\}$. Then we can extend the construction of the cellular basis to the case where $W$ is not crystallographic. Furthermore, if $\bH$ is a multi-parameter algebra, we will see that there always exists at least one cellular structure on $\bH$. Finally, one may also hope that the new construction of $\bJ$ can be extended to Hecke algebras associated to complex reflection groups.