Automorphisms of Coxeter groups and Lusztig's conjectures for Hecke algebras with unequal parameters

TL;DR

This paper investigates the relationships between certain algebraic invariants of Coxeter groups and their fixed point subgroups under automorphisms, assuming Lusztig's conjectures hold, with implications for Hecke algebras with unequal parameters.

Contribution

It compares the ${f a}$-function, Duflo involutions, and Kazhdan-Lusztig cells of Coxeter groups and their automorphism-fixed subgroups under Lusztig's conjectures.

Findings

Comparison of ${f a}$-functions for groups and fixed subgroups

Analysis of Duflo involutions under automorphisms

Behavior of Kazhdan-Lusztig cells in this context

Abstract

Let $(W,S)$ be a Coxeter system, let $G$ be a finite solvable group of automorphisms of $(W,S)$ and let $\varphi$ be a weight function which is invariant under $G$. Let $\varphi_G$ denote the weight function on $W^G$ obtained by restriction from $\varphi$. The aim of this paper is to compare the ${\mathbf{a}}$-function, the set of Duflo involutions and the Kazhdan-Lusztig cells associated to $(W,\varphi)$ and to $(W^G,\varphi_G)$, provided that Lusztig's Conjectures hold.