Conjugacy classes of involutions and Kazhdan-Lusztig cells

TL;DR

This paper explores the conjugacy of involutions within two-sided cells across Coxeter groups, generalizing Schützenberger's symmetric group result and proving Kottwitz's conjecture for classical types, advancing understanding of cell structures.

Contribution

It generalizes Schützenberger's involution conjugacy result to 'smooth' two-sided cells and proves Kottwitz's conjecture for classical Coxeter groups, leaving only type E8 unresolved.

Findings

Involutions in a two-sided cell are conjugate in symmetric groups.

Schützenberger's result extends to 'smooth' two-sided cells in Coxeter groups.

Kottwitz's conjecture is proved for classical types.

Abstract

According to an old result of Sch\"utzenberger, the involutions in a given two-sided cell of the symmetric group $\SG_n$ are all conjugate. In this paper, we study possible generalisations of this property to other types of Coxeter groups. We show that Sch\"utzenberger's result is a special case of a general result on "smooth" two-sided cells. Furthermore, we consider Kottwitz' conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types; combined with previous work, this leaves type $E_8$ as the only remaining open case.