Conjugacy classes of involutions and Kazhdan–Lusztig cells

Abstract

According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group S n \mathfrak {S}_n are all conjugate. In this paper, we study possible generalizations of this property to other types of Coxeter groups. We show that Schützenberger’s result is a special case of a general result on “smooth” two-sided cells. Furthermore, we consider Kottwitz’s 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 which, combined with further recent work, settles this conjecture in general.