Conjugacy classes of involutions and Kazhdan-Lusztig cells

According to an old result of Sch"utzenberger, the involutions in a given two-sided cell of the symmetric group ${\text{SG}}_n$ 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.