Conjugacy classes of involutions and Kazhdan–Lusztig cells

Abstract

Abstract. According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group 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.

How to compute the Frobenius–Schur indicator of a unipotent character of a finite Coxeter system

“…From this and the results of [14,26], it follows that Kottwitz's conjecture holds for all finite irreducible Coxeter groups except possibly those of type BC n , D n , E 7 , and E 8 . Recent work of Bonnafé and Geck [8,19] has established the conjecture in all of these remaining cases except E 8 .…”

“…In Section 5 below, we show ourselves that the conjecture holds in all of the non-crystallographic cases H 3 , H 4 , and I 2 (m). Two preprints of Geck and Bonnafé [8,19], appearing after the completion of this article, establish several more cases, leaving the conjecture open only in type E 8 .…”

“…1,8,21,39, 74, 2 2k , k 2 , or k 2 + k + 2 (where k is a positive integer), and the families of size 74, k 2 , and k 2 + k + 2 occur only in types H 4 , I 2 (2k + 1), and I 2 (2k + 2), respectively. If (W, S) has one of the classical types A n , BC n , or D n , then the families in Uch(W ) each correspond to setsM (Z/2Z) k for various integers k ≥ 0.…”

On the Kazhdan–Lusztig cells in type $E_8$

“…By work of Kottwitz himself, Casselman [5], Bonnafé and the first-named author [4], [19], [21], this conjecture is already known to hold except possibly for W of type E 8 . The verification for type E 8 is now a matter of combining various pieces of known information.…”

“…The original motivation for this work was a conjecture due to Kottwitz [31], concerning the characters of left cell representations and intersections of left cells with conjugacy classes of involutions. By work of Kottwitz himself, Casselman [5], Bonnafé and the first-named author [4], [19], [21], this conjecture was known to hold except possibly for type E 8 . The algorithms developed in this paper allow us to verify Kottwitz's conjecture for type E 8 in a straightforward way (by an almost automatic procedure).…”