Conjugacy classes of involutions and Kazhdan–Lusztig cells

Bonnafé, Cédric; Geck, Meinolf

doi:10.1090/S1088-4165-2014-00456-4

Conjugacy classes of involutions and Kazhdan–Lusztig cells
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by Cédric Bonnafé and Meinolf Geck

Represent. Theory 18 (2014), 155-182

DOI: https://doi.org/10.1090/S1088-4165-2014-00456-4

Published electronically: July 22, 2014

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Abstract:

According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group $\mathfrak {S}_n$ 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.

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