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Journal of Lie Theory 25 (2015), No. 2, 395--429
Copyright Heldermann Verlag 2015

On Kottwitz's Conjecture for Twisted Involutions

Meinolf Geck
IAZ - Lehrstuhl für Algebra, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany


Motivated by problems on nilpotent orbital integrals for real Lie groups, Kottwitz (2000) formulated a conjecture concerning the relationship between Kazhdan-Lusztig cells of a finite Coxeter group $W$ and its conjugacy classes of $\diamond$-twisted involutions, where $\diamond$ is an involutory graph automorphism of $W$. In this paper, we study this relationship in type $D_n$ and all cases where $\diamond$ is non-trivial. Combined with work of Kottwitz himself, Casselmann, Marberg, and joint work of Bonnaf\'e, Halls and the author, this completes the proof of Kottwitz's Conjecture for all $W,\,\diamond$.

Keywords: Coxeter groups, twisted involutions, Kazhdan-Lusztig cells.

MSC: 20F55; 20G40, 22E50

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