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

Abstract

For each finite, irreducible Coxeter system $(W,S)$, Lusztig has associated a set of "unipotent characters" $\Uch(W)$. There is also a notion of a "Fourier transform" on the space of functions $\Uch(W) \to \RR$, due to Lusztig for Weyl groups and to Brou\'e, Lusztig, and Malle in the remaining cases. This paper concerns a certain $W$-representation $\varrho_{W}$ in the vector space generated by the involutions of $W$. Our main result is to show that the irreducible multiplicities of $\varrho_W$ are given by the Fourier transform of a unique function $\epsilon : \Uch(W) \to \{-1,0,1\}$, which for various reasons serves naturally as a heuristic definition of the Frobenius-Schur indicator on $\Uch(W)$. The formula we obtain for $\epsilon$ extends prior work of Casselman, Kottwitz, Lusztig, and Vogan addressing the case in which $W$ is a Weyl group. We include in addition a succinct description of the irreducible decomposition of $\varrho_W$ derived by Kottwitz when $(W,S)$ is classical, and prove that $\varrho_{W}$ defines a Gelfand model if and only if $(W,S)$ has type $A_n$, $H_3$, or $I_2(m)$ with $m$ odd. We show finally that a conjecture of Kottwitz connecting the decomposition of $\varrho_W$ to the left cells of $W$ holds in all non-crystallographic types, and observe that a weaker form of Kottwitz's conjecture holds in general. In giving these results, we carefully survey the construction and notable properties of the set $\Uch(W)$ and its attached Fourier transform.