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

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)→R, due to Lusztig for Weyl groups and to Broué, Lusztig, and Malle in the remaining cases. This paper concerns a certain W-representation ϱW in the vector space generated by the involutions of W. Our main result is to show that the irreducible multiplicities of ϱW are given by the Fourier transform of a unique function ϵ:Uch(W)→{−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 ϵ 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 ϱW derived by Kottwitz when (W,S) is classical, and prove that ϱW defines a Gelfand model if and only if (W,S) has type An, H3, or I2(m) with m odd. We show finally that a conjecture of Kottwitz connecting the decomposition of ϱW 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.