Frobenius–Schur indicators of unipotent characters and the twisted involution module

Abstract

Let W W be a finite Weyl group and σ \sigma a non-trivial graph automorphism of W W . We show a remarkable relation between the σ \sigma -twisted involution module for W W and the Frobenius–Schur indicators of the unipotent characters of a corresponding twisted finite group of Lie type. This extends earlier results of Lusztig and Vogan for the untwisted case and then allows us to state a general result valid for any finite group of Lie type. Inspired by recent work of Marberg, we also formally define Frobenius–Schur indicators for “unipotent characters” of twisted dihedral groups.