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

TL;DR

This paper establishes a connection between twisted involution modules in Weyl groups and Frobenius--Schur indicators of unipotent characters in twisted finite groups of Lie type, extending previous untwisted results.

Contribution

It introduces a general relation between twisted involution modules and Frobenius--Schur indicators, broadening the understanding of these concepts in twisted Lie type groups.

Findings

Relation between twisted involution modules and Frobenius--Schur indicators established

Extension of Lusztig-Vogan results to twisted cases

Definition of Frobenius--Schur indicators for twisted dihedral groups

Abstract

Let $W$ be a finite Weyl group and $\sg$ be a non-trivial graph automorphism of $W$. We show a remarkable relation between the $\sg$-twisted involution module for $W$ and the Frobenius--Schur indicators of the unipotent characters of a corresponding twisted finite group of Lie type. This extends earlier results of Lusztig-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.