The Real Chevalley Involution

TL;DR

This paper investigates the Chevalley involution in real reductive groups, establishing its uniqueness up to conjugacy and exploring implications for self-dual representations and Frobenius-Schur indicators.

Contribution

It proves the existence and uniqueness of the Chevalley involution in real reductive groups and applies this to characterize self-dual representations and Frobenius-Schur indicators.

Findings

Existence of a unique Chevalley involution up to conjugacy.

Conditions for all irreducible representations to be self-dual.

Determination of Frobenius-Schur indicators for these groups.

Abstract

We consider the Chevalley involution in the context of real reductive groups. We show that if G(R) is the real points of a connected reductive group, there is an involution, unique up to conjugacy by G(R), taking any semisimple element to a conjugate of its inverse. As applications we give a condition for every irreducible representation of G(R) to be self-dual, and to the Frobenius Schur indicator for such groups.