Real fundamental Chevalley involutions and conjugacy classes

TL;DR

This paper proves that for real connected reductive groups, the fundamental Chevalley involution maps each element to a conjugate of its inverse, extending to Lie algebras.

Contribution

It establishes a conjugacy property of the Chevalley involution for real reductive groups and their Lie algebras, a result not previously shown.

Findings

For every element g in G(R), C(g) is conjugate to g^{-1}.

The result extends to the Lie algebra setting.

Provides a new understanding of involutions in real algebraic groups.

Abstract

Let $\mathsf G$ be a connected reductive linear algebraic group defined over $\mathbb R$, and let $C: \mathsf G\rightarrow \mathsf G$ be a fundamental Chevalley involution. We show that for every $g\in \mathsf G(\mathbb R)$, $C(g)$ is conjugate to $g^{-1}$ in the group $\mathsf G(\mathbb R)$. Similar result on the Lie algebras is also obtained.