Reality Properties of Conjugacy Classes in algebraic Groups

TL;DR

This paper investigates the properties of conjugacy classes in algebraic groups, focusing on the concepts of real and strongly real elements, and establishes conditions under which real elements are strongly real across various types of algebraic groups and fields.

Contribution

It proves that strongly regular $k$-real elements are strongly $k$-real in connected adjoint semisimple groups over perfect fields, and extends reality results to classical groups and groups of type $G_2$.

Findings

Strongly regular $k$-real elements are strongly $k$-real in certain groups.

Semisimple $k$-real elements are strongly $k$-real in classical groups with mild exceptions.

Real elements in groups of type $G_2$ are strongly $k$-real under specific characteristic conditions.

Abstract

Let $G$ be an algebraic group defined over a field $k$. We call $g\in G$ {\bf real} if $g$ is conjugate to $g^{-1}$ and $g\in G(k)$ as {\bf $k$-real} if $g$ is real in $G(k)$. An element $g\in G$ is {\bf strongly real} if $\exists h\in G$, $h^{2}=1$ (i.e. $h$ is an {\bf involution}) such that $hgh^{-1}=g^{-1}$. Clearly, strongly real elements are real and are product of two involutions. Let $G$ be a connected adjoint semisimple group over a perfect field $k$, with -1 in the Weyl group. We prove that any strongly regular $k$-real element in $G(k)$ is strongly $k$-real (i.e. is a product of two involutions in $G(k)$). For classical groups, with some mild exceptions, over an arbitrary field $k$ of characteristic not 2, we prove that $k$-real semisimple elements are strongly $k$-real. We compute an obstruction to reality and prove some results on reality specific to fields $k$ with $cd(k)\leq 1$. Finally, we prove that in a group $G$ of type $G_2$ over $k$, characteristic of $k$ different from 2 and 3, any real element in $G(k)$ is strongly $k$-real. This extends our results in \cite{st}, on reality for semisimple and unipotent real elements in groups of type $G_2$.