Reality Properties of Conjugacy Classes in G_2

TL;DR

This paper investigates the reality of conjugacy classes in algebraic groups of type G_2 over various fields, characterizing when elements are conjugate to their inverses and providing examples of nonreal elements.

Contribution

It characterizes reality for semisimple and unipotent elements in G_2 over different fields, including conditions for semisimple elements to be real and examples of nonreal elements.

Findings

Semisimple elements are real iff they are products of two involutions.

All unipotent elements are products of two involutions.

Reality fails for semisimple elements over $\\Q$ and $\Q_p$, but holds over fields with $cd(k)\leq 1$.

Abstract

Let $G$ be an algebraic group over a field $k$. We call $g\in G(k)$ {\bf real} if $g$ is conjugate to $g^{-1}$ in $G(k)$. In this paper we study reality for groups of type $G_2$ over fields of characteristic different from 2. Let $G$ be such a group over $k$. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element in $G(k)$ is real if and only if it is a product of two involutions in $G(k)$. Every unipotent element in $G(k)$ is a product of two involutions in $G(k)$. We discuss reality for $G_2$ over special fields and construct examples to show that reality fails for semisimple elements in $G_2$ over $\Q$ and $\Q_p$. We show that semisimple elements are real for $G_2$ over $k$ with $cd(k)\leq 1$. We conclude with examples of nonreal elements in $G_2$ over $k$ finite, with characteristic $k$ not 2 or 3, which are not semisimple or unipotent.