Real Elements in Spin Groups

TL;DR

This paper extends the characterization of real elements from classical groups to semisimple elements in Spin groups over fields with characteristic not 2, focusing on dimensions congruent to 0, 1, or 2 modulo 4.

Contribution

It provides a new criterion for identifying real elements in Spin groups, generalizing previous results known for classical groups.

Findings

Semisimple elements in Spin groups are real if and only if they can be expressed as a product of two elements with squares ±1.

The characterization depends on the dimension of the underlying vector space modulo 4.

The results unify the understanding of real elements across different algebraic groups.

Abstract

Let $F$ be a field of characteristic $\neq 2$. Let $G$ be an algebraic group defined over $F$. An element $t\in G(F)$ is called {\bf real} if there exists $s\in G(F)$ such that $sts^{-1}=t^{-1}$. A semisimple element $t$ in $GL_n(F), SL_n(F), O(q), SO(q), Sp(2n)$ and the groups of type $G_2$ over $F$ is real if and only if $t=\tau_1\tau_2$ where $\tau_1^2=\pm 1=\tau_2^2$ (ref. \cite{st1,st2}). In this paper we extend this result to the semisimple elements in $Spin$ groups when $\dim(V)\equiv 0,1,2 \imod 4$.