Real Elements in Spin Groups

Abstract

Let $F$ be a field of characteristic $\ne 2$. 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 $st{s}^{-1}=t^{-1}$. A semisimple element $t$ in $G{L}_n(F),S{L}_n(F),O(q),SO(q),Sp(2n)$ 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 $\dim(V)\equiv 0,1,2 \imod 4$.

11 pages