Real Elements in Spin 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 $t=τ_1τ_2$ where $τ_1^2=\pm 1=τ_2^2$ (ref. \cite{st1,st2}). In this paper we extend this result to the semisimple elements in $Spin$ groups when $\dim(V)\equiv 0,1,2 \imod 4$.