Isomorphy Classes of $k$-Involutions of $\text{SO}(n,k,\beta )$, $n>2$

Abstract

A first characterization of the isomorphism classes of $k$-involutions for any reductive algebraic group defined over a perfect field was given in \cite{Helm2000} using $3$ invariants. In \cite{HWD04,Helm-Wu2002} a full classification of all $k$-involutions on $\text{SL}(n,k)$ for $k$ algebraically closed, the real numbers, the $p$-adic numbers or a finite field was provided. In this paper, we find analogous results to develop a detailed characterization of the $k$-involutions of $\text{SO}(n,k,\beta )$, where $\beta$ is any non-degenerate symmetric bilinear form and $k$ is any field not of characteristic $2$. We use these results to classify the isomorphy classes of $k$-involutions of $\text{SO}(n,k,\beta )$ for some bilinear forms and some fields $k$.

43 Pages

References in corpus (1)

• Formule de Plancherel pour les fonctions de Whittaker sur un groupe réductif $p$-adique