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

TL;DR

This paper classifies the isomorphism classes of $k$-involutions for special orthogonal groups over various fields, extending previous classifications to include detailed cases for different bilinear forms and fields.

Contribution

It provides a detailed characterization and classification of $k$-involutions of $ ext{SO}(n,k,eta)$ for various fields and bilinear forms, extending known results beyond previously studied groups.

Findings

Classification of $k$-involutions for $ ext{SO}(n,k,eta)$ over various fields.

Development of invariants to distinguish isomorphism classes.

Extension of previous classifications to new cases of bilinear forms and fields.

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 $k$.