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

TL;DR

This paper classifies the isomorphism classes of involutions in the symplectic group $ ext{SP}(2n, k)$ over fields with characteristic not 2, extending previous classifications for other groups and fields.

Contribution

It provides a detailed classification of involutions in $ ext{SP}(2n, k)$, building on prior work for $ ext{SL}(n,k)$ and general algebraic groups.

Findings

Complete classification of involutions in $ ext{SP}(2n, k)$

Extension of involution classification to arbitrary fields not of characteristic 2

Connection to isomorphism classes of involutions in symplectic groups

Abstract

A first characterization of the isomorphism classes of $k$-involutions for any reductive algebraic groups defined over a perfect field was given by Helminck in 2000 using $3$ invariants. In 2004, Helminck, Wu, and Dometrius gave a full classification of all 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 build on these results to develop a detailed characterization of the involutions of $\text{SP}(2n, k)$. We use these results to classify the isomorphy classes of involutions of $\text{SP}(2n, k)$ where $k$ is any field not of characteristic 2.