On the Conjugacy Classes in the orthogonal and symplectic groups over algebraically closed fields

TL;DR

This paper provides a new, simplified proof that conjugacy classes in orthogonal and symplectic groups over algebraically closed fields are characterized by elementary divisors, under the assumption of large characteristic.

Contribution

It introduces a novel, straightforward proof of conjugacy classification in isometry groups using the Jacobson-Morozov lemma, extending understanding in algebraic group theory.

Findings

Conjugacy in isometry groups is determined by elementary divisors.

The proof applies to fields with large characteristic, including zero.

The approach simplifies previous methods for classifying conjugacy classes.

Abstract

Let $\F$ be an algebraically closed field. Let $\V$ be a vector space equipped with a non-degenerate symmetric or symplectic bilinear form $B$ over $\F$. Suppose the characteristic of $\F$ is \emph{large}, i.e. either zero or greater than the dimension of $\V$. Let $I(\V, B)$ denote the group of isometries. Using the Jacobson-Morozov lemma we give a new and simple proof of the fact that two elements in $I(\V,B)$ are conjugate if and only if they have the same elementary divisors.