Hesselink normal forms of unipotent elements in some representations of classical groups in characteristic two

TL;DR

This paper classifies unipotent elements in certain representations of classical groups over fields of characteristic two, extending understanding of their conjugacy classes and Hesselink normal forms.

Contribution

It explicitly determines conjugacy classes of unipotent elements in specific representations of classical groups in characteristic two, including tensor product cases.

Findings

Classifies unipotent elements in specific symplectic representations.

Describes conjugacy classes in tensor product symplectic groups.

Provides Hesselink normal forms for these unipotent elements.

Abstract

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic two. Any non-trivial self-dual irreducible $K[G]$-module $W$ admits a non-degenerate $G$-invariant alternating bilinear form, thus giving a representation $f: G \rightarrow \operatorname{Sp}(W)$. In the case where $G = \operatorname{SL}_n(K)$ and $W$ has highest weight $\varpi_1 + \varpi_{n-1}$, and in the case where $G = \operatorname{Sp}_{2n}(K)$ and $W$ has highest weight $\varpi_2$, we determine for every unipotent element $u \in G$ the conjugacy class of $f(u)$ in $\operatorname{Sp}(W)$. As a part of this result, we describe the conjugacy classes of unipotent elements of $\operatorname{Sp}(V_1) \otimes \operatorname{Sp}(V_2)$ in $\operatorname{Sp}(V_1 \otimes V_2)$.