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

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 $\operatorname{Sp}(V_1 \otimes V_2)$.