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 $f:G\to \mathrm{Sp}(W)$. In the case where $G={\mathrm{SL}}_n(K)$ and $W$ has highest weight ${\varpi }_1+{\varpi }_{n-1}$, and in the case where $G={\mathrm{Sp}}_{2n}(K)$ and $W$ has highest weight ${\varpi }_2$, we determine for every unipotent element $u\in G$ the conjugacy class of $f(u)$ in $\mathrm{Sp}(W)$. As a part of this result, we describe the conjugacy classes of unipotent elements of $\mathrm{Sp}(V_1)\otimes \mathrm{Sp}(V_2)$ in $\mathrm{Sp}(V_1\otimes V_2)$.

to appear in J. Algebra