Adjoint Jordan blocks for simple algebraic groups of type $C_{\ell}$ in characteristic two

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $K$ with Lie algebra $\mathfrak{g}$. For unipotent elements $u \in G$ and nilpotent elements $e \in \mathfrak{g}$, the Jordan block sizes of $\operatorname{Ad}(u)$ and $\operatorname{ad}(e)$ are known in most cases. In the cases that remain, the group $G$ is of classical type in bad characteristic, so $\operatorname{char} K = 2$ and $G$ is of type $B_{\ell}$, $C_{\ell}$, or $D_{\ell}$. In this paper, we consider the case where $G$ is of type $C_{\ell}$ and $\operatorname{char} K = 2$. As our main result, we determine the Jordan block sizes of $\operatorname{Ad}(u)$ and $\operatorname{ad}(e)$ for all unipotent $u \in G$ and nilpotent $e \in \mathfrak{g}$. In the case where $G$ is of adjoint type, we will also describe the Jordan block sizes on $[\mathfrak{g}, \mathfrak{g}]$.