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 $[\mathfrak{ g}, \mathfrak{ g}]$.
Categories: math.GR math.RT