On extensions of the Jacobson-Morozov theorem to even characteristic

Abstract

Let G be a simple algebraic group over an algebraically closed field k of characteristic 2. We consider analogues of the Jacobson-Morozov theorem in this setting. More precisely, we classify those nilpotent elements with a simple 3-dimensional Lie overalgebra in $\mathfrak{g}:=\text{Lie}(G)$ and also those with overalgebras isomorphic to the algebras $\text{Lie}({\text{SL}}_2)$ and $\text{Lie}({\text{PGL}}_2)$. This leads us to calculate the dimension of Lie automiser ${\mathfrak{n}}_{\mathfrak{g}}(k\cdot e)/{\mathfrak{c}}_{\mathfrak{g}}(e)$ for all nilpotent orbits; in even characteristic this quantity is very sensitive to isogeny.

22 pages