On ${\mathfrak{sl}}_2$-triples for classical algebraic groups in positive characteristic

Abstract

Let $k$ be an algebraically closed field of characteristic $p>2$, let $n\in {\mathbb{Z}}_{>0}$, and take $G$ to be one of the classical algebraic groups ${\mathrm{GL}}_n(k)$, ${\mathrm{SL}}_n(k)$, ${\mathrm{Sp}}_n(k)$, ${\mathrm{O}}_n(k)$ or ${\mathrm{SO}}_n(k)$, with $\mathfrak{g}=\mathrm{Lie}G$. We determine the maximal $G$-stable closed subvariety $\mathcal{V}$ of the nilpotent cone $\mathcal{N}$ of $\mathfrak{g}$ such that the $G$-orbits in $\mathcal{V}$ are in bijection with the $G$-orbits of ${\mathfrak{sl}}_2$-triples $(e,h,f)$ with $e,f\in \mathcal{V}$. This result determines to what extent the theorems of Jacobson--Morozov and Kostant on ${\mathfrak{sl}}_2$-triples hold for classical algebraic groups over an algebraically closed field of "small" odd characteristic.

20 pages, minor changes