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

TL;DR

This paper investigates the structure of nilpotent orbits and $rak{sl}_2$-triples in classical algebraic groups over algebraically closed fields of characteristic greater than 2, extending classical theorems to positive characteristic.

Contribution

It identifies the maximal $G$-stable subvariety of the nilpotent cone where $G$-orbits correspond to $rak{sl}_2$-triples, clarifying the applicability of Jacobson--Morozov and Kostant theorems in small odd characteristic.

Findings

Characterizes the maximal $G$-stable subvariety of the nilpotent cone.

Establishes the correspondence between $G$-orbits and $rak{sl}_2$-triples in this subvariety.

Clarifies the extent of classical theorems 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 $\mathfrak g = \operatorname{Lie} G$. 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.