Spherical subgroups in simple algebraic groups

TL;DR

This paper completes the classification of spherical subgroups in simple algebraic groups over fields of positive characteristic, revealing one new case in characteristic 2 and using deformation techniques to relate characteristics.

Contribution

It provides the first complete classification of spherical subgroups in positive characteristic, including a new instance in characteristic 2, and introduces a deformation method to transfer results from characteristic zero.

Findings

Only one new instance in characteristic 2 identified.

Deformation techniques connect sphericality across characteristics.

Classification aligns with Kr"amer's list in characteristic zero.

Abstract

Let $G$ be a simple algebraic group. A closed subgroup $H$ of $G$ is called spherical provided it has a dense orbit on the flag variety $G/B$ of $G$. Reductive spherical subgroups of simple Lie groups were classified by Kr\"amer in 1979. In 1997, Brundan showed that each example from Kr\"amer's list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, there is no classification of all such instances in positive characteristic to date. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic $2$ which has no counterpart in Kr\"amer's classification. As one of our key tools, we prove a general deformation result for subgroup schemes allowing us to deduce the sphericality of subgroups in positive characteristic from this property for subgroups in characteristic $0$.