Unipotent elements forcing irreducibility in linear algebraic groups

TL;DR

This paper generalizes a result about unipotent elements in simple algebraic groups, showing that certain unipotent elements force connected reductive subgroups containing them to be irreducible, with specific exceptions.

Contribution

It extends previous work by proving that unipotent elements of order p typically force subgroups to be irreducible, except for two known cases, and explores additional examples for elements of order greater than p.

Findings

Unipotent elements of order p generally prevent subgroups from being contained in proper parabolic subgroups.

Two specific exceptions occur in the case (G, p) = (C_2, 2).

Additional examples are provided for elements of order greater than p involving indecomposable tilting modules.

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p > 0$. We consider connected reductive subgroups $X$ of $G$ that contain a given distinguished unipotent element $u$ of $G$. A result of Testerman and Zalesski (Proc. Amer. Math. Soc., 2013) shows that if $u$ is a regular unipotent element, then $X$ cannot be contained in a proper parabolic subgroup of $G$. We generalize their result and show that if $u$ has order $p$, then except for two known examples which occur in the case $(G, p) = (C_2, 2)$, the subgroup $X$ cannot be contained in a proper parabolic subgroup of $G$. In the case where $u$ has order $> p$, we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.