Overgroups of regular unipotent elements in reductive groups

TL;DR

This paper investigates reductive subgroups containing regular unipotent elements in algebraic groups, establishing their irreducibility under certain conditions and extending results to Lie algebras and finite groups of Lie type.

Contribution

It generalizes previous results by proving that such subgroups are G-irreducible, with concise, conceptual, and uniform proofs across different algebraic structures.

Findings

Reductive subgroups with regular unipotent elements are G-irreducible under certain hypotheses.

Results extend to Lie algebras and finite groups of Lie type.

Proofs are short, conceptual, and uniform.

Abstract

We study reductive subgroups $H$ of a reductive linear algebraic group $G$ -- possibly non-connected -- such that $H$ contains a regular unipotent element of $G$. We show that under suitable hypotheses, such subgroups are $G$-irreducible in the sense of Serre. This generalizes results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.