Overgroups of regular unipotent elements in simple algebraic groups

TL;DR

This paper studies the structure of certain subgroups in simple algebraic groups, showing that positive-dimensional reductive subgroups containing regular unipotent elements are generally not contained in proper parabolic subgroups, extending previous results.

Contribution

It extends prior work by characterizing overgroups of regular unipotent elements in simple algebraic groups, especially regarding their containment in parabolic subgroups.

Findings

Positive-dimensional reductive subgroups containing regular unipotent elements are mostly not inside proper parabolic subgroups.

Connected components of such subgroups are typically tori.

The result generalizes earlier findings by Testerman and Zalesski.

Abstract

We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.