On good $A_1$ subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups

TL;DR

This paper provides a unified proof of a theorem about the irreducibility of certain subgroups containing distinguished unipotent elements in algebraic groups, introduces new properties of good A_1 subgroups, and extends results to overgroups, finite groups of Lie type, and Lie algebras.

Contribution

It offers a simplified proof of Korhonen's theorem using good A_1 subgroups, introduces new properties of these subgroups, and generalizes the theorem to broader contexts including overgroups, finite groups, and Lie algebras.

Findings

New properties of good A_1 subgroups established

Extended Korhonen's theorem to overgroups of unipotent elements

Generalized results to cases with different element orders and Lie algebra analogues

Abstract

Suppose $G$ is a simple algebraic group defined over an algebraically closed field of good characteristic $p$. In 2018 Korhonen showed that if $H$ is a connected reductive subgroup of $G$ which contains a distinguished unipotent element $u$ of $G$ of order $p$, then $H$ is $G$-irreducible in the sense of Serre. We present a short and uniform proof of this result under an extra hypothesis using so-called good $A_1$ subgroups of $G$, introduced by Seitz. In the process we prove some new results about good $A_1$ subgroups of $G$ and their properties. We also formulate a counterpart of Korhonen's theorem for overgroups of $u$ which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of $u$ under a mild condition on $p$ depending on the rank of $G$, and we present an analogue of Korhonen's theorem for Lie algebras.