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

Michael Bate, Sören Böhm, Benjamin Martin and Gerhard Röhrle

Vol. 336 (2025), No. 1-2, 29–61

DOI: 10.2140/pjm.2025.336.29

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.

Dedicated to the fond memory of Gary Seitz

Keywords

$G$-complete reducibility, $G$-irreducibility, distinguished unipotent elements, distinguished nilpotent elements, finite groups of Lie type, good $A_1$ subgroups

Mathematical Subject Classification

Primary: 14L24, 20G15

Milestones

Received: 13 June 2024

Revised: 20 October 2024

Accepted: 21 October 2024

Published: 26 May 2025