$G$-complete reducibility and saturation

Michael Bate, Sören Böhm, Alastair Litterick, Benjamin Martin and Gerhard Röhrle

Vol. 337 (2025), No. 1, 1–24

DOI: 10.2140/pjm.2025.337.1

Abstract

Let $H\subseteq G$ be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic $p>0$. In our first main theorem we show that if a closed subgroup $K$ of $H$ is $H$-completely reducible, then it is also $G$-completely reducible in the sense of Serre, under some restrictions on $p$, generalising the known case for $G=\mathrm{GL}<!--FUNCTION\; APPLICATION-->(V)$. Our proof uses R. W. Richardson’s notion of reductive pairs to reduce to the $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(V)$ case. We study Serre’s notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if $K$ is $H$-completely reducible, then the saturation of $K$ in $G$ is completely reducible in the saturation of $H$ in $G$ (which is again a connected reductive subgroup of $G$), under suitable restrictions on $p$, again generalising the known instance for $G=\mathrm{GL}<!--FUNCTION\; APPLICATION-->(V)$. We also study saturation of finite subgroups of Lie type in $G$. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in the case $G=\mathrm{GL}<!--FUNCTION\; APPLICATION-->(V)$.

Keywords

$G$-complete reducibility, saturation, finite groups of Lie type

Mathematical Subject Classification

Primary: 14L24, 20G15

Milestones

Received: 20 October 2024

Revised: 27 March 2025

Accepted: 2 April 2025

Published: 2 June 2025