$G$-complete reducibility and semisimple modules

TL;DR

This paper provides concise proofs of key results relating $G$-complete reducibility to semisimple modules in algebraic groups, extending classical theorems and establishing new criteria for subgroup properties.

Contribution

It offers simplified proofs of fundamental results in $G$-complete reducibility and introduces new conditions linking semisimplicity of modules to subgroup properties.

Findings

Semisimplicity of $V$ characterizes $G$-complete reducibility of $H$ under certain conditions.

Reductivity of $H^ ext{circ}$ is equivalent to $H$ being $G$-completely reducible.

New criteria for $H$ being $G$-completely reducible based on semisimplicity of $V ensor V^*$.

Abstract

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field %$k$ of characteristic $p > 0$. Our first aim in this note is to give concise and uniform proofs for two fundamental and deep results in the context of Serre's notion of $G$-complete reducibility, at the cost of less favourable bounds. Here are some special cases of these results: Suppose that the index $(H:H^\circ)$ is prime to $p$ and that $p > 2\dim V-2$ for some faithful $G$-module $V$. Then the following hold: (i) $V$ is a semisimple $H$-module if and only if $H$ is $G$-completely reducible; (ii) $H^\circ$ is reductive if and only if $H$ is $G$-completely reducible. We also discuss two new related results: (i) if $p \ge \dim V$ for some $G$-module $V$ and $H$ is a $G$-completely reducible subgroup of $G$, then $V$ is a semisimple $H$-module -- this generalizes Jantzen's semisimplicity theorem (which is the case $H = G$); (ii) if $H$ acts semisimply on $V \otimes V^*$ for some faithful $G$-module $V$, then $H$ is $G$-completely reducible.