Complete Reducibility in Good Characteristic

TL;DR

This paper classifies non-$G$-completely reducible subgroups of exceptional algebraic groups over fields of good characteristic, detailing their conjugacy classes, actions, and centralisers, thus advancing understanding of subgroup structure.

Contribution

It provides a complete classification of non-$G$-cr simple subgroups and their centralisers in exceptional groups over good characteristic fields.

Findings

Classified all non-$G$-cr simple connected subgroups of $G$.

Determined the action of these subgroups on the adjoint module.

Identified maximal connected reductive subgroups not maximal among all connected subgroups.

Abstract

Let $G$ be a simple algebraic group of exceptional type, over an algebraically closed field of characteristic $p \ge 0$. A closed subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) if whenever $H$ is contained in a parabolic subgroup $P$ of $G$, it is contained in a Levi subgroup of $P$. In this paper we determine the $G$-conjugacy classes of non-$G$-cr simple connected subgroups of $G$ when $p$ is good for $G$. For each such subgroup $X$, we determine the action of $X$ on the adjoint module $L(G)$ and the connected centraliser of $X$ in $G$. As a consequence we classify all non-$G$-cr connected reductive subgroups of $G$, and determine their connected centralisers. We also classify the subgroups of $G$ which are maximal among connected reductive subgroups, but not maximal among all connected subgroups.