Reducible subgroups of exceptional algebraic groups

TL;DR

This paper completes the classification of connected G-completely reducible subgroups in exceptional algebraic groups, focusing on their Levi subgroup structures and providing detailed classifications for groups like F4.

Contribution

It determines the L0-irreducible connected reductive subgroups within Levi factors of exceptional groups, advancing the understanding of G-cr subgroup classifications.

Findings

Classified all G-cr semisimple subgroups in type F4.

Identified properties of reducible G-cr subgroups.

Extended previous classifications to exceptional groups.

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field. 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 factor of $P$. In this paper we complete the classification of connected $G$-cr subgroups when $G$ has exceptional type, by determining the $L_{0}$-irreducible connected reductive subgroups for each simple classical factor $L_{0}$ of a Levi subgroup of $G$. As an illustration, we determine all reducible, $G$-cr semisimple subgroups when $G$ has type $F_4$ and various properties thereof. This work complements results of Lawther, Liebeck, Seitz and Testerman, and is vital in classifying non-$G$-cr reductive subgroups, a project being undertaken by the authors elsewhere.