G-complete reducibility and the exceptional algebraic groups

TL;DR

This paper classifies various reductive subgroups of exceptional algebraic groups G of types G2 and F4 over algebraically closed fields, focusing on their conjugacy classes, reducibility properties, and maximality, including special cases related to characteristic p.

Contribution

It provides a complete classification of conjugacy classes of reductive subgroups in G2 and F4, including non-G-completely reducible and maximal subgroups, extending previous work and analyzing subgroup actions.

Findings

Classified all conjugacy classes of reductive subgroups in G2 and F4.

Identified infinite collections of maximal reductive subgroups not contained in any maximal reductive subgroup.

Determined subgroup structures related to characteristic p and root element generation.

Abstract

Let $G=G(K)$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p>0$. A subgroup $X$ of $G$ is said to be $G$-completely reducible if, whenever it is contained in a parabolic subgroup of $G$, it is contained in a Levi subgroup of that parabolic. A subgroup $X$ of $G$ is said to be $G$-irreducible if $X$ is in no parabolic subgroup of $G$; and $G$-reducible if it is in some parabolic of $G$. In this thesis, we consider the case that $G$ is of exceptional type. When $G$ is of type $G_2$ we find all conjugacy classes of closed, connected, reductive subgroups of $G$. When $G$ is of type $F_4$ we find all conjugacy classes of closed, connected, reductive $G$-reducible subgroups $X$ of $G$. Thus we also find all non-$G$-completely reducible closed, connected, reductive subgroups of $G$. When $X$ is closed, connected and simple of rank at least two, we find all conjugacy classes of $G$-irreducible subgroups $X$ of $G$. Together with the work of Amende in [Ame05] classifying irreducible subgroups of type $A_1$ this gives a complete classification of the simple subgroups of $G$. Amongst the classification of subgroups of $G=F_4(K)$ we find infinite collections of subgroups $X$ of $G$ which are maximal amongst all reductive subgroups of $G$ but not maximal subgroups of $G$; thus they are not contained in any maximal reductive subgroup of $G$. The connected, semisimple subgroups contained in no maximal reductive subgroup of $G$ are of type $A_1$ when $p=3$ and of semisimple type $A_1^2$ or $A_1$ when $p=2$. Some of those which occur when $p=2$ act indecomposably on the 26-dimensional irreducible representation of $G$. We also use this classification to find all subgroups of $G=F_4$ which are generated by short root elements of $G$, by utilising and extending the results of [LS94].