Algebraic groups and $G$-complete reducibility: a geometric approach

TL;DR

This paper introduces the concept of $G$-complete reducibility in algebraic groups, connecting it to geometric invariant theory and expanding understanding of subgroup structures.

Contribution

It provides an introduction to $G$-complete reducibility and presents a geometric invariant theory approach to studying algebraic subgroup structures.

Findings

$G$-completely reducible subgroups generalize classical complete reducibility.

The approach links subgroup properties to geometric invariant theory.

Applications to algebraic group subgroup classification.

Abstract

The notion of a \emph{$G$-completely reducible} subgroup is important in the study of algebraic groups and their subgroup structure. It generalizes the usual idea of complete reducibility from representation theory: a subgroup $H$ of a general linear group $G= {\rm GL}_n(k)$ is $G$-completely reducible if and only if the inclusion map $i\colon H\rightarrow {\rm GL}_n(k)$ is a completely reducible representation of $H$. In these notes I give an introduction to the theory of complete reducibility and its applications, and explain an approach to the subject using geometric invariant theory.