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

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 $G={\mathrm{GL}}_n(k)$ is $G$-completely reducible if and only if the inclusion map $i:H\to {\mathrm{GL}}_n(k)$ 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.

Notes based on lectures given at the International Workshop on "Algorithmic problems in group theory, and related areas", held at the Oasis Summer Camp near Novosibirsk from July 26 to August 4, 2016. 28 pages