Rationality problems for complete reducibility of subgroups of reductive algebraic groups

Abstract

Let k be a field. Let G be a connected reductive algebraic group defined over k. Following Serre, a closed subgroup H is of G is called G-completely reducible over k (G-cr over k for short) if whenever H is contained in a k-defined parabolic subgroup P of G, H is contained in a k-defined Levi subgroup of P. This thesis is a compilation of four independent papers concerning rationality problems for complete reducibility of subgroups of G and various related problems ... We obtain various general results concerning complete reducibility over an arbitrary k via geometric invariant theory (GIT for short) and the theory of spherical building (in particular the recently proved center conjecture of Tits). GIT and the center conjecture give a very short proof for many results. We also consider non-connected G, and obtain analogous results. Various open problems concerning complete reducibility and related problems are discussed.