Martin, BLeemans, DUchiyama, Tomohiro2016-07-1520162016http://hdl.handle.net/2292/29443Let 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.Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher.https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmhttp://creativecommons.org/licenses/by-nc-sa/3.0/nz/ThesisCopyright: The Authorhttp://purl.org/eprint/accessRights/OpenAccessQ111963490