Complete Reducibility and Geometric Invariant Theory

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In this proposal we aim to study J.-P. Serre's notion of G-complete reducibility (G-cr) using tools from geometric invariant theory (GIT). In a recent joint paper by Bate, Martin and the author it was shown that Serre's concept of G-cr is equivalent to Richardson's notion of strong reductivity. This equivalence allowed us to use methods from GIT in the study of G-cr subgroups of reductive algebraic groups G such as the Hilbert-Mumford Theorem to derive new criteria for G-cr subgroups. The aim of the proposed research is to extend and deepen this geometric investigation.The general guiding principle of this work is to undertake a comprehensive study of the behaviour of G-cr subgroups under natural group-theoretic operations, such as taking normal subgroups, taking quotients, taking centralisers, taking normalisers, forming semi-direct products and applying group homomorphisms, etc. Although by earlier work some results are known, a systematic study is needed.Another general question we aim to address is the following: Let K,H, and G be reductive groups with K contained in H and H contained G. What conditions on K,H,G and the ground field that ensure that if K is G-cr, then K is H-cr, and vice versa? This involves extending several results form earlier joint work with Bate and Martin.We want to further investigate the connection between reductive paris and complete reducibility; this should be an effective replacement for characteristic restrictions in earlier work on G-cr subgroups. Also we want to develop some criteria for some converse results.Moreover, we intend to study further rationality properties of G-cr subgroups and generalisations of our results to non-connected reductive groups.In the context of his original building-theoretic approach J.-P. Serre observed that the notion of G-complete reducibility makes sense for semi-algebraic actions. We want to extend our earlier results to this setting.