Closed orbits and uniform 𝑆-instability in geometric invariant theory

In this paper we consider various problems involving the action of a reductive group G G on an affine variety V V . We prove some general rationality results about the G G -orbits in V V . In addition, we extend fundamental results of Kempf and Hesselink regarding optimal destabilizing parabolic subgroups of G G for such general G G -actions. We apply our general rationality results to answer a question of Serre concerning the behaviour of his notion of G G -complete reducibility under separable field extensions. Applications of our new optimality results also include a construction which allows us to associate an optimal destabilizing parabolic subgroup of G G to any subgroup of G G . Finally, we use these new optimality techniques to provide an answer to Tits’ Centre Conjecture in a special case.