Abstract. Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of G. A separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that some of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very good for G, then any subgroup of G is separable; we deduce that under these hypotheses on G, a subgroup H of G is G-completely reducible provided Lie G is semisimple as an H-module.Recently, Guralnick has proved that if H is a reductive subgroup of G and C is a conjugacy class of G, then C ∩H is a finite union of H-conjugacy classes. For generic p -when certain extra hypotheses hold, including separability -this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig's deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick's result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1.

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

Abstract. Let H be a reductive subgroup of a reductive group G over an algebraically closed field k. We consider the action of H on G n , the n-fold Cartesian product of G with itself, by simultaneous conjugation. We give a purely algebraic characterization of the closed H-orbits in G n , generalizing work of Richardson which treats the case H = G. This characterization turns out to be a natural generalization of Serre's notion of Gcomplete reducibility. This concept appears to be new, even in characteristic zero. We discuss how to extend some key results on G-complete reducibility in this framework. We also consider some rationality questions.