Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 2

Let k be a separably closed field. Let G be a reductive algebraic k-group. In this paper, we study Serre's notion of complete reducibility of subgroups of G over k. In particular, using the recently proved center conjecture of Tits, we show that the centralizer of a k-subgroup H of G is G-completely reducible over k if it is reductive and H is G-completely reducible over k. We also show that a regular reductive k-subgroup of G is G-completely reducible over k. Various open problems concerning complete reducibility are discussed. We present examples where the number of overgroups of irreducible subgroups and the number of G(k)-conjugacy classes of k-anisotropic unipotent elements are infinite.