G ‐complete reducibility and semisimple modules

Let G be a connected reductive algebraic group defined over an algebraically closed field of characteristic p > 0. Our first aim in this note is to give concise and uniform proofs for two fundamental and deep results in the context of Serre's notion of G ‐complete reducibility, at the cost of less favourable bounds. Here are some special cases of these results: Suppose that the index ( H : H ° ) is prime to p and that p > 2 dim V −2 for some faithful G ‐module V . Then the following hold: (i) V is a semisimple H ‐module if and only if H is G ‐completely reducible; (ii) H ° is reductive if and only if H is G ‐completely reducible. We also discuss two new related results. (i) If p ⩾ dim V for some G ‐module V and H is a G ‐completely reducible subgroup of G , then V is a semisimple H ‐module; this generalizes Jantzen's semisimplicity theorem (which is the case H = G ). (ii) If H acts semisimply on V ⊗ V * for some faithful G ‐module V , then H is G ‐completely reducible.