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 dimV − 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 ≥ dimV 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.