On complete reducibility of tensor products of simple modules over simple algebraic groups

Let G be a simply connected simple algebraic group over an al- gebraically closed field k of characteristic p > 0. The category of rationalG-modules is not semisimple. We consider the question of when the tensorproduct of two simple G-modules L(λ) and L(μ) is completely reducible. Using some technical results about weakly maximal vectors (i.e. maximal vectors for the action of the Frobenius kernel G1 of G) in tensor products, we obtain a reduction to the case where the highest weights λ and μ are p-restricted. In this case, we also prove that L(λ)⊗L(μ) is completely reducible as a G-module if and only if L(λ) ⊗ L(μ) is completely reducible as a G1 -module.