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

Let G G be a simply connected simple algebraic group over an algebraically closed field k k of characteristic p > 0 p>0 . The category of rational G G -modules is not semisimple. We consider the question of when the tensor product of two simple G G -modules L ( λ ) L(\lambda ) and  L ( μ ) L(\mu ) is completely reducible. Using some technical results about weakly maximal vectors (i.e. maximal vectors for the action of the Frobenius kernel G 1 G_1 of G G ) in tensor products, we obtain a reduction to the case where the highest weights λ \lambda and  μ \mu are p p -restricted. In this case, we also prove that L ( λ ) ⊗ L ( μ ) L(\lambda )\otimes L(\mu ) is completely reducible as a G G -module if and only if L ( λ ) ⊗ L ( μ ) L(\lambda )\otimes L(\mu ) is completely reducible as a G 1 G_1 -module.