Tensor products of primitive modules

Let F be a field and. for i = 1, 2, let G(i) be a group and V-i an irreducible, primitive, finite dimensional FG(i)-module. Set G = G(1) x G(2) and V = V-1 circle times (F) V-2. The main aim of this paper is to determine sufficient conditions for V to be primitive as a G-module. In particular this turns out to be the case if V-1 and V-2 are absolutely irreducible and V-1 is absolutely quasi-primitive, Thus we extend a result of N.S. Heckster, who has shown that V is primitive whenever G is finite and F is the complex field. We also give a characterization of absolutely quasi-primitive modules. Ultimately, our results rely on the classification of finite simple groups.