I am looking for a reference for the following assertion that I believe to be true. All representations are assumed to be finite-dimensional.

Let $G_1$ and $G_2$ be affine algebraic group schemes over a field $k$ such that every simple $G_2$-representation $V_2$ has endomorphism ring $\mathrm{End}_{G_2}(V_2) = k$. Then for any two simple representations $V_1$ and $V_2$ of $G_1$ and $G_2$, respectivey, $$V_1\otimes V_2$$ is a simple $G_1\times G_2$-representation. Moreover, any simple $G_1\times G_2$-representation is of this form.

The corresponding statements regarding complex representations of compact Lie groups are given as Lemmas 3.66 and 3.67 in Adams’ Lectures on Lie Groups, and may probably also be found in any other relevant textbook.

For the given statement concerning affine algebraic groups, I believe to have a proof based on the following lemma in Jantzen's *Representations of Algebraic Groups* (see 6.15 (2)):
Let $V$ and $W$ be representations of an affine algebraic group scheme $G$ over a field $k$. Suppose $V$ restricts to a simple representation of a normal subgroup scheme $N\triangleleft G$ such that $\mathrm{End}_{N}(V) = k$. Then $(\mathrm{soc}_N W)_V$, the $V$-isotypical part of the $N$-socle of $W$, is a $G$-sub-representation of $W$, and we have an isomorphism of $G$-modules $\mathrm{Hom}_N(V,W)\otimes V \cong (\mathrm{soc}_N W)_V$.

However, I have not been able to locate the above statement regarding products as such in the literature, which I still find a little puzzling.

isa $G_1\times G_2$-representation, but there is no assertion of simplicity etc. $\endgroup$ad hocproof using the group algebra and complete reducibility. (P.S. The term "Communicative Algebra" actually appears prominently by mistake in a French chapter of the Bourbaki series on commutative algebra.) $\endgroup$am. $\endgroup$1more comment