I'm writing up some notes, and I realize I don't have a counterexample for something I suspect is false.


**Dubious claim**: If $(\pi, V)$ and $(\rho, W)$ are irreducible representations of two groups $G$ and $H$, respectively, then the "external" tensor product $\pi \boxtimes \rho$ is an irreducible representation of $G \times H$.


Of course this is true and well-known in the usual cases, e.g., when $G$ and $H$ are finite groups. The proof I know uses the converse to Schur's Lemma, or something similar. 


Is there a nice counterexample for complex representations of some infinite groups? Published somewhere?



External tensor product of irreducible representations is not irreducible?
You can generate examples from standard counterexamples to (generalisations of) Schur's lemma.


Let E/F be a field extension. Let $G=H=E^\times$, acting on the F-vector space E. Then the external tensor product is not irreducible, for example the kernel of the multiplication map $E\otimes\_F E\to E$ is a submodule.


