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?
Best Answer
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.