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?