Let $G$ and $H$ be two locally compact Hausdorff topological group. Let $(\pi,V)$ be an irreducible unitary representation of $G$ and let $(\rho,W)$ be an irreducible unitary representation of $H$. Let $(\pi \otimes\rho, V \otimes W)$ denote the unitary representation of $G \times H$ on $V \otimes W$ defined by $$[\pi \otimes \rho](g,h)(v \otimes w):= \pi(g)v\otimes \rho(h)w, (g\in G, h \in H, v \in V, w \in W).$$
Is it true that $(\pi \otimes\rho, V \otimes W)$ is an irreducible representation of $G \times H$ ? If it is so, how to show it ?
Remark:
- Answer to the above question is yes when both $G$ and $H$ are compact or abelian.
2)In the above question we don't necessarily assume that both $G$ and $H$ are compact or abelian.