Proposition 5

Let $\pi \in \widehat{G}$. The space $\boldsymbol{\varrho }_G(\pi )$ is a subrepresentation of $\boldsymbol{\varrho }_G$ which is isomorphic to $\overline{\pi}\boxtimes \pi$. In particular, it is an irreducible representation of $G\times G$.

By prop. 7 of V, p. 422 (applied with $Z =\{e\}$), the space $\boldsymbol{\varrho }_G(\pi )$ is a subrepresentation of $\boldsymbol{\varrho }_G$ and the linear mapping from $\overline{E}_{\pi}\otimes E_{\pi}$ into $L^2(G)$ which associates to $x\otimes y$ the matrix coefficient $g\mapsto \langle x|\pi (g)y\rangle$ is a $(G\times$ G)-isomorphism of $\overline{\pi}\boxtimes \pi$ onto $\boldsymbol{\varrho }_G(\pi )$. The subrepresentation $\boldsymbol{\varrho }_G(\pi )$ is therefore irreducible (prop. 3 of V, p. 461).