Proposition 7

Let $\pi$ be an irreducible unitary representation of G in a Hilbert space E which is square-integrable modulo the center. Let $\chi$ be the restriction to Z of the central character of $\pi$.

There exist a real number $c >0$ and a unique isometric $(G\times G)$-morphism $w$ of the unitary representation $\overline{\pi}\boxtimes \pi$ into $L^2_{\chi}(G)$ such that, for all $(x, y)\in \overline{E}\times E$, the element $w(x\otimes y)$ is the class in $L^2_{\chi}(G)$ of the function $c^{1/2}f_{x,y}$.

For all $(x, y)\in E\times E$, one has $f_{x,y}\in \mathscr{L}_{\chi}^2(G)$ (lemma 9). Let $v$ denote the unique linear mapping of $\overline{E}\otimes E$ into $L^2_{\chi}(G)$ such that $v(x\otimes y)$ is the class of $f_{x,y}$ for all $(x, y)\in E\times E$.

We shall prove below the following lemma.