Lemma 10

There exists a real number $c >0$ such that the linear mapping $w=c^{1/2}v$ is isometric.

This lemma being assumed valid, let us remark that formula (10), p. 420, may be written

for all $(g_1, h_1)\in G\times G$ and all $(x, y)\in \overline{E}\times E$. The isometric linear mapping $w$ of $\overline{E}\otimes E$ into $L^2_{\chi}(G)$ admits a continuous extension, still denoted by $w$, to $E\widehat{\otimes}_2E$. By continuity and linearity, formula (11) implies that $w$ is a $(G\times$ G)-morphism of $\overline{\pi}\boxtimes \pi$ into $L^2_{\chi}$(G), which concludes the proof of the proposition.

Let us prove the lemma. For every $x\in E$, let $u_x$ denote the linear mapping $y\mapsto v(x\otimes y) =f_{x,y}$ of E into $L^2_{\chi}(G)$. We have $u_x\in$ Hom$_G(\pi ,\boldsymbol{\gamma }_{G,\chi})$ (formula (10)). By cor. 5 of V, p. 388, there exists a real number $\lambda_x\geqslant 0$ such that $\lambda_xu_x$ is isometric.

Let $x$ and $y$ be in E. We have

since $G/Z$ is unimodular (lemma 7 of V, p. 417). Noting that $\check{f}_{x,y}=f_{y,x}$, we obtain

This means that the positive real number $\lambda_x/\|x\|^2$ is independent of the choice of the nonzero element $x$ of E. It is strictly positive since, for every nonzero $x$, the function $f_{x,x}$ is continuous and takes the value $\|x\|^2>0$ at $e$, whence $\|u_x(x)\|=\|f_{x,x}\|>0$. Let us denote this real number by $c^{-1}$.

For every $(x, y)\in E\times E$, it follows that

Using EVT, V, p. 29, cor. 1, we deduce that the linear mapping $w=c^{1/2}v$ of $\overline{E}\widehat{\otimes}_2E$ into $L^2_{\chi}(G)$ is isometric, as required.