Corollary

Let $\pi$ be an irreducible unitary representation of G and let $\chi$ be the restriction to Z of its central character. The representation $\pi$ is square-integrable modulo the center if and only if it is isomorphic to a subrepresentation of the representation $\boldsymbol{\gamma }_{G,\chi}$ of G in $L^2_{\overline{\chi}}(G)$ (resp. of the representation $\boldsymbol{\delta }_{G,\chi}$ of G in $L^2_{\chi}(G)$).

Let us prove the assertion concerning $\boldsymbol{\gamma }_{G,\chi}$, the second being proved in a similar way.

Suppose first that there exists a subrepresentation E of the representation $\boldsymbol{\gamma }_{G,\chi}$ isomorphic to $\pi$. Since the space E is nonzero, there exists a function $f_1\in \mathscr{K}_{\overline{\chi}}(G)$ whose class $\widetilde{f}_1$ is not orthogonal to E. We have $f_1\in \mathscr{L}_{\overline{\chi}}^1(G)$. Let $\widetilde{f}_{1,E}$ denote the orthogonal projection of $\widetilde{f}_1$ on E; it is a nonzero element of E. Let moreover $\widetilde{f}_2\in E$ be nonzero. The mapping $h:g\mapsto \langle \widetilde{f}_{1,E}|\boldsymbol{\gamma }_{G,\chi}(g)\widetilde{f}_2\rangle$ is a matrix coefficient of $\pi$. For every $g\in G$, we have $h(g) =\langle \widetilde{f}_1|\boldsymbol{\gamma }_{G\overline{,\chi}}(g)\widetilde{f}_2\rangle$ since $\widetilde{f}_1-\widetilde{f}_{1,E}$ is orthogonal to E. By Lemma 8 of V, p. 418, the function $h$ belongs to $\mathscr{L}_{\chi}^2$(G), hence Proposition 6 implies that $\pi$ is square-integrable modulo the center.

Conversely, suppose that $\pi$ is square-integrable modulo the center. Let $x_0\in E$ be a nonzero vector. By Prop. 7 and formula (10), the mapping $y\mapsto \check{f}_{x_0,y}$ is an injective G-morphism of $\pi$ into $\boldsymbol{\gamma }_{G,\chi}$.