Lemma 9

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 $\pi$. For all $x$ and $y$ in E, one has $f_{x,y}\in \mathscr{L}_{\chi}^2(G)$.

The function $f_{x,y}$ is continuous, hence $\mu$-measurable. One has $f_{x,y}\in \mathscr{F}_{\chi}(G)$ by formula (9), p. 420. Moreover, by INT, VII, p. 64, § 2, No.$^o8$, cor. 1, c), one has $N_2(f_{x,y})<+\infty$ since $C/Z$ is compact and $|f_{x,y}| \in \mathscr{L}^2(G/C)$. The assertion therefore follows from proposition 5 of V, p. 413.