Exercise 14, § 2

Let Z be a subgroup of the center C of G such that $C/Z$ is compact and $\nu$ a Haar measure on $G/Z$. Let $\pi$ be an irreducible unitary representation of G in a complex Hilbert space E. Suppose that $\pi$ is square-integrable modulo the center. Let $u\in \mathscr{L}(E)$ be a nuclear endomorphism (Def. 4 of IV, p. 169).

a) For every $x$ and $y$ in E, the function

belongs to $\mathscr{F}(G/Z)$.

b) For every $x$ and $y$ in E, one has

$\int_{G/Z}\langle x|\pi (g)u\pi (g^{-1})y\rangle d\nu (g) =\frac{1}{c_Z(\pi)}\langle x|y\rangle$ Tr($u$).