Exercise 15, § 2

Let Z be a subgroup of the center C of G such that $C/Z$ is compact, and let $\nu$ be a Haar measure on $G/Z$. Let $\pi$ be an irreducible unitary representation of G in a complex Hilbert space E. Let us denote by $\chi$ the restriction to Z of the central character of $\pi$.

Let $f_{x,y}$ denote the matrix coefficient $g\mapsto \langle x|\pi (g)y\rangle$ for $(x, y)\in E^2$. Let $p\geqslant 1$ be a real number and q the conjugate exponent of $p$.

a) Let $y\in E$ be such that the matrix coefficients $f_{x,y}$ belong to $\mathscr{L}_{\chi}^p(G)$. The mapping $x\mapsto f_{x,y}$ is a continuous G-morphism of $\pi$ into $\boldsymbol{\delta }^{(p)}_{G,\chi}$.

b) Let $\varrho$ be an irreducible unitary representation of G in a complex Hilbert space F whose central character has restriction to Z equal to $\chi$. Suppose that $p >1$ and that the matrix coefficients of $\pi$ belong to $\mathscr{L}_{\chi}^p(G)$ and those of $\varrho$ belong to $\mathscr{L}_{\chi}^q(G)$. Let $x_0\in E$ and $y_0\in F$. For every $x\in E$, there exists a continuous linear form $\ell_x$ on F such that

for every $y\in F$.

c) There exists a continuous linear mapping $u$ from E into F such that $\ell_x(y) =$ $\langle u(x)|y\rangle$ for every $(x, y)\in E\times F$.

d) We have

for $(x, x', y, y')\in E^2\times F^2$.

e) Suppose that $p= 1$. The matrix coefficients of $\pi$ are orthogonal to those of $\varrho$.