Exercise 11, § 3

Let G be a locally compact group.

Let $\varrho$ and $\pi$ be unitary representations of G. One says that $\varrho$ is weakly contained in $\pi$, and one writes $\varrho \leqslant \pi$, if the set of diagonal matrix coefficients of $\varrho$ is contained in the closure of the set of finite sums of diagonal matrix coefficients of $\pi$ in the space $\mathscr{C}(G)$ endowed with the topology of compact convergence.

a) The relation “ $\varrho$ and $\pi$ are unitary representations of G and $\varrho \leqslant \pi$ ” is a preorder relation. If $\varrho '$ (resp. $\pi '$) is isomorphic to $\varrho$ (resp. to $\pi$ ) then $\varrho \leqslant \pi$ if and only if $\varrho '\leqslant \pi '$.

b) If $\varrho$ is isomorphic to a subrepresentation of $\pi$, then $\varrho \leqslant \pi$. If I and J are non-empty finite sets, and if $\varrho \leqslant \pi$, then $\bigoplus_{i\in I}\varrho \leqslant \bigoplus_{j\in J}\pi$.

c) Let X be a subset of the space E of $\varrho$ which generates $\pi$. One has $\varrho \leqslant \pi$ if and only if every matrix coefficient $g\mapsto \langle x|\varrho (g)x\rangle$ with $x\in X$ belongs to the closure of the set of finite sums of diagonal matrix coefficients of $\pi$ in the space $\mathscr{C}(G)$ endowed with the topology of compact convergence. (Now prove that the set Y of elements of E satisfying this property is a closed vector subspace of E.)

d) Let H be a closed subgroup of G. Let $\pi$ and $\varrho$ be unitary representations of G such that $\varrho \leqslant \pi$; then Res$^G_H(\varrho )\leqslant$ Res$^G_H(\pi )$.

e) Let H be a closed subgroup of G and $\nu$ a Haar measure on H. Let $\pi$ and $\varrho$ be unitary representations of H such that $\varrho \leqslant \pi$; then Ind$^G_H(\varrho )\leqslant$ Ind$^G_H(\pi )$. (Let $\varrho '=$ Ind$^G_H(\varrho )$ and $\pi '=$ Ind$^G_H(\pi )$; by using question c), reduce to approximating matrix coefficients $g\mapsto \langle f|\varrho '(g)f\rangle$ for $f$ of the form

where $\varphi \in \mathscr{K}(G)$ and $y$ belongs to the space of $\varrho .)$