Exercise 6, § 2

One assumes that the group G is unimodular. We denote by $\widehat{G}_d\subset \widehat{G}$ the set of classes of irreducible unitary square integrable representations of G. For $\pi \in \widehat{G}_d$, we denote by $p_{\pi}$ the orthogonal projector of $L^2(G)$ whose image is the $\pi$-isotypical component of the right regular representation of G.

(a) Let $f\in \mathscr{L}^2(G)$ and let $\pi \in \widehat{G}_d$. We denote by E the space of $\pi$. The mapping $g\mapsto f(g)\pi (g)$ belongs to $L^1(G;\mathscr{L}(E))$; the endomorphism

is a Hilbert–Schmidt endomorphism, denoted $v=\pi (f)$.

(b) Let $f\in \mathscr{L}^2(G)$. We have $\|\pi (\overline{f})\|_2=\|p_{\pi}(f)\|$.

(c) Let $L^2_d(G)$ be the closed subspace of $L^2(G)$ generated by the $\pi$-isotypical components of $L^2(G)$ for $\pi \in \widehat{G}_d$. It is the Hilbertian sum of the images of the orthogonal projectors $p_{\pi}$ for $\pi \in \widehat{G}_d$. For every $f\in L^2_d$(G), one has

where $c(\pi )$ is the formal degree of $\pi$ relative to $\{e\}$.