Lemma 4

Let $\pi$ be an irreducible representation of G. The mapping $f\mapsto \pi (f)$ defines by passing to the subspaces an isometric isomorphism of $\boldsymbol{\varrho }_G(\overline{\pi})$ into End(E$_{\pi}$).

Put $\varepsilon_{\pi}(g, h)u=\pi (g)\circ u\circ \pi (h^{-1})$ for every $(g, h)\in G\times G$ and for every $u\in$ End(E$_{\pi}$). The mapping $\varepsilon_{\pi}$ is a continuous representation of $G\times G$ in End(E$_{\pi}$). It is unitary. In fact, since $\pi$ itself is unitary, one obtains

$\|\varepsilon_{\pi}(g, h)u\|^2_2=$ dim($\pi$ ) Tr$((\pi (g)u\pi (h^{-1}))^*\pi (g)u\pi (h^{-1}))$

= dim($\pi$ ) Tr($\pi (h)u^*u\pi (h)^{-1}$) $=$ dim($\pi$ ) Tr($u^*u$) $=\|u\|^2_2$ for every $(g, h)\in G\times G$ and every $u\in$ End(E$_{\pi}$).

The mapping Ψ defined by $f\mapsto \pi (f)$ is a $(G\times$ G)-morphism of $\boldsymbol{\varrho }_G(\overline{\pi})$ into End(E$_{\pi}$), since

for every $(g, h)\in G\times G$ and every $f\in \boldsymbol{\varrho }_G(\overline{\pi})$. Since $\boldsymbol{\varrho }_G(\overline{\pi})$ is an irreducible representation (th. 1, a) of V, p. 462), there exists $\lambda \in \mathbf{C}^*$ such that the mapping $\lambda \Psi$ is zero or isometric (cor. 5, a) of V, p. 388).

Let $f=$ dim($\pi$ )$\overline{\chi}_{\pi}\in \boldsymbol{\varrho }_G(\overline{\pi})$. Then $\pi (f) = 1_{E_{\pi}}$ (cor. 2 of V, p. 465), whence $\|\pi (f)\|_2=$ dim($\pi$ ) $=\|f\|$ (cor. 3 of V, p. 459). Consequently, the mapping Ψ is isometric. Since $\boldsymbol{\varrho }_G(\overline{\pi})$ and End(E$_{\pi}$) are of the same dimension, Ψ is an isometric isomorphism.

We denote by $F^2(\widehat{G})$ the Hilbert sum of the Hilbert spaces End(E$_{\pi}$). The norm of an element $x\in F^2(\widehat{G})$ is again denoted by $\|x\|_2$.

In LIE, IX, p. 79, this space is denoted $L^2(\widehat{G})$, a notation which we prefer to avoid here so as not to create confusion with the space $\ell^2(\widehat{G})$.

Let $(u_{\pi})$ be an element of $F^2(\widehat{G})$. Since

$^{\pi\in\widehat{G}}\sum_{\pi\in\widehat{G}}\|u_{\pi}\|^2_2=\sum_{\pi\in\widehat{G}}$ dim($\pi$ ) Tr($u^*_{\pi}u_{\pi}$)$<+\infty$

and $\|u_{\pi}\|^2\leqslant$ Tr($u^*_{\pi}u_{\pi}$) $($cf. EVT, V, p. 52, formula (33)), the norm in $\mathscr{L}(E_{\pi})$ of the endomorphism $u_{\pi}$ tends to 0 at infinity. One can therefore identify $F^2(\widehat{G})$ with a subspace of $F_0(\widehat{G})$.

Let $\pi \in \widehat{G}$ and $u\in$ End(E$_{\pi}$). We denote by $\mathscr{F}_{\pi}(u)$ the function on G defined by $\mathscr{F}_{\pi}(u)(g) =\langle \pi (g)|u\rangle =$ dim($\pi$)Tr($\pi (g)^*u$) for every $g\in G$. This is a continuous function on G. If G is a compact real Lie group, then the function $\mathscr{F}_{\pi}(u)$ is analytic on G (LIE, III, § 8, n$^o1$, th. 1).

Since G is compact, one can identify $L^2(G)$ with a subspace of $L^1(G)$.