Proposition 9

The family $(\chi_{\pi})_{\pi\in\widehat{G}}$ of the characters of the irreducible unitary representations of G is an orthonormal basis of $L^2(G^{\sharp})$.

By the Peter-Weyl theorem (th. 1 of V, p. 462), the closed subspace $L^2(G^{\sharp})$ of $L^2(G)$ formed by the elements invariant under the unitary representation $g\mapsto \boldsymbol{\varrho }_G(g, g)$ of G is the Hilbert sum of the subspaces $L^2(G^{\sharp})\cap \boldsymbol{\varrho }_G(\pi )$ for $\pi \in \widehat{G}$. The proposition therefore follows from the preceding lemma and cor. 3 of V, p. 459.