Lemma 3

Let $\pi$ be an irreducible unitary representation of G. The vector space $L^2(G^{\sharp})\cap \boldsymbol{\varrho }_G(\pi )$ has dimension one and is generated by the character of $\pi$.

Consider the unitary representation $\sigma$ of G in the space $\boldsymbol{\varrho }_G(\pi )$ defined by $\sigma (g) =\boldsymbol{\varrho }_G(g, g)$. Since the unitary representation $\boldsymbol{\varrho }_G(\pi )$ of $G\times G$ is isomorphic to $\overline{\pi}\boxtimes \pi$ (prop. 5 of V, p. 462), its character is given by

for every $g\in G$. The space $L^2(G^{\sharp})\cap \boldsymbol{\varrho }_G(\pi )$ is the subspace of the elements invariant under this representation, and its dimension is equal to

(cor. 3 of V, p. 466 and cor. 3 of V, p. 459). Since the character of $\pi$ is a nonzero element of $L^2(G^{\sharp})\cap \boldsymbol{\varrho }_G(\pi )$, it is a basis of this space.