Proposition 8

One has

(1) $\chi_{\pi}*\chi_{\sigma}= 0$ for all $\pi , \sigma$ belonging to $\widehat{G}, \pi \not =\sigma$,

(2) $\chi_{\pi}*\chi_{\pi}=$ dim($1\pi$ )$\chi_{\pi}$ for every $\pi$ belonging to $\widehat{G}$.

Let $\pi$ and $\sigma$ be irreducible representations of G. One has

dim($\pi$ )$(\chi_{\pi}*\chi_{\sigma}) =$ dim($\pi$ )$\boldsymbol{\gamma }_G(\chi_{\pi})\chi_{\sigma}$

(lemma 4 of V, p. 407). The function dim($\pi$ )$\boldsymbol{\gamma }_G(\chi_{\pi})\chi_{\sigma}$ is the orthogonal projection of $\chi_{\sigma}$ on the $\overline{\pi}$-isotypical component of $\boldsymbol{\gamma }_G$ (cor. 2 of V, p. 465), that is, on $\boldsymbol{\varrho }_G(\pi )$ (cor. 4 of V, p. 463). Since $\chi_{\sigma}$ belongs to $\boldsymbol{\varrho }_G(\sigma )$, the result follows from theorem 1 of V, p. 462.