Corollary 4

a) Let $\varrho$ be a finite-dimensional continuous representation of G. One has

$\chi_{\varrho}=\sum_{\pi\in\widehat{G}}$ dim(Hom$_G(\pi , \varrho )$)$\chi_{\pi}=\sum_{\pi\in\widehat{G}}$ dim(Hom$_G(\varrho , \pi )$)$\chi_{\pi}$.

b) Let $\varrho_1$ and $\varrho_2$ be finite-dimensional continuous representations of G. One has

$\langle \chi_{\varrho_1}|\chi_{\varrho_2}\rangle =$ dim(Hom$_G(\varrho_1, \varrho_2)$) $=$ dim(Hom$_G(\varrho_2, \varrho_1)$).

c) A finite-dimensional continuous representation $\varrho$ of G is irreducible if and only if $\|\chi_{\varrho}\|^2= 1$.

Since $\varrho$ is semisimple, it is isomorphic to the direct sum of its $\pi$-isotypic components $M_{\pi}(\varrho )$ for $\pi \in \widehat{G}$. Denoting by $m_{\pi}(\varrho )$ the multiplicity of $\pi$ in $\varrho$ (Def. 10 of V, p. 398), one then has

Assertion a) therefore results from the fact that

$m_{\pi}(\varrho ) =$ dim Hom$_G(\pi , \varrho ) =$ dim Hom$_G(\varrho , \pi )$

(formula (1) of V, p. 377 and Corollary 2 of V, p. 387).

By bilinearity and orthonormality of the characters, assertion a) implies that

$\langle \chi_{\varrho_1}|\chi_{\varrho_2}\rangle =\sum_{\pi\in\widehat{G}}$ dim(Hom$_G(\pi , \varrho_1)$) dim(Hom$_G(\pi , \varrho_2)$), and on the other hand one has canonical isomorphisms

Hom$_G(\varrho_1, \varrho_2)\rightarrow \bigoplus_{(\pi_1,\pi_2)\in\widehat{G}\times\widehat{G}}$ Hom$_G(M_{\pi_1}(\varrho_1),M_{\pi_2}(\varrho_2))$

$\rightarrow \bigoplus_{\pi\in\widehat{G}}$ Hom$_G(M_{\pi}(\varrho_1),M_{\pi}(\varrho_2))$

(formula (1) of V, p. 377), whence it follows that

dim(Hom$_G(\varrho_1, \varrho_2)$) $=\sum_{\pi\in\widehat{G}}m_{\pi}(\varrho_1)m_{\pi}(\varrho_2)$,

whence assertion b). Assertion c) also results from a) and Schur’s Lemma (prop. 6 of V, p. 386).