Remark

When G is finite, Schur’s orthogonality relation (resp. the orthogonality formula for characters) coincides with that of A, VIII, p. 399 (resp. that of A, VIII, p. 400, prop. 4). On the other hand, the “second orthogonality relation” for the characters of finite groups (A, VIII, p. 402, formula (32)) has no exact analogue when G is compact.

For finite G, the particular case of the second orthogonality formula corresponding to the conjugacy class of $e$ is the formula

Card G1 $\sum_{\pi\in\widehat{G}}$ dim($\pi$ )$\chi_{\pi}(g) =$ 10 sisinon$g=,e$ which is also interpreted as the calculus of the character of the regular representation $\boldsymbol{\gamma }_G$ of G, and which is equivalent to the formula Tr($\boldsymbol{\gamma }_G(f)$) $=f(e)$ for every function $f$ from G into $\mathbf{C}$.

Let G again be any compact group. For every function $f\in \mathscr{C}$(G), the endomorphism $\boldsymbol{\gamma }_G(f)$ of $L^2(G)$ coincides with the endomorphism $\varphi \mapsto f*\varphi$, and it is of finite trace (lemma 4 of V, p. 407 and corollary 2 of III, p. 33). By loc. cit. and Theorem 2 of IV, p. 177, one also has

Tr($\boldsymbol{\gamma }_G(f)$) $=\int_Gf(x^{-1}x)d\mu(x) =f(e)$.