Proposition 6

Let $f$ be a central function on G and $u$ be a central function on H. We have

(33) $\langle$Ind$^G_H(u), f\rangle_G=\langle u$, Res$^G_H(f)\rangle_H$.

The characters of the simple representations of G form a basis of Z(K[G]) (VIII, p. 411, Proposition 5), and the same holds for H. It therefore suffices to establish (33) in the case when $f$ is the character $\chi_{\pi}$ of a simple representation $\pi$ of G and $u$ is the character $\chi_{\sigma}$ of a simple representation $\sigma$ of H. In this case, Ind$^G_H(u)$ is the character of the representation Ind$^G_H(\sigma )$ of G, and, by VIII, p. 410, Corollary, we have

$\langle$Ind$^G_H(u), f\rangle_G=$ (dim$_K$ Hom$_G$(Ind$^G_H(\sigma ), \pi$ ))$\cdot 1$.

We show the relation

$\langle u$, Res$^G_H(f)\rangle_H=$ (dim$_K$ Hom$_H(\sigma$, Res$^G_H(\pi )$))$\cdot 1$

likewise, and equality (33) follows by Frobenius reciprocity.