Exercise 7, § 21

For any representation V of G of finite dimension, we denote by $\varphi (V)$ the dimension of the subspace of V fixed by G; we extend $\varphi$ by linearity to a $\mathbf{Z}$-linear form on $R_K(G)$. Let $(V_{\lambda})_{\lambda\in\widehat{G}}$ be the family of irreducible representations of G, and let $\omega$ be the element $\sum_{\lambda\in\widehat{G}}[V_{\lambda}] [V_{\lambda}^*]$ of $R_K(G)$. Prove the formula $\varphi (\omega^n) =\sum_Cd(C)^{n-1}$ for every integer $n\geqslant 0$ (apply Proposition 8 of VIII, p. 415 and the second orthogonality relation for characters).