Remark

Let $(V, \pi )$ be a linear representation of G. By Maschke’s theorem, the K[G]-module V is semisimple. For any $\lambda \in \widehat{G}$, denote by $V^{\lambda}$ the isotypical component of type $\lambda$ of the K[G]-module V; we have $V =\bigoplus_{\lambda\in\widehat{G}}V^{\lambda}$. By Proposition 15 of VIII, p. 147 and formula (17), the projector of V with image $V^{\lambda}$ associated with this decomposition of V is equal to

Applying this formula to $\pi =\pi_{\lambda}$, we obtain that the element $d_{\lambda}\cdot 1$ of K is not zero. We will see further on (VIII, p. 420, Corollary 2) that $d_\mu$ divides the cardinal of G.

Let $\lambda$ be an element of $\widehat{G}$. The mapping $\pi_{\lambda}$ from K[G] to End$_K(V_{\lambda})$ induces an isomorphism of $(K[G]$, K[G])-bimodules from $e_{\lambda}K[G]$ to End$_K(V_{\lambda})$. By Proposition 15 of VIII, p. 147, the K[G]-submodule $e_{\lambda}K[G]$ is the isotypical component of K[G] of type $\lambda$. It is also the isotypical component of type $\lambda^{\vee}$ for the right regular representation of G (VIII, p. 143, Proposition 11) as well as the isotypical component of type $\lambda \times \lambda^{\vee}$ for the biregular representation (VIII, p. 381, Proposition 4).