Remark 3

Let $\tau : G\rightarrow \mathbf{G}\mathbf{L}(V)$ be an irreducible representation. There exists an element $\nu$ of X(C(G)) such that $\tau (s)v=\nu (s)v$ for all $s\in C(G), v\in V:$ indeed, $\tau (C(G))$ is contained in the commutant of $\tau$(G), which is equal to $\mathbf{C}^*.1_V($Algebra, Chap. VIII, §3, no. 2, Th. 1). For every weight $\lambda$ of $\tau$, the restriction of $\lambda$ to C(G) is equal to $\nu$.