Lemma 1

If G is commutative, $\widetilde{V}$ is the direct sum of the $\widetilde{V}_{\lambda}(G)$ for $\lambda \in X(G)$.

Since $\rho$ is semi-simple (§1, no. 1), it suffices to prove the lemma in the case in which $\rho$ is simple. In that case, the commutant Z of $\rho (G)$ in End( $\widetilde{V}$) reduces to homotheties (Algebra, Chap. VIII, §3, no. 2, Th. 1); thus, the image of the homomorphism $\widetilde{\rho}$ is contained in the subgroup $\mathbf{C}^*.1_V$ of GL( $\widetilde{V}$), and there exists $\lambda \in X(G)$ such that $\widetilde{V} = \widetilde{V}_{\lambda}(G)$.