Proposition 6

The homomorphism $\rho : G\rightarrow \mathbf{G}\mathbf{L}(V)$ is injective if and only if $P(\rho ,T)$ generates the $\mathbf{Z}$-module X(T).

The homomorphism $\rho$ is injective if and only if its restriction to T is injective (§2, no. 6, Prop. 9). Further, since the canonical homomorphism $\mathbf{G}\mathbf{L}(V)\rightarrow \mathbf{G}\mathbf{L}( \widetilde{V})$ is injective, we can replace $\rho$ by $\widetilde{\rho}$. It then follows from (7) that the kernel of the restriction of $\rho$ to T is the intersection of the kernels of the elements of $P(\rho ,T)$. Thus, the conclusion follows from Prop. 2 of no. 1.

The linear representation $L(\rho )$ of $\mathfrak{t}$ in $\mathfrak{g}\mathfrak{l}( \widetilde{V})$ extends to a homomorphism of $\mathbf{C}$-Lie algebras

Moreover, recall (no. 1) that we have associated to every element $\lambda$ of X(T) a linear form $\delta (\lambda )$ on $\mathfrak{t}_{\mathbf{C}}$ such that

(exp$_Tx$)$^{\lambda}=e^{\delta(\lambda)(x)},x\in \mathfrak{t}$. (10)

Recall finally (Chap. VII, §1, no. 1) that, for any map $\mu:\mathfrak{t}_{\mathbf{C}}\rightarrow \mathbf{C}$, we denote by $\widetilde{V}_\mu(\mathfrak{t}_{\mathbf{C}})$ the vector subspace of $\widetilde{V}$ consisting of the $v$ such that $(\widetilde{L}(\rho )(u))(v) =\mu(u).v$ for all $u\in \mathfrak{t}_{\mathbf{C}}$.

We now deduce from (7) and loc. cit., Prop. 3: