Remark 1

Let $\mathfrak{n}_+$ be the subalgebra of $\mathfrak{g}_{\mathbf{C}}$ that is the sum of the $\mathfrak{g}^{\alpha}$ for $\alpha >0$. Let $\tau : G\rightarrow \mathbf{G}\mathbf{L}(V)$ be an irreducible representation, $\lambda \in X_{++}$ its highest weight and $\tau ':\mathfrak{g}_{\mathbf{C}}\rightarrow \mathfrak{g}\mathfrak{l}(V)$ the representation induced by $\tau$. Then $V_{\lambda}(T)$ is the subspace consisting of the vectors $v$ in V such that $\tau '(x)v= 0$ for all $x\in \mathfrak{n}_+$.

Indeed, this follows from the corresponding statement for the $\mathfrak{g}_{\mathbf{C}}$-modules $\mathbf{C}(\nu )\otimes E(\mu)$ (Chap. VIII, §6, no. 2, Prop. 3).