Lemma 2

Let $\varphi$ be a linear representation of the complex Lie algebra $\mathfrak{g}_{\mathbf{C}}$ on a finite dimensional complex vector space V. There exists a representation $\tau$ of G on V such that $L(\tau )_{(\mathbf{C})}=\varphi$ if and only if $\varphi$ is semi-simple and the weights of $\mathfrak{t}_{\mathbf{C}}$ on V belong to $\delta (X(T))$.

If there exists a representation $\tau$ of G such that $L(\tau )_{(\mathbf{C})}=\varphi$, then $\varphi$ is semi-simple because G is connected and $\tau$ is semi-simple (Chap. III, §6, no. 5, Cor. 2 of Prop. 13), and the weights of $\mathfrak{t}_{\mathbf{C}}$ on V belong to the image of $\delta$. Thus, the condition is necessary; we shall prove that it is sufficient. If $\varphi$ is semi-simple, V is the direct sum of the $V_\mu(\mathfrak{t}_{\mathbf{C}})$, where $\mu$ belongs to the set of weights of $\mathfrak{t}_{\mathbf{C}}$ on V (Chap. VII, §2, no. 4, Cor. 3 of Th. 2); if all the weights belong to the image of $\delta$, there exists a representation $\tau_T$ of T on V such that $L(\tau_T)_{(\mathbf{C})}=\varphi |\mathfrak{t}_{\mathbf{C}}:$ indeed, it suffices to put $\tau_T(t)v=t^{\lambda}v$ for $t\in T$ and $v\in V_{\delta(\lambda)}(\mathfrak{t}_{\mathbf{C}})$. The lemma now follows from Prop. 8 of §2, no. 6.