Theorem 1

a) Let $\tau : G\rightarrow \mathbf{G}\mathbf{L}(V)$ be an irreducible representation of G. Then the set of weights of $\tau$ (relative to T) has a largest element $\lambda$, which is dominant, and the space $V_{\lambda}(T)$ is of dimension 1.

b) Two irreducible representations of G are equivalent if and only if their highest weights are equal.

c) For every dominant element $\lambda$ of X(T), there exists an irreducible representation of G of highest weight $\lambda$.

By Lemma 2, the equivalence classes of irreducible representations of G correspond bijectively to the classes of finite dimensional irreducible representations of $\mathfrak{g}$ whose weights belong to $\delta (X(T))$.

Denote by $\mathscr{C}\mathfrak{g}_{\mathbf{C}}$ the centre and by $\mathscr{D}\mathfrak{g}_{\mathbf{C}}$ the derived Lie algebra of $\mathfrak{g}_{\mathbf{C}}$, so that $\mathfrak{g}_{\mathbf{C}}=\mathscr{C}\mathfrak{g}_{\mathbf{C}}\oplus \mathscr{D}\mathfrak{g}_{\mathbf{C}}$. For every bilinear form $\mu$ on $\mathfrak{t}_{\mathbf{C}}\cap \mathscr{D}\mathfrak{g}_{\mathbf{C}}$, denote by $E(\mu)$ the simple $\mathscr{D}\mathfrak{g}_{\mathbf{C}}$-module introduced in Chap. VIII, §6, no. 3; for every linear form $\nu$ on $\mathscr{C}\mathfrak{g}_{\mathbf{C}}$, denote by $\mathbf{C}(\nu )$ the 1-dimensional $\mathscr{C}\mathfrak{g}_{\mathbf{C}}$-module over $\mathbf{C}$ associated to it. Then the $\mathfrak{g}_{\mathbf{C}}$-modules $\mathbf{C}(\nu )\otimes E(\mu)$ are simple, and by Chap. VIII, §7, no. 2, Cor. 2 of Th. 1 and Algebra, Chap. VIII, §11, no. 1, Th. 1, every finite dimensional simple $\mathfrak{g}_{\mathbf{C}}$-module is isomorphic to one of the modules $\mathbf{C}(\nu )\otimes E(\mu)$; moreover (loc. cit.)$\mathbf{C}(\nu )\otimes E(\mu)$ is finite dimensional if and only if $\mu(H_{\alpha})$ is a positive integer for every simple root $\alpha$. If we denote by $\nu +\mu$ the linear form on $\mathfrak{t}_{\mathbf{C}}$ that induces $\nu$ on $\mathscr{C}\mathfrak{g}_{\mathbf{C}}$ and $\mu$ on $\mathfrak{t}_{\mathbf{C}}\cap \mathscr{D}\mathfrak{g}_{\mathbf{C}}$, then $(\nu +\mu)(H_{\alpha}) =\mu(H_{\alpha})$; moreover, the weights of $\mathbf{C}(\nu )\otimes E(\mu)$ are the $\nu +\lambda$, where $\lambda$ is any weight of $E(\mu)$, and are thus of the form $\nu +\mu-\theta$, with $\theta \in \delta (X_+)$ (Chap. VIII, §6, no. 2, Lemma 2).

We conclude that the $\mathfrak{g}$-module $\mathbf{C}(\nu )\otimes E(\mu)$ is finite dimensional if and only if $(\nu +\mu)(H_{\alpha})$ is a positive integer for every simple root $\alpha$, and that its weights belong to $\delta (X(T))$ if and only if $\nu +\mu$ belongs to $\delta (X(T))$. The conjunction of these two conditions means that $\nu +\mu$ belongs to $\delta (X_{++})$; in that case, $\nu +\mu$ is the highest weight of $\mathbf{C}(\nu )\otimes E(\mu)$. Thus, we have constructed for every dominant weight $\lambda$ of X(T) an irreducible representation of G of highest weight $\lambda$, and have thus obtained, up to equivalence, all the irreducible representations of G. Since the vectors of weight $\nu +\mu$ in $\mathbf{C}(\nu )\otimes E(\mu)$ form a subspace of dimension 1, the proof is complete.