Theorem 1

Tag 012W, permanent. Cite it as [Bourbaki, Tag 012W].

Lie Groups and Lie Algebras, chapter VIII, § 6. Modules over a split semi-simple Lie algebra

Let $\lambda \in \mathfrak{h}^*$ . The $\mathfrak{g}$ -module $E(\lambda )$ is simple and has highest weight $\lambda$ . Every simple $\mathfrak{g}$ -module of highest weight $\lambda$ is isomorphic to $E(\lambda )$ .

The first assertion follows from Prop. 5 (iii). The second follows from Prop. 4.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Exercise 10, § 8

Cites

Prop. 4
Prop. 5