Corollary 2

Tag 013E, permanent. Cite it as [Bourbaki, Tag 013E].

Lie Groups and Lie Algebras, chapter VIII, § 7. Finite dimensional modules over a split semi-simple Lie algebra

Let $w\in W$ . For all $\lambda \in \mathfrak{h}^*$ , the vector subspaces $V^{\lambda}$ and $V^{w\lambda}$ have the same dimension. The set of weights of V is stable under W.

Indeed, $w$ is of the form $s^*$ with $s\in$ Aut $_e(\mathfrak{g},\mathfrak{h}) ($ §2, no. 2, Cor. of Th. 2).

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

Lemma 6, § 7
Proposition 3, § 7
Proposition 7, § 7

Cites

§2, no. 2, Cor. of Th. 2