Theorem 1 (Character Formula of H. Weyl)

Let M be a finite dimensional simple $\mathfrak{g}$-module, and $\lambda$ its highest weight. Then

$(\sum_{w\in W}\varepsilon (w)e^{w\rho})$.ch $M =\sum_{w\in W}\varepsilon (w)e^{w(\lambda+\rho)}$.

With the notations of Lemma 3, the central character of M is $\chi_{\lambda}($§6, no. 4, Prop. 7). Hence, by Lemmas 2 and $3,d$.ch M is a $\mathbf{Z}$-linear combination of the $e^{\mu+\rho}$ such that

On the other hand, by §7, no. 7, Lemma $7,d$.ch M is anti-invariant, and its unique maximal term is $e^{\lambda+\rho}$, hence the theorem.