Corollary 2

Let $E,E'$ be finite dimensional simple $\mathfrak{g}$-modules, and $\chi , \chi '$ their central characters. If $\chi =\chi '$, E and $E'$ are isomorphic.

Let $\lambda , \lambda '$ be the highest weights of $E,E'$. By §6, no. 4, Prop. $7,\chi_{\lambda}=\chi =$ $\chi '=\chi_{\lambda'}$, so there exists $w\in W$ such that $\lambda '+\rho =w(\lambda +\rho )$. Since $\lambda +\rho$ and $\lambda '+\rho$ belong to the chamber defined by B, we have $w= 1$. Thus, $\lambda =\lambda '$, hence the corollary.