Exercise 10, § 8

(In this exercise, and in the following three, we retain the notations of §6.) Let $\lambda \in \mathfrak{h}^*$.

a) Let $N,N'$ be $\mathfrak{g}$-submodules of $Z(\lambda )$ such that $N'\subset N$ and $N/N'$ is simple. Show that there exists $\mu\in \lambda -Q_+$ such that $N/N'$ is isomorphic to $E(\mu)$ (apply Th. 1 of §6), and such that $\mu+\rho \in W.(\lambda +\rho )$ (apply Cor. 1 of Th. 2). b) Show that $Z(\lambda )$ admits a Jordan-Hölder sequence. (Apply $a$) and the fact that the weights of $Z(\lambda )$ are of finite multiplicity.)