Exercise 11, § 8

Let $\lambda \in \mathfrak{h}^*$ and let V be a non-zero $\mathfrak{g}$-submodule of $Z(\lambda )$. Show that there exists $\mu\in \mathfrak{h}^*$ such that V contains a simple $\mathfrak{g}$-submodule isomorphic to $Z(\mu)$.

(Let A be the set of $\nu \in \mathfrak{h}^*$ such that V contains a $\mathfrak{g}$-submodule isomorphic to $Z(\nu )$. By using Prop. 6 of no. 6, show first that $A\not=\emptyset$. Then show that A is finite, and consider an element $\mu$ of A such that $(\mu-Q_+)\cap A =\{\mu\}.$)