Exercise 5, § 8

We retain the notations of §6 and §7. Let $\lambda \in \mathfrak{h}^*$. If, for any $w\in W,w\not= 1$, we have $(\lambda +\rho )-w(\lambda +\rho )\notin Q_+$, then $Z(\lambda )$ is simple.

(Use Cor. 1 (ii) of Th. 2.)