Corollary 2

Let $\rho =\frac{1}{2}\sum_{\alpha\in R_+}\alpha$, and $\lambda , \mu\in \mathfrak{h}^*$. Assume that $\lambda +\rho$ is a dominant weight in R, and that there exists $w\in W$ with $\mu+\rho =w(\lambda +\rho )$. Then $Z(\mu)$ is isomorphic to a submodule of $Z(\lambda )$.

The assertion is clear when $w= 1$. Assume that it is established whenever $w$ is of length $< q$. If $w$ is of length $q$, there exists $\alpha \in B$ such that $w=s_{\alpha}w^{'-1}$, with $l(w') =q-1$. We have $w'(\alpha )\in R_+$ (Chap. VI, §1, no. 6, Cor. 2 of Prop. 17), and hence $w^{'-1}(\lambda +\rho )(H_{\alpha}) = (\lambda +\rho )(H_{w'\alpha})$ is an integer $\geq 0$. Put

By the induction hypothesis, $Z(\mu')$ is isomorphic to a submodule of $Z(\lambda )$. On the other hand, by Chap. VI, §1, no. 10, Prop. 29 (ii),

Moreover, $\rho (H_{\alpha}) = 1$ (Chap. VI, §1, Prop. 29 (iii)), so $\mu'(H_{\alpha})+1\in \mathbf{N}$. Cor. 1 now implies that $Z(\mu)$ is isomorphic to a submodule of $Z(\mu')$, and hence also to a submodule of $Z(\lambda )$.