Exercise 13, § 8

a) Let $\mathfrak{a}$ be a nilpotent Lie algebra, $x\in \mathfrak{a}, n\in \mathbf{N}, p\in \mathbf{N}$. There exists $l\in \mathbf{N}$ such that $x^ly_1. . . y_n\in U(\mathfrak{a})x^p$ for all $y_1, . . . , y_n\in \mathfrak{a}$.

b) Let $\lambda , \mu\in \mathfrak{h}^*$, and $\alpha \in B$ be such that

Assume that $\lambda \in P$. Let $p=\lambda (H_{\alpha})\in \mathbf{Z}$. Show that:

$b_1)$ If $p\leq 0$, then $Z(\lambda -\rho )\subset Z(s_{\alpha}\lambda -\rho )$.

$b_2)$ If $p >0$, then $Z(s_{\alpha}\mu-\rho )\subset Z(s_{\alpha}\lambda -\rho )\subset Z(\lambda -\rho )$.

(Use $a$), and §6, Cor. 1 of Prop. 6.)

c) Let $\lambda \in \mathfrak{h}^*, \alpha \in R$, and $m=\lambda (H_{\alpha})$. Assume that $m\in \mathbf{N}$. Show that

(Prove this first for $\lambda \in P$ by using $b$), and then in the general case by using Exerc. $12d).)^{17}$