Exercise 23, § 7

Let $U^0$ be the commutant of $\mathfrak{h}$ in $U(\mathfrak{g})$ (cf. §6, no. 4).

a) Let V be a (finite dimensional) $\mathfrak{g}$-module. Show that the $V^{\lambda},\lambda \in P$, are stable under $U^0$, and that, if V is simple and $V^{\lambda}\not= 0, V^{\lambda}$ is a simple $U^0$-module (use the decomposition $U(\mathfrak{g}) =\oplus U^{\lambda}$, loc. cit.).

b) Show that, for every element $c\not= 0$ of $U^0$, there exists a finite dimensional simple representation $\rho$ of $U^0$ such that $\rho (c)\not= 0$. (Use $a$) and Chap. I, §7, Exerc. $3a).)$