Exercise 4, § 6

We use the notations of no. 4.

a) Let $m\in \mathbf{N}$. Give $\mathbf{N}^m$ the product order. Show that, for any subset S of $\mathbf{N}^m$, the set of minimal elements of S is finite.

b) Let $\alpha_1, . . . , \alpha_m$ be distinct elements of $R,X_i\in \mathfrak{g}^{\alpha_i}$ -- $\{0\}, S$ the set of non-zero sequences $(p_i)\in \mathbf{N}^m$ such that $\sum p_i\alpha_i= 0, M$ the set of minimal elements of S. Then $\mathfrak{h}$ and the $X_1^{p_1}. . . X_m^{p_m}$, where $(p_1, . . . , p_m)\in M$, generate the algebra $U^0$.

c) Show that $U^0$ is both a left- and right-noetherian algebra. (Give $U^0$ the filtration induced by that of $U(\mathfrak{g})$, and show, by using $a$) and b), that gr $U^0$ is commutative of finite type.)

d) Show that, for all $\lambda \in \mathfrak{h}^*, U^{\lambda}$ is a left (resp. right) $U^0$-module of finite type.

e) Let V be a simple $\mathfrak{g}$-module such that $V =\bigoplus_{\lambda\in\mathfrak{h}^*}V_{\lambda}$. If one of the $V_{\lambda}\not= 0$

is finite dimensional, then all of the $V_{\lambda}$ are finite dimensional. (Use $d$)$.)$