Exercise 7, § 7

If $\lambda \in P_{++}$, denote by $\rho_{\lambda}: U(\mathfrak{g})\rightarrow$ End(E($\lambda$ )) the representation defined by the simple module $E(\lambda )$. We have Im($\rho_{\lambda}$) $=$ End(E($\lambda$ )); put $\mathfrak{m}_{\lambda}=$ Ker($\rho_{\lambda}$). a) Show that the $\mathfrak{m}_{\lambda}$ are pairwise distinct, and that they are the only two-sided ideals $\mathfrak{m}$ of $U(\mathfrak{g})$ such that $U(\mathfrak{g})/\mathfrak{m}$ is a finite dimensional simple $k$-algebra.

b) If I is a finite subset of $P_{++}$, put $\mathfrak{m}_I=\bigcap_{\lambda\in I}\mathfrak{m}_{\lambda}$. Show that the canonical map $U(\mathfrak{g})/\mathfrak{m}_I\rightarrow \prod_{\lambda\in I}U(\mathfrak{g})/\mathfrak{m}_{\lambda}$ is an isomorphism, and that every two-sided

ideal of $U(\mathfrak{g})$ of finite codimension is equal to exactly one of the $\mathfrak{m}_I$.

c) Show that the principal anti-automorphism of $U(\mathfrak{g})$ transforms $\mathfrak{m}_{\lambda}$ to $\mathfrak{m}_{\lambda^*}$, where $\lambda^*=-w_0\lambda$ (cf. Prop. 11).