Exercise 18, § 3

Let $\mathfrak{g}$ be a Lie algebra, $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$ and $U\mathfrak{g}$ (resp. $U\mathfrak{h}$) the enveloping algebra of $\mathfrak{g}$ (resp. $\mathfrak{h}$). A linear form $\varphi$ on $U\mathfrak{g}$ is said to be central if it vanishes on $[U\mathfrak{g},U\mathfrak{g}]$, i.e. if $\varphi (a.b) =\varphi (b.a)$ for all $a, b\in U\mathfrak{g}$.

a) Let $x, y\in \mathfrak{g}$. Assume that there exists $s\in$ Aut$_e(\mathfrak{g})$ such that $s(x) =y$. Show that

for all $n\in \mathbf{N}$ and for every central linear form $\varphi$ on $U\mathfrak{g}$.

b) Let $\varphi$ be a central linear form on $U\mathfrak{g}$ whose restriction to $U\mathfrak{h}$ vanishes. Show that $\varphi = 0$. (We can assume that $k$ is algebraically closed. Deduce from $a$) that we then have $\varphi (x^n) = 0$ for all $n\in \mathbf{N}$ and all regular $x\in \mathfrak{g}$; use a density argument to remove the assumption of regularity.)

c) Show that $U\mathfrak{g}= [U\mathfrak{g},U\mathfrak{g}] + U\mathfrak{h}$.

d) Let V be a semi-simple $\mathfrak{g}$-module. Show that V is semi-simple as an $\mathfrak{h}$-module. In particular, $V^{\lambda}(\mathfrak{h}) = V_{\lambda}(\mathfrak{h})$ for all $\lambda \in \mathfrak{h}^*$.

e) Let $V'$ be a semi-simple $\mathfrak{g}$-module. Assume that V and $V'$ are isomorphic as $\mathfrak{h}$-modules. Show that they are isomorphic as $\mathfrak{g}$-modules. (If $a\in U\mathfrak{h}$, remark that Tr($a_V$) $=$ Tr($a_{V'}$). Deduce, by using $b$) or c), that Tr($x_V$) $=$ Tr($x_{V'}$) for all $x\in U\mathfrak{g}$, and conclude by using Algebra, Chap. VIII.)

If $k$ is algebraically closed, the assumption that“VandV$'$ are $\mathfrak{h}$-isomorphic” is equivalent to saying that dim $V_{\lambda}(\mathfrak{h}) =$ dim $V'_{\lambda}(\mathfrak{h})$ for all $\lambda \in \mathfrak{h}^*$.