I have the setup:
$L$ is a finite dimensional semisimple Lie algebra over an algebraically closed field $F$ of characteristic $0$ and $H$ a Cartan subalgebra of $L$.
So I'm trying to work out this problem:
Let $V$ be an infinite dimensional simple $L$-module. Prove: $V$ has a non-zero weight space if and only if dim$U(H).v < \infty$ for all $v\in V$.
Assuming $V$ has a non-zero weight space, then the sum of weight spaces $V' = \sum_{\lambda \in H^{*}}V_\lambda$ is a submodule of $V$. This means that $V'=V$ since $V$ is simple and at least one weight space is non-zero. So all elements $v\in V$ is a sum of weight vectors, i.e. I should be able to determine dim$U(H).v$ by considering $U(H).v_\lambda$ for the weight vectors.
I don't really know how to proceed from here. The universal enveloping algebra is still quite confusing to me. I know how to pass from $L$-modules to $U(L)$-modules and see their action, but how do I pass from $U(L)$-modules to $U(H)$-modules? If that is even what I need.
Some help would be greatly appreciated, I feel quite stuck.