TITLE: Weight spaces of representations of finite dimensional simple Lie algebras
QUESTION [5 upvotes]: This question has probably been asked before on this website, but I could not find any solution and neither can I solve this question. So again I am asking the following question:
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$ with Cartan subalgebra $\mathfrak{h}$ and $V$ be an irreducible weight module of $\mathfrak{g}$ with respect to $\mathfrak{h}$, i.e. $V= \oplus_{\lambda \in \mathfrak{h}^*} V_{\lambda}$ where $V_{\lambda} = \{v \in V : h.v = \lambda(h)v$ $\forall h \in \mathfrak{h}\}$. If $V_{\mu}$ is a non-zero finite dimensional weight space for some $\mu \in \mathfrak{h}^*$, then show that $V_{\lambda}$ is finite dimensional $\forall \lambda \in \mathfrak{h}^*$.
Any help or reference will be highly appreciated.

REPLY [2 votes]: The requisite property follows from the following key proposition:

$U_{\lambda}$ is a finitely generated right $U_0$-module. 

Notation The subscripts denote the grading of the universal enveloping algebra $U=U({\frak g})$ with respect to the adjoint action of the Cartan subalgebra ${\frak h},\, \displaystyle U=\bigoplus_{\lambda\in P}U_{\lambda},$ with the grading abelian group the root lattice $P.$ The subspace $U_0$ is $U^{\frak h}$, the subalgebra of ${\frak h}$-invariants in $U.$
Similarly, $\displaystyle S=\bigoplus_{\lambda\in P}S_{\lambda}$ for the symmetric algebra $S=S({\frak g})$ and $S_0$ is the subalgebra of ${\frak h}$-invariants in $S.$
Proof of the property The action of $U_0$ stabilizes each weight subspace of $V$ and the action of $U_{\lambda}$ increases the weight by $\lambda$. Let $W=UV_{\mu}$, where a weight subspace $V_{\mu}$ is non-zero and finite-dimensional. Since $V$ is simple and $W$ is a non-zero submodule of $V$, $W=V$. Note that $U_{\lambda}V_{\mu}$ is $W_{\lambda+\mu}$, the weight subspace of $W$ of weight $\lambda+\mu$. Therefore $$\displaystyle V=\bigoplus_{\lambda\in P}U_{\lambda}V_{\mu}=\bigoplus_{\lambda\in P}W_{\lambda+\mu}$$ is the weight decomposition of $W=V$. For any $\lambda\in P$, $U_{\lambda}=X_{\lambda}U_0$ with a finite set $X_{\lambda}$, according to the proposition, and $U_{0}V_{\mu}=V_{\mu}$. It follows that each weight subspace $W_{\lambda+\mu}$ is finite-dimensional: $$W_{\lambda+\mu}=U_{\lambda}V_{\mu}=X_{\lambda}U_{0}V_{\mu}=X_{\lambda}V_{\mu}.$$ 
Proof of the proposition The algebra $U$ is almost commutative (i.e. its associated graded algebra ${\rm gr\,}U=S$ is commutative and generated by degree 1 part) and the adjoint action of $\frak h$ on $U$ is semisimple and preserves the filtration, so that ${\rm gr\,}U_{0}=S_{0}$ and ${\rm gr\,}U_{\lambda}=S_{\lambda}$. Hence it suffices to prove corresponding statement for the associated graded algebra: 

$S_{\lambda}$ is a finitely generated $S_0$-module. 

Recall that $S$ is a polynomial ring, the symmetric algebra of ${\frak g}$, and it is graded by $P$, i.e. it is a multigraded ring, and the last statement is a general property of multigraded rings. A good reference for these rings is the Miller-Sturmfels book "Combinatorial Commutative Algebra".