Weight spaces of representations of finite dimensional simple Lie algebras

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".