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.



Weight spaces of representations of finite dimensional simple Lie algebras
The requisite property follows from the following key proposition:



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> $U\_{\lambda}$ is a finitely generated right $U\_0$-module. 
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*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: 



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> $S\_{\lambda}$ is a finitely generated $S\_0$-module. 
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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".


