Weight spaces of representations of finite dimensional simple Lie algebras

Question

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.

Answer 1

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".

Answer 2

EDIT: I misunderstood at first what your basic question is but now understand it better. One cautionary case comes from older work of Richard Block here, which includes the rank 1 simple Lie algebra and provides a classification of all irreducible representations of it (which had been regarded by Dixmier and others as an impossible task).

However, Block's classification and construction do not provide an immediate answer to your question. It is a natural case to begin with, but is already complicated to study.

In general there are plenty of irreducible $U(\mathbb{g})$-modules which are infinite dimensional but still in the BGG category $\mathcal{O}$. This category consists of modules (in characteristic 0) which satisfy several basic axioms including finite generation and being a direct sum of weight spaces. All modules in this category have finite dimensional weight spaces. Verma modules for example are generated by a one dimensional highest weight space.

The key property of the universal enveloping algebra of a semisimple Lie algebra is being noetherian and (via a PBW basis) having a nice "triangular" decomposition. Anyway, the concise 1976 paper (accessible online in the original Russian version) is one source, and my more leisurely but belated AMS textbook (and in later chapters survey) *Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$ (2008) gives a more comprehensive view of material including the Kazhdan-Lusztig conjecture of 1979 (soon a theorem). See especially Chapter 1.

P.S. Your auestion would be more clearly focused if you said "simple Lie algebra" in the header, or even "finite dimensional simple Lie algebra". This should be taken over any algebraically closed field such as $\mathbb{C}$ of characteristic 0.