Finite weight-space detection of subquotients

Given: The setting above and the hypotheses in the statement.

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. The category $\mathcal{O}$ is closed under submodules, quotients and finite direct sums and is an abelian category. If $0\to A\to E\to B\to 0$ is exact, $A,B\in \mathcal{O}$, and $E$ is $\mathfrak{h}$-semisimple, then $E\in \mathcal{O}$. The middle-term weight hypothesis is essential. (Category O is abelian and extension closed among weight modules)

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. Every nonzero $M\in \mathcal{O}$ contains a nonzero weight vector killed by ${\mathfrak{n}}^{+}$. (A nonzero O-object has a highest-weight vector)

Let ${\chi }_{\lambda }$ and ${\chi }_{\mu }$ be the central characters obtained from highest weights $\lambda$ and $\mu$. Then ${\chi }_{\lambda }={\chi }_{\mu }\text{if and only if}\mu \in W\cdot \lambda ,$ where $W\cdot \lambda :=\{w(\lambda +\rho )-\rho :w\in W\}$. (Central characters are dot-Weyl orbits)

Every central element acts on a cyclic highest-weight module by a scalar. In particular, each cyclic highest-weight module has a well-defined central character in the sense of def-central-character-of-a-lie-algebra-module. (Central elements act by scalars on cyclic highest-weight modules)

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. Every $M\in \mathcal{O}$ decomposes canonically into finitely many nonzero generalized central-character submodules: $M={⨁}_{\chi }{M}_{\chi }\mathrm{.}$ For each summand there is a single $N\ge 1$ such that ${\mathfrak{m}}_{\chi }^N{M}_{\chi }=0$. The decomposition of zero is empty. (Generalized central-character summands)

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. Let $M$ be a finitely generated $\mathfrak{h}$-semisimple $\mathfrak{g}$-module. Then $M\in \mathcal{O}$ if and only if $\mathrm{supp}M\subset {\bigcup }_{i=1}^r({\lambda }_i-Q^{+})$ for some finite list of weights. In either case every $M_{\mu }$ is finite dimensional. The list may be empty for $M=0$; finite generation is an independent hypothesis. (The support description of category O with finite generation)

Let a cyclic highest-weight module have highest vector of weight $\mu$. Then every $z\in Z(U(\mathfrak{g}))$ acts by the scalar $\mathrm{pr}(z)(\mu )={\chi }_{\mu }(z)$. (The Harish-Chandra projection computes the highest-weight scalar)