Every object of O has finite length

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 $w\cdot \lambda =w(\lambda +\rho )-\rho$. Suppose $M\in {\mathcal{O}}_{{\chi }_{\lambda }}$. Every nonzero subquotient $T$ of $M$ has $T_{\mu }\mathrm{\ne }0$ for some $\mu \in W\cdot \lambda$. In particular the number of strict inclusions in any finite chain of submodules of $M$ is at most $d_{\lambda }(M)={\sum }_{\mu \in W\cdot \lambda }\dim M_{\mu },$ where distinct weights in the orbit are counted once. (Finite weight-space detection of subquotients)

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 is the categorical direct sum $\mathcal{O}={⨁}_{\chi }{\mathcal{O}}_{\chi }$: objects have finite support in the index $\chi$, morphisms between distinct components vanish, and the canonical component projections are exact. (Generalized central-character decomposition of O)

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)

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)

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)