The support description of category O with finite generation

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 BGG category $\mathcal{O}$ is the full subcategory of left $U(\mathfrak{g})$-modules $M$ satisfying all three conditions: $M$ is finitely generated; $M={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{M}_{\mu }$ where $M_{\mu }=\{v:hv=\mu (h)v\text{ for all }h\in \mathfrak{h}\}$; and $U({\mathfrak{n}}^{+})v$ is finite dimensional for each $v\in M$. Its morphisms are all $\mathfrak{g}$-linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category 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 $w\cdot \lambda =w(\lambda +\rho )-\rho$. Every $M\in \mathcal{O}$ has a finite-dimensional, $\mathfrak{b}$-stable, $\mathfrak{h}$-semisimple generating subspace $E$. There is a flag $0=E_0\subset E_1\subset \cdots \subset E_r=E$ of $\mathfrak{b}$-submodules whose quotients are one dimensional and annihilated by ${\mathfrak{n}}^{+}$. For $M=0$ take $E=0$ and the empty flag. (Finite Borel-stable generators and weight flags)

Let $x_1,\dots ,x_r$ be an ordered basis of a finite-dimensional complex Lie algebra $\mathfrak{g}$. Then the monomials $x_1^{a_1}{x}_2^{a_2}\cdots x_r^{a_r}(a_i\in {\mathbb{N}}_0)$ form a basis of $U(\mathfrak{g})$. In particular, multiplication identifies $\mathrm{gr}U(\mathfrak{g})$ with the symmetric algebra $S(\mathfrak{g})$ on the symbols of the $x_i$. (PBW gives an ordered monomial basis for the enveloping algebra)