Generalized central-character subcategories

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 $Z=Z(U(\mathfrak{g}))$ and let $\chi :Z\to \mathbb{C}$ be a unital complex-algebra character, as in Central character of a Lie algebra module. Put ${\mathfrak{m}}_{\chi }=\ker \chi$ and, for $M$ in The classical BGG category O, define

$M_{\chi }=\{v\in M:{\mathfrak{m}}_{\chi }^{N}v=0\text{ for some integer }N\ge 1\}\mathrm{.}$

Here ${\mathfrak{m}}_{\chi }^{N}v=0$ means every element of that ideal kills $v$; the exponent may initially depend on $v$. The full subcategory ${\mathcal{O}}_{\chi }$ consists of the objects with $M=M_{\chi }$. This is a generalized central-character condition, weaker than scalar central action. It does not by definition assert that ${\mathcal{O}}_{\chi }$ is an indecomposable block.