Central-character cuts of a typed module are typed by the matching weights

Given: The Axiom of Choice, a Verma-filtered module $M\in \mathcal{O}$ with a fixed filtration $0=M_0\subseteq M_1\subseteq \cdots \subseteq M_n=M$ and weights ${\psi }_1,\dots ,{\psi }_n$ with $M_j\mathrm{/}{M}_{j-1}\cong M({\psi }_j)$, and a central character $\chi$.

Every $M\in \mathcal{O}$ decomposes canonically as $M={⨁}_{{\chi }^{\mathrm{\prime }}}{M}_{{\chi }^{\mathrm{\prime }}}$ into finitely many generalised central-character submodules, and the canonical projections $M\to M_{{\chi }^{\mathrm{\prime }}}$ are exact functors (Generalized central-character summands, Generalized central-character decomposition of O).

If $0\to A\to B\to C\to 0$ is an exact sequence in $\mathcal{O}$, applying the exact projection functor gives an exact sequence $0\to A_{\chi }\to B_{\chi }\to C_{\chi }\to 0$, and the quotients of a filtration are computed by $(B_j\mathrm{/}{B}_{j-1}{)}_{\chi }\cong B_{j,\chi }\mathrm{/}{B}_{j-1,\chi }$ (F1, Type of a module with a Verma filtration).

Every cyclic highest-weight module has a well-defined central character; on $M(\psi )$ every $z\in Z(U(\mathfrak{g}))$ acts by the scalar ${\chi }_{\psi }(z)$, and therefore $M(\psi {)}_{{\chi }^{\mathrm{\prime }}}=M(\psi )$ if ${\chi }^{\mathrm{\prime }}={\chi }_{\psi }$ while $M(\psi {)}_{{\chi }^{\mathrm{\prime }}}=0$ if ${\chi }^{\mathrm{\prime }}\mathrm{\ne }{\chi }_{\psi }$ (Central elements act by scalars on cyclic highest-weight modules, Generalized central-character subcategories).

${\chi }_{\psi }={\chi }_{{\psi }^{\mathrm{\prime }}}$ if and only if ${\psi }^{\mathrm{\prime }}$ lies in the dot-Weyl orbit of $\psi$ (Central characters are dot-Weyl orbits).