Central-character summands refine into linkage blocks

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 $w\cdot \lambda =w(\lambda +\rho )-\rho$. For a weight $\lambda$, define ${\mathrm{\Phi }}_{\lambda }=\{\alpha \in \mathrm{\Phi }:⟨\lambda +\rho ,{\alpha }^{\vee }⟩\in \mathbb{Z}\},W_{\lambda }=⟨s_{\alpha }:\alpha \in {\mathrm{\Phi }}_{\lambda }⟩\subset W\mathrm{.}$ Reflections and coroots are those of def-root-reflections-and-the-weyl-group-action, with the shift from def-weyl-vector-rho-for-a-chosen-positive-system. The integral-reflection linkage class through $\lambda$ is $W_{\lambda }\cdot \lambda$. The word integral includes zero and negative integral pairings. If ${\mathrm{\Phi }}_{\lambda }$ is empty the generated group is $\{1\}$. This definition uses generating reflections; no identification with a root-lattice-coset stabilizer is assumed. (The integral Weyl group of a weight)

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 set ${\mathrm{\Phi }}_{\lambda }$ is preserved by its root reflections. If $\mu \in W_{\lambda }\cdot \lambda$, then ${\mathrm{\Phi }}_{\mu }={\mathrm{\Phi }}_{\lambda }$ and $W_{\mu }=W_{\lambda }$. The equivalence classes generated by moves $\eta \mapsto s_{\alpha }\cdot \eta$ with $\alpha \in {\mathrm{\Phi }}_{\eta }$ are exactly $W_{\lambda }\cdot \lambda$. Every strong-linkage chain stays in one such class. (Integral reflection linkage is an equivalence relation)

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 $w\cdot \lambda =w(\lambda +\rho )-\rho$. A short exact sequence $0\to L(\mu )\to E\to L(\lambda )\to 0$ in $\mathcal{O}$ splits whenever $\lambda$ and $\mu$ belong to distinct integral-reflection linkage classes. (Simple extensions cannot cross linkage classes)

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 $w\cdot \lambda =w(\lambda +\rho )-\rho$. Partition the isomorphism classes of simple objects of $\mathcal{O}$ into parts ${\mathcal{P}}_t$. Suppose every extension of two simples from different parts splits, in either order. Then each $M\in \mathcal{O}$ has a unique decomposition $M={⨁}_t{M}_t$ into submodules whose composition factors lie in ${\mathcal{P}}_t$, with finitely many nonzero terms. This decomposition is functorial, and maps between modules supported on disjoint collections of parts are zero. (Splitting finite-length modules across separated simple classes)

For $\alpha \in {\mathrm{\Phi }}^{+}$, if $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\in {\mathbb{Z}}_{>0}$, then $M(s_{\alpha }\cdot \lambda )\hookrightarrow M(\lambda )$. (Verma embedding for an arbitrary positive root)

The proper submodule $J(\lambda )$ which is the sum of all proper submodules is the unique maximal submodule of $M(\lambda )$. The quotient $L(\lambda ):=M(\lambda )\mathrm{/}J(\lambda )$ is simple and is its unique simple quotient. (A Verma module has a unique simple quotient)

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)