A singular integral A2 central-character summand

Given: The Axiom of Choice, the setting above, and the hypotheses in the example.

The category $\mathcal{O}$ is the categorical direct sum of its generalized central-character subcategories ${\mathcal{O}}_{\chi }$, with exact component projections (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 $w\cdot \lambda =w(\lambda +\rho )-\rho$. 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)

Under the given Choice premise, 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)

Let $\mathrm{\Phi }$ be the root system from thm-the-root-set-is-a-reduced-crystallographic-root-system. For a root $\alpha$, define the corresponding coroot by ${\alpha }^{\vee }:=\frac{2H_{\alpha }}{\alpha (H_{\alpha })}\in \mathfrak{h},$ where $H_{\alpha }$ is the vector from def-killing-dual-vector-attached-to-a-root. The associated root reflection is the linear map $s_{\alpha }(\lambda ):=\lambda -\lambda ({\alpha }^{\vee })\alpha (\lambda \in {\mathfrak{h}}^{\ast })\mathrm{.}$ The subgroup of $\mathrm{GL}({\mathfrak{h}}^{\ast })$ generated by these reflections is the Weyl group $W$, acting on ${\mathfrak{h}}^{\ast }$ in the usual way. (Root reflections and the Weyl group action)

Every simple object of $\mathcal{O}$ is $L(\eta )$ for a unique highest weight $\eta$ (The simple objects of O). Its generalized-character membership means each $z-\chi (z)$ acts locally nilpotently (Generalized central-character subcategories), while $z$ acts on any cyclic highest-weight module by the scalar ${\chi }_{\eta }(z)$ (The Harish-Chandra projection computes the highest-weight scalar).

Under the given Choice premise, every object of $\mathcal{O}$ has a finite composition series (Every object of O has finite length). Every Verma module $M(\eta )$ has a unique simple quotient $L(\eta )$ (A Verma module has a unique simple quotient).

A positive integral simple-root pairing gives a nonzero Verma embedding in the reflected direction (Simple-reflection embeddings of Verma modules).