Nonintegral sl2 central characters split into two simple blocks

Given: The setting above and the hypotheses in the example.

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 linkage class $C=W_{\lambda }\cdot \lambda$, let ${\mathcal{O}}_C$ be the full subcategory of objects all of whose simple composition factors have labels in $C$. Then $\mathcal{O}={⨁}_C{\mathcal{O}}_C$, and each nonzero ${\mathcal{O}}_C$ is indecomposable as a categorical direct summand. These are precisely the blocks. Each ${\mathcal{O}}_C$ lies in ${\mathcal{O}}_{{\chi }_{\lambda }}$; a central-character summand can contain several blocks. Independently, grouping weights by cosets of the root lattice $Q$ gives a canonical coarser decomposition by weight cosets. (Central-character summands refine into linkage blocks)

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 short exact sequence $0\to M(\lambda )\to E\stackrel{p}{\to }M(\lambda )\to 0$ in $\mathcal{O}$ splits. (Verma self-extensions in O split)

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 $M,N\in \mathcal{O}$, ${\mathrm{Hom}}_{\mathcal{O}}(M,N)$ is finite dimensional. For every weight $\lambda$, ${\mathrm{End}}_{\mathcal{O}}(L(\lambda ))=\mathbb{C}\mathrm{id}$. (Finite-dimensional Hom spaces in O)

$M(\lambda )$ is simple if and only if $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\mathrm{\notin }{\mathbb{Z}}_{>0}$ for every $\alpha \in {\mathrm{\Phi }}^{+}$. (The Verma irreducibility criterion from Shapovalov determinants)

Let $Z=Z(U(\mathfrak{g}))$, let $\chi :Z\to \mathbb{C}$ be a unital complex-algebra character, and put ${\mathfrak{m}}_{\chi }=\ker \chi$. For $M\in \mathcal{O}$, the generalized central-character submodule is $M_{\chi }=\{v\in M:{\mathfrak{m}}_{\chi }^{N}v=0\text{ for some }N\ge 1\}$, and ${\mathcal{O}}_{\chi }$ consists of the objects with $M=M_{\chi }$. (Generalized central-character subcategories)

The simple objects of $\mathcal{O}$ are exactly the modules $L(\mu )$, $\mu \in {\mathfrak{h}}^{\ast }$, and $L(\mu )\cong L(\nu )$ if and only if $\mu =\nu$. (The simple objects of O)

If a cyclic highest-weight module has highest weight $\mu$, then every $z\in Z(U(\mathfrak{g}))$ acts on it by the scalar $\mathrm{pr}(z)(\mu )={\chi }_{\mu }(z)$. (The Harish-Chandra projection computes the highest-weight scalar)

The highest-weight central characters satisfy ${\chi }_{\mu }={\chi }_{\lambda }$ if and only if $\mu \in W\cdot \lambda$. (Central characters are dot-Weyl orbits)

Every object of $\mathcal{O}$ has a finite composition series and is both Noetherian and Artinian. The length of zero is zero. (Every object of O has finite length)