The regular integral sl2 block

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 $w\cdot \lambda =w(\lambda +\rho )-\rho$. Restricted Chevalley duality is an exact contravariant equivalence $D:\mathcal{O}\to {\mathcal{O}}^{\mathrm{o}\mathrm{p}}$, with a natural isomorphism $D^2\cong \mathrm{id}$. It preserves each weight-space dimension, the formal character, and every simple composition multiplicity. (Restricted duality is exact and involutive on 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 $w\cdot \lambda =w(\lambda +\rho )-\rho$. The costandard object $\mathrm{\nabla }(\lambda )$ has a unique simple submodule, isomorphic to $L(\lambda )$. Its socle, the sum of all simple submodules, is that submodule. (The simple socle of a costandard object)

If $⟨\lambda +\rho ,{\alpha }_i^{\vee }⟩\in {\mathbb{Z}}_{>0}$, there is an embedding $M(s_i\cdot \lambda )\hookrightarrow M(\lambda )$. (Simple-reflection embeddings of Verma modules)

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)

$M(\lambda )$ 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)

The weights of $M(\lambda )$ are exactly $\lambda -\beta$ for $\beta \in Q^{+}$; every weight space is finite dimensional, and $M(\lambda {)}_{\lambda }=\mathbb{C}{v}_{\lambda }$. (Weights of a Verma module lie below lambda)

For every highest weight $\eta$, $D(L(\eta ))\cong L(\eta )$ as $\mathfrak{g}$-modules. (Restricted self-duality of simple highest-weight modules)

The standard and costandard objects are $\mathrm{\Delta }(\eta )=M(\eta )$ and $\mathrm{\nabla }(\eta )=D(M(\eta ))$. (Standard and costandard objects)