The regular integral block and its antidominant weight

Assume the Axiom of Choice, inherited from the published block theory (Central-character summands refine into linkage blocks, Standard and costandard objects, The Axiom of Choice).

Work in category $\mathcal{O}$ of a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$ with the conventions of The classical BGG category O: a Cartan subalgebra $\mathfrak{h}$, a positive system with simple roots ${\alpha }_i$, the Weyl group $W$ of Root reflections and the Weyl group action, the Weyl vector $\rho$, and the dot action $w\cdot \lambda =w(\lambda +\rho )-\rho$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$ and $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$.

The fixed weight. Fix a weight $\lambda$ which is integral, dot-regular and dominant: $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\in \mathbb{Z}$ for every root $\alpha$ (so the integral Weyl group $W_{\lambda }$ of The integral Weyl group of a weight is all of $W$), and $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\mathrm{\ne }0$, respectively $>0$, for every root, respectively every positive root (Dot-Weyl facets and single-wall translation data). The set ${\mathrm{\Phi }}_{\lambda }$ of The integral Weyl group of a weight is therefore the full root system and $W_{\lambda }=W$.

The block. The regular point $\lambda +\rho$ lies in an open chamber. An element fixing it preserves that chamber, so Simple transitivity on Weyl chambers makes its stabilizer trivial; hence the linkage class $C=W_{\lambda }\cdot \lambda =W\cdot \lambda$ has exactly $\mathrm{\mid }W\mathrm{\mid }$ elements. The regular integral block is the full subcategory ${\mathcal{O}}_{\lambda }:={\mathcal{O}}_C$ of objects of $\mathcal{O}$ whose simple composition factors have labels in $C$, as in Central-character summands refine into linkage blocks; it is nonzero, abelian and extension-closed as a block, and $K_0({\mathcal{O}}_{\lambda })$ is the subgroup of $K_0(\mathcal{O})$ of The Grothendieck group and character of O spanned by the classes of objects of the block.

Its labelled objects. For $w\in W$ define the standard and costandard objects and the simple object by $\mathrm{\Delta }(w\cdot \lambda ):=M(w\cdot \lambda ),\mathrm{\nabla }(w\cdot \lambda ):=D(M(w\cdot \lambda )),L(w\cdot \lambda ):=M(w\cdot \lambda )\mathrm{/}J(w\cdot \lambda ),$ where $M$ is the Verma module of Verma modules, $D$ the restricted dual of Standard and costandard objects, and $J(w\cdot \lambda )$ the sum of all proper submodules, so that $L(w\cdot \lambda )$ is the unique simple quotient of $\mathrm{\Delta }(w\cdot \lambda )$ by A Verma module has a unique simple quotient. By The simple objects of O the simple objects of ${\mathcal{O}}_{\lambda }$ are exactly the $L(w\cdot \lambda )$ for $w\in W$, pairwise non-isomorphic. Let $P(w\cdot \lambda )$ denote a projective cover of $L(w\cdot \lambda )$, which may be chosen inside the linkage class of $w\cdot \lambda$ and is unique up to isomorphism and indecomposable (Category O has enough projectives); every such $P(w\cdot \lambda )$ carries a finite Verma flag (Projectives in category O have finite Verma flags, Finite Verma flags and their multiplicities).

Triangularity of the Verma-flag matrix. For all $v,w\in W$, $[\mathrm{\Delta }(w\cdot \lambda ):L(v\cdot \lambda )]\mathrm{\ne }0⟹v\cdot \lambda \le w\cdot \lambda ,[\mathrm{\Delta }(w\cdot \lambda ):L(w\cdot \lambda )]=1,$ where the multiplicity is the composition multiplicity of Standard and costandard objects. Indeed $(P(v\cdot \lambda ):\mathrm{\Delta }(w\cdot \lambda ))=[\mathrm{\Delta }(w\cdot \lambda ):L(v\cdot \lambda )]$ by BGG reciprocity and $(P(v\cdot \lambda ):\mathrm{\Delta }(w\cdot \lambda ))\mathrm{\ne }0$ forces $w\cdot \lambda -v\cdot \lambda \in Q^{+}$ while $(P(v\cdot \lambda ):\mathrm{\Delta }(v\cdot \lambda ))=1$ by The triangular restriction on projective Verma flags.

The antidominant weight. Let $w_0\in W$ be the longest element. Since $⟨w_0\cdot \lambda +\rho ,{\alpha }^{\vee }⟩=⟨\lambda +\rho ,w_0^{-1}{\alpha }^{\vee }⟩<0$ for every positive root $\alpha$, the antidominant Verma module $\mathrm{\Delta }(w_0\cdot \lambda )$ is simple (Antidominant regular Verma modules are simple), so $\mathrm{\Delta }(w_0\cdot \lambda )=L(w_0\cdot \lambda )$ and this is the unique standard object of the block that is simple: for $w\mathrm{\ne }{w}_0$ the regular weight $w\cdot \lambda$ is not antidominant, so some positive root $\alpha$ has $⟨w\cdot \lambda +\rho ,{\alpha }^{\vee }⟩\in {\mathbb{Z}}_{>0}$, and Verma embedding for an arbitrary positive root embeds $M(s_{\alpha }\cdot (w\cdot \lambda ))$ as a proper submodule of $\mathrm{\Delta }(w\cdot \lambda )$. The weight $w_0\cdot \lambda$ is called the antidominant weight of the block and $\mathrm{\Delta }(w_0\cdot \lambda )$ its antidominant Verma module.

Grothendieck group. By Simple and standard bases of K0(O) applied to the finite label set $C=W\cdot \lambda$, both the classes $[\mathrm{\Delta }(w\cdot \lambda )]$ and the classes $[L(w\cdot \lambda )]$, $w\in W$, form $\mathbb{Z}$-bases of $K_0({\mathcal{O}}_{\lambda })$, and the transition matrix between them is unitriangular for any linear order of $W$ extending the order on the dot orbit induced by $\le$.

Every statement above uses the published block, Verma, duality and projective-cover theory named in the citations, and that theory assumes the Axiom of Choice; the definition itself selects nothing further.