Truncation at a finite downward-closed ideal of a linkage class

Assume the Axiom of Choice (The Axiom of Choice). Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$ with the conventions of The classical BGG category O: positive roots ${\mathrm{\Phi }}^{+}$, simple roots ${\alpha }_i$, $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, the root order $\mu \le \lambda$ meaning $\lambda -\mu \in Q^{+}$, Weyl vector $\rho$, the dot action $w\cdot \lambda =w(\lambda +\rho )-\rho$, and category $\mathcal{O}$.

For a weight $\lambda$ let $C=W_{\lambda }\cdot \lambda$ be its integral-reflection linkage class in the sense of The integral Weyl group of a weight; it is contained in the full dot orbit $W\cdot \lambda$, hence finite because $W$ is finite (The Weyl group is finite and faithful, with the identification of the abstract reflections with the $s_{\alpha }$ of The roots form a reduced crystallographic Euclidean root system). A finite downward-closed ideal of $C$ is a finite subset $\mathrm{\Gamma }\subseteq C$ such that $\nu \in \mathrm{\Gamma }\text{ and }\mu \in C\text{ and }\mu \le \nu ⟹\mu \in \mathrm{\Gamma },$ where $\le$ is the root order of The classical BGG category O and not the strong linkage order $\uparrow$ of The strong linkage order on weights. It is a lower set in the restriction of the partial order $\le$ to $C$. The empty ideal is allowed; every nonempty such ideal has minimal elements.

For such a $\mathrm{\Gamma }$, the truncation ${\mathcal{O}}_{\mathrm{\Gamma }}$ is the full subcategory of $\mathcal{O}$ whose objects are those $X$ all of whose simple composition factors are $L(\mu )$ with $\mu \in \mathrm{\Gamma }$; composition factors are those of Composition series and composition factors of an object and the simple objects of $\mathcal{O}$ are the $L(\mu )$ of The simple objects of O. Because every object of $\mathcal{O}$ has finite length (Every object of O has finite length) and composition factors of a composition series are independent of the chosen series (Jordan-Holder theorem in an abelian category), membership in ${\mathcal{O}}_{\mathrm{\Gamma }}$ depends only on the isomorphism class of the object and not on a chosen composition series. Consequently ${\mathcal{O}}_{\mathrm{\Gamma }}$ contains the zero object and is closed in $\mathcal{O}$ under finite direct sums, subobjects, quotients and extensions (Category O is abelian and extension closed among weight modules); it is the truncation of the finite label poset of one linkage class.

The Verma-placement claim below also has a direct justification. The highest weight line of $M(\mu )$ is one-dimensional and generates the whole module; hence it cannot be distributed among two nonzero direct summands. The linkage-block decomposition of Central-character summands refine into linkage blocks therefore places this Verma in the block of its unique simple quotient $L(\mu )$ (A Verma module has a unique simple quotient), so all its composition labels lie in $C$. A label $\eta$ of a composition factor is a weight of $M(\mu )$: in a short exact sequence of $\mathfrak{h}$-semisimple modules, a weight vector in the quotient lifts in that same weight by extracting that component of any finite weight decomposition of a lift. Iterating through a composition series and using Weights of a Verma module lie below lambda gives $\eta \le \mu$. Thus $\mu \in \mathrm{\Gamma }$ and lower closure within $C$ force every such $\eta \in \mathrm{\Gamma }$, proving $M(\mu )\in {\mathcal{O}}_{\mathrm{\Gamma }}$ without a bound by one greatest label.

Two boundaries are part of the definition. First, ${\mathcal{O}}_{\mathrm{\Gamma }}$ is a condition on the highest-weight labels of simple composition factors, not a bound on all weights of a Verma module: a Verma module $M(\mu )$ with $\mu \in \mathrm{\Gamma }$ lies in ${\mathcal{O}}_{\mathrm{\Gamma }}$ although its weights run over the whole cone $\mu -Q^{+}$. Second, $\mathrm{\Gamma }$ is a finite ideal inside a single linkage class $C$, not the infinite lower ideal generated by $\lambda$ in the whole weight lattice; in particular a discrete series truncation is a different construction.