TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Generalized central-character decomposition of O

Statement

The category is the categorical direct sum $O = ⨁_{χ} O_{χ}$ : objects have finite support in the index $χ$ , morphisms between distinct components vanish, and the canonical component projections are exact.

Facts & Assumptions

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

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every $M \in O$ decomposes canonically into finitely many nonzero generalized central-character submodules: $M = ⨁_{χ} M_{χ} .$ For each summand there is a single $N \geq 1$ such that $m_{χ}^{N} M_{χ} = 0$ . The decomposition of zero is empty. (Generalized central-character summands)

Proof

1.1F1

Use the finite intrinsic decomposition of each object. If $f : M \to N$ is a module map, $m_{χ}^{N} v = 0$ implies $m_{χ}^{N} f (v) = 0$ , so $f (M_{χ}) \subset N_{χ}$ . Consequently all off-diagonal components of a map vanish, and maps between objects decompose uniquely into their same-character components.

2.1F1algebrastep 1.1

For an exact sequence $0 \to A \to B \to C \to 0$ , the image and kernel equalities restrict to each character. In particular, to lift $c \in C_{χ}$ , lift it to $b \in B$ and decompose $b = \sum_{ψ} b_{ψ}$ . Preservation of characters and the direct decomposition of $C$ imply that $b_{χ}$ maps to $c$ . This proves surjectivity and hence exactness of every projection.

3.1F1algebrastep 2.1∎

The functor taking a finitely supported family to its direct sum and the functor $M \mapsto (M_{χ})_{χ}$ are inverse up to the canonical isomorphisms. An empty family gives zero, and a single nonzero component is fixed by its projection.

Depends on

Generalized central-character summands

Used by

The residue translation functors on the sum of parabolic blocks Definition
The singular integral block and its dot stabilizer Definition
Translation functors by tensoring and projection Definition
A singular integral A2 central-character summand Example
Nonintegral sl2 central characters split into two simple blocks Example
The regular integral sl2 block Example
Central-character cuts of a typed module are typed by the matching weights Lemma
Exact projections onto linkage blocks preserve projectives Lemma
The regular coset-dominant Harish–Chandra tensor comparison Lemma
Weak BGG resolution of the trivial module Lemma
Translation functors are exact and biadjoint Proposition
Every object of O has finite length Theorem
Parabolic category O is a highest weight category Theorem
Translation to and from a single wall on standard modules Theorem
Translation to the wall on Verma, simple and projective objects Theorem
Weak BGG resolution Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

§6 Theorem 6.2(1), pp.9–10 (standard reference, not scraped)