TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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BGG reciprocity

Statement

Assume the Axiom of Choice (The Axiom of Choice). For all weights $λ$ and $μ$ , $(P (λ) : Δ (μ)) = [Δ (μ) : L (λ)] = [M (μ) : L (λ)],$ where $P (λ)$ is the projective cover of $L (λ)$ , the left-hand multiplicity is the Verma-flag multiplicity of Projectives in category O have finite Verma flags, and the right-hand multiplicity is the simple composition multiplicity of the Verma module $M (μ)$ (Standard and costandard objects).

Facts & Assumptions

Given: The Axiom of Choice, weights $λ, μ$ , the projective cover $P (λ)$ of $L (λ)$ , and the costandard object $\nabla (μ) = D (M (μ))$ .

[F1]

$P (λ)$ is Verma-filtered, so $\dim_{C} {Hom}_{O} (P (λ), \nabla (μ)) = (P (λ) : Δ (μ))$ and ${Ext}_{O}^{1} (P (λ), \nabla (μ)) = 0$ (Projectives in category O have finite Verma flags, Hom to costandards counts Verma-flag factors).

[F2]

For every finite-length object $X$ one has $\dim_{C} {Hom}_{O} (P (λ), X) = [X : L (λ)]$ (Hom from a projective counts simple composition factors).

[F3]

Restricted duality $D$ is an exact contravariant involution preserving composition multiplicities and $D (L (λ)) ≅ L (λ)$ ; hence $[\nabla (μ) : L (λ)] = [D (M (μ)) : L (λ)] = [M (μ) : L (λ)]$ (Restricted duality is exact and involutive on O, Restricted self-duality of simple highest-weight modules).

Proof

technique · direct: convert the flag multiplicity into a Hom dimension, then count with the projective-cover Hom formula and duality

1.1F1given

By [F1], $(P (λ) : Δ (μ)) = \dim_{C} {Hom}_{O} (P (λ), \nabla (μ))$ .

1.2F2F3given

Since $\nabla (μ) = D (M (μ))$ is an object of $O$ of finite length, [F2] gives $\dim_{C} {Hom}_{O} (P (λ), \nabla (μ)) = [\nabla (μ) : L (λ)]$ , and by [F3] this equals $[M (μ) : L (λ)]$ .

2.1step 1.1step 1.2∎

Combining steps 1.1 and 1.2 gives $(P (λ) : Δ (μ)) = [M (μ) : L (λ)]$ ; since $Δ (μ) = M (μ)$ , the middle and right multiplicities agree, so all three quantities are equal.

Depends on

Used by

Injectives have costandard filtrations Corollary
The triangular restriction on projective Verma flags Corollary
A Verma module need not be projective in its block Counterexample
The sl2 reciprocity matrices Example
The two projectives in the principal sl2 block Example

Dependency tree · two levels

29 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

Lin Chen, lecture notes (Spring 2024), Lecture 9, Theorem 2.2 and its proof (standard reference, not scraped)
Pavel Etingof, Representations of Lie Groups (18.757, Fall 2023), Theorem 20.6 (BGG reciprocity) (standard reference, not scraped)