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Kostant cohomology and BGG characters give the same Weyl numerator

Statement

Assume the Axiom of Choice and let $λ \in Λ^{+}$ . Put $D = \prod_{α \in Φ^{+}} (1 - e^{- α})$ and $N_{λ} = \sum_{w \in W} (- 1)^{ℓ (w)} e^{w \cdot λ}$ . In the Grothendieck group of the linkage block $O_{λ}$ , the BGG resolution gives $[L (λ)] = \sum_{w \in W} (- 1)^{ℓ (w)} [M (w \cdot λ)]$ by The BGG resolution of a finite-dimensional simple module. Applying its formal-character map and the Verma character formula The formal character of a Verma module gives $ch L (λ) = D^{- 1} N_{λ}$ . Separately, the spaces $H^{k} (n^{+}, L (λ))$ are finite-dimensional $h$ -modules, and Kostant's nilradical cohomology theorem gives $\sum_{k \geq 0} (- 1)^{k} {ch}_{h} H^{k} (n^{+}, L (λ)) = N_{λ}$ , as recorded in The Kostant Euler character recovers the Weyl numerator. Thus the character calculations identify the same finite Weyl numerator after clearing the Verma denominator in the BGG character formula. No equality between classes in different Grothendieck groups is asserted, and no spectral sequence is constructed.

Facts & Assumptions

Given: The Axiom of Choice; $λ \in Λ^{+}$ ; the linkage block $O_{λ}$ with its Grothendieck group and formal-character map; the finite sums $D$ and $N_{λ}$ .

[F1]

In the Grothendieck group of the linkage block, $[L (λ)] = \sum_{w} (- 1)^{ℓ (w)} [M (w \cdot λ)]$ , and applying the character homomorphism with the Verma character $ch M (μ) = e^{μ} \prod_{α > 0} (1 - e^{- α})^{- 1}$ gives $ch L (λ) = \sum_{w} (- 1)^{ℓ (w)} e^{w \cdot λ} D^{- 1} = D^{- 1} N_{λ}$ (The Euler-character identity for a finite-dimensional simple module, The BGG resolution of a finite-dimensional simple module, The Bruhat graph and the BGG Verma sum in degree k, The Grothendieck group and character of O, The formal character of a Verma module, The classical BGG category O, Verma and finite-dimensional weight modules belong to O).

[F2]

Independently of [F1], the Kostant decomposition computes the alternating sum of the finite-dimensional $h$ -module characters of the nilradical cohomology: $\sum_{k} (- 1)^{k} ch H^{k} (n^{+}, L (λ)) = \sum_{w} (- 1)^{ℓ (w)} e^{w \cdot λ} = N_{λ}$ (The Kostant Euler character recovers the Weyl numerator, Kostant's nilradical cohomology theorem, Integral, dominant, and strictly dominant weights).

Proof

technique · compute the numerator twice, once from the BGG class and once from the cohomology decomposition, and compare after clearing the denominator

1.1F1

The BGG side: [F1] gives $ch L (λ) = D^{- 1} N_{λ}$ in the formal-character target of the category- $O$ character map, where $D = \prod_{α > 0} (1 - e^{- α})$ is the Verma denominator.

1.2F2

The cohomological side: by [F2] the alternating sum of the characters of the finite-dimensional $h$ -modules $H^{k} (n^{+}, L (λ))$ equals the same finite numerator $N_{λ}$ , computed directly from the cohomology decomposition and using no Weyl-character input.

2.1F1F2step 1.1step 1.2∎

Clearing the common denominator $D$ in step 1.1 and comparing with step 1.2 identifies the same finite sum $N_{λ}$ : $D \cdot D^{- 1} N_{λ} = N_{λ} = \sum_{k} (- 1)^{k} ch H^{k} (n^{+}, L (λ))$ . Both sides live in the common completed formal-character target after this clearing, and no equality of classes in different Grothendieck groups is used: the BGG class lives in $K_{0} (O_{λ})$ , while the cohomology spaces are finite-dimensional $h$ -modules and their alternating character is computed there. No spectral sequence is constructed.

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Sources

Peter Woit, Lie Algebra Cohomology and the Borel–Weil–Bott Theorem (Math G4344, Spring 2012), printed pp.1–7 (standard reference, not scraped)
Faisal Al-Faisal, On the Representation Theory of Semisimple Lie Groups (University of Waterloo MMath thesis, 2010), §3.5 printed pp.77–84 (standard reference, not scraped)