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The Euler-character identity for a finite-dimensional simple module

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $λ \in Λ^{+}$ . In the Grothendieck group of the linkage block of $λ$ (The Grothendieck group and character of O, Simple and standard bases of K0(O)) the finite alternating sum of Verma classes equals the class of the simple module:

$[L (λ)] = \sum_{w \in W} (- 1)^{ℓ (w)} [M (w \circ λ)] .$

Equivalently, applying the character homomorphism and the Verma character $ch M (μ) = e^{μ} \prod_{α \in Φ^{+}} (1 - e^{- α})^{- 1}$ (The formal character of a Verma module),

$ch L (λ) = \sum_{w \in W} (- 1)^{ℓ (w)} e^{w \circ λ} \prod_{α \in Φ^{+}} (1 - e^{- α})^{- 1},$

the Weyl numerator identity in the form needed by the Weyl character formula. Proof: an exact finite complex has vanishing alternating sum of classes.

Facts & Assumptions

Given: The Axiom of Choice, a dominant integral weight $λ \in Λ^{+}$ , the BGG resolution of $L (λ)$ , and the Grothendieck group of the linkage block of $λ$ with its character homomorphism.

[F1]

$0 \to C_{∣ Φ^{+} ∣} (λ) \to \dots \to C_{1} (λ) \to C_{0} (λ) \to L (λ) \to 0$ is an exact sequence in $O$ , and $C_{i} (λ) = ⨁_{ℓ (w) = i} M (w \circ λ)$ (The BGG resolution of a finite-dimensional simple module, The Bruhat graph and the BGG Verma sum in degree k).

[F2]

The Grothendieck group $K (O)$ is the abelian group with generators the classes of objects and relations $[B] = [A] + [C]$ for every short exact sequence $0 \to A \to B \to C \to 0$ ; consequently an exact sequence $0 \to A_{n} \to \dots \to A_{0} \to B \to 0$ gives $[B] = \sum_{i = 0}^{n} (- 1)^{i} [A_{i}]$ , and $[A \oplus B] = [A] + [B]$ . The classes of the simple modules $[L (μ)]$ and of the Verma modules $[M (μ)]$ each form a basis of $K (O)$ (The Grothendieck group and character of O, Simple and standard bases of K0(O)).

[F3]

The formal character $ch$ is additive on exact sequences and hence defines a homomorphism from $K (O)$ to the group of formal characters; $ch M (μ) = e^{μ} \prod_{α \in Φ^{+}} (1 - e^{- α})^{- 1}$ (The formal character of a Verma module, The Grothendieck group and character of O).

[F4]

All $M (w \circ λ)$ and $L (λ)$ lie in the linkage block of $λ$ ; the block decomposition splits $O$ into a direct sum of subcategories, and the corresponding projection of Grothendieck groups is additive on classes. Hence an identity between classes of objects of the block that holds in $K (O)$ holds in the Grothendieck group of the block (Central-character summands refine into linkage blocks, The Grothendieck group and character of O).

Proof

1.1F1F2F4

The resolution of [F1] is a finite exact sequence $0 \to C_{∣ Φ^{+} ∣} (λ) \to \dots \to C_{0} (λ) \to L (λ) \to 0$ . By the additivity of [F2] applied successively to its short exact sequences, $[L (λ)] = \sum_{i = 0}^{∣ Φ^{+} ∣} (- 1)^{i} [C_{i} (λ)]$ ; by the direct-sum rule and [F1], $[C_{i} (λ)] = \sum_{ℓ (w) = i} [M (w \circ λ)]$ . Substituting gives $[L (λ)] = \sum_{w \in W} (- 1)^{ℓ (w)} [M (w \circ λ)]$ . All the modules involved lie in the linkage block of $λ$ , so by [F4] this identity holds in the Grothendieck group of that block.

2.1F3step 1.1

Applying the character homomorphism of [F3] to the identity of step 1.1 and using the Verma character gives $ch L (λ) = \sum_{w \in W} (- 1)^{ℓ (w)} ch M (w \circ λ) = \sum_{w \in W} (- 1)^{ℓ (w)} e^{w \circ λ} \prod_{α \in Φ^{+}} (1 - e^{- α})^{- 1}$ , the common factor $\prod_{α \in Φ^{+}} (1 - e^{- α})^{- 1}$ being independent of $w$ .

3.1step 1.1step 2.1∎

Steps 1.1 and 2.1 are exactly the two asserted identities: the alternating sum of Verma classes in the Grothendieck group of the linkage block, and the Weyl numerator form of the character identity.

Depends on

Used by

The BGG resolution has length the number of positive roots Corollary

Dependency tree · two levels

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Sources

Fan Zhou, The classical and the functorial BGG resolutions (Columbia thesis 2021), Part I Sec. 3.3, pp. 11-13 (standard reference, not scraped)
P. Etingof, Lie Groups and Lie Algebras II (18.755), Sec. 26.1 and Sec. 26.3 (Weyl character formula, Theorem 26.4), pp. 139-142 (standard reference, not scraped)