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The Kostant Euler character recovers the Weyl numerator

Statement

Assume the Axiom of Choice. In the setting of Kostant's nilradical cohomology theorem, define the formal character of a finite-dimensional $h$ -module by $ch M = \sum_{μ} \dim (M_{μ}) e^{μ}$ . Then $\sum_{k \geq 0} (- 1)^{k} ch H^{k} (n^{+}, V) = \sum_{w \in W} (- 1)^{ℓ (w)} e^{w \cdot λ} .$ The right-hand side is the finite alternating character computed directly from Kostant's decomposition. The separate BGG/Weyl-character identity at The Euler-character identity for a finite-dimensional simple module identifies this same finite sum as the Weyl numerator. Thus the cohomological computation of the sum itself does not use Weyl-character input; its interpretation as the numerator is the comparison supplied by that identity.

Facts & Assumptions

Given: The setting of Kostant's nilradical cohomology theorem and the finite sums $\sum_{μ} \dim (M_{μ}) e^{μ}$ of formal exponentials for finite-dimensional $h$ -modules $M$ .

[L1]

$H^{k} (n^{+}, V) ≅ ⨁_{ℓ (w) = k} C_{w \cdot λ}$ as $h$ -modules, so each $H^{k}$ is finite dimensional, multiplicity free, and has weights exactly $w \cdot λ$ for the length- $k$ elements (Kostant's nilradical cohomology theorem).

[L2]

For a finite-dimensional module, the formal character is the finite sum of $e^{μ}$ over the weights with multiplicity, and distinct weight spaces contribute distinct monomials (Weight and weight space, The Grothendieck group and character of O).

[L3]

The BGG identity $ch L (λ) = \sum_{w} (- 1)^{ℓ (w)} e^{w \circ λ} \prod_{α > 0} (1 - e^{- α})^{- 1}$ holds in the formal-character ring, with the Verma character supplied by The formal character of a Verma module; it is a comparison identity, not an input to the computation below (The Euler-character identity for a finite-dimensional simple module).

Proof

technique · substitute the multiplicity-free weight list into the alternating sum

1.1L1L2

By [L1] and [L2] the formal character of $H^{k} (n^{+}, V)$ is $\sum_{ℓ (w) = k} e^{w \cdot λ}$ , the sum of one monomial for each Weyl element of length $k$ , with no multiplicities and no other terms.

2.1L1step 1.1

Substituting step 1.1 into the alternating sum and interchanging the two finite sums over $k$ and $w$ gives $\sum_{k \geq 0} (- 1)^{k} ch H^{k} (n^{+}, V) = \sum_{k \geq 0} (- 1)^{k} \sum_{ℓ (w) = k} e^{w \cdot λ} = \sum_{w \in W} (- 1)^{ℓ (w)} e^{w \cdot λ}$ , the displayed finite identity; the sums are finite because the cochain complex vanishes above $\dim n^{+} = ∣ Φ^{+} ∣$ and $W$ is finite.

3.1L3step 2.1∎

The computation in step 2.1 used only the cohomology decomposition of [L1], not the Weyl character formula; the separate identity [L3] is what names the resulting finite sum $\sum_{w} (- 1)^{ℓ (w)} e^{w \cdot λ}$ as the Weyl numerator after clearing the Verma denominator, and no spectral sequence or BGG input enters the computation itself.

Depends on

Used by

Kostant cohomology and BGG characters give the same Weyl numerator Proposition

Dependency tree · two levels

39 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

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)