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Kostant's nilradical cohomology theorem

Statement

Assume the Axiom of Choice. Let $g$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $h$ , chosen positive system $Φ^{+}$ and $n^{+} = ⨁_{α > 0} g_{α}$ , and let $λ \in Λ^{+}$ be dominant integral with $V = L (λ)$ . Then for every $k \geq 0$ there is an isomorphism of $h$ -modules $H^{k} (n^{+}, V) ≅ ⨁_{w \in W : ℓ (w) = k} C_{w \cdot λ},$ where $C_{μ}$ is the one-dimensional $h$ -module of weight $μ$ and $w \cdot λ = w (λ + ρ) - ρ$ . Equivalently, $H^{k} (n^{+}, V)$ is multiplicity free with weights exactly the pairwise distinct weights $w \cdot λ$ over the length- $k$ elements, and $\dim H^{k} (n^{+}, V) = # {w \in W : ℓ (w) = k}$ .

Facts & Assumptions

Given: The Axiom of Choice; the finite-dimensional data $(g, h, Φ^{+}, λ, V = L (λ))$ and the operators $d, δ, □$ of The Chevalley–Eilenberg Laplacian is scalar on weight components.

[F1]

The harmonic projection identifies $H^{k} (n^{+}, V)$ with the harmonic cochains $\ker □ \cap C^{k} (n^{+}, V)$ , and these are spanned by the extremal cochains: $\ker □ \cap C^{k} = ⨁_{ℓ (w) = k} C γ_{w}$ (Each extremal harmonic space is one-dimensional).

[F2]

Each $γ_{w}$ is a nonzero cocycle of weight $w \cdot λ = w (λ + ρ) - ρ$ (The extremal weight cochain of a Weyl element is closed and unique).

[F3]

The identifications $H^{k} ≅ \ker □ \cap C^{k}$ and the direct sum are isomorphisms of $h$ -modules: the $h$ -action of The normalizer acts on Lie algebra cohomology commutes with $d$ , the harmonic projection is obtained from the $h$ -stable subspaces $im d$ and $\ker □$ , and the displayed identity for $□$ in The Chevalley–Eilenberg Laplacian is scalar on weight components exhibits $□$ as a polynomial in operators $Θ (H)$ that commute with one another, so $□$ commutes with the $h$ -action as well.

[F4]

The dot weights are pairwise distinct: $λ + ρ$ is regular, with trivial stabilizer, by (Positive coroot pairings of a dominant integral weight, Integral, dominant, and strictly dominant weights, The Weyl vector rho for a chosen positive system); and each weight space $V_{w λ}$ is one-dimensional (Extremal Weyl-orbit weights, Highest-weight classification, Weight and weight space, Length and longest Weyl-group element, Chevalley–Eilenberg cochains, Lie algebra cohomology).

Proof

technique · combine the one-dimensional harmonic spaces with the weight computation for the extremal cochains

1.1F1F3

By [F1] the space $H^{k} (n^{+}, V)$ is isomorphic to $⨁_{ℓ (w) = k} C γ_{w}$ , and by [F3] this isomorphism is $h$ -linear.

2.1F2F4step 1.1

Each summand $C γ_{w}$ is the one-dimensional $h$ -module of weight $w \cdot λ$ by [F2], and the weights $w \cdot λ$ for distinct Weyl elements $w$ are pairwise distinct by the regularity of $λ + ρ$ recorded in [F4]. Hence the direct sum in step 1.1 is exactly $⨁_{ℓ (w) = k} C_{w \cdot λ}$ as an $h$ -module.

3.1F4step 2.1∎

Counting dimensions in step 2.1 gives $\dim H^{k} (n^{+}, V) = # {w \in W : ℓ (w) = k}$ and shows that the multiplicity of every occurring weight is one; in particular $H^{k} (n^{+}, V)$ is multiplicity free with the stated weights, and all groups are finite dimensional because $W$ is finite.

Depends on

Used by

Kostant cohomology in degrees zero and top Corollary
The Kostant Euler character recovers the Weyl numerator Corollary
Degree-one Kostant classes correspond to simple reflections Example
Kostant cohomology for sl2 Example
Kostant cohomology for the trivial sl3 module Example
Kostant cohomology and BGG characters give the same Weyl numerator Proposition

Dependency tree · two levels

83 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

Faisal Al-Faisal, On the Representation Theory of Semisimple Lie Groups (University of Waterloo MMath thesis, 2010), §3.4 printed pp.73–77 (standard reference, not scraped)
Peter Woit, Lie Algebra Cohomology and the Borel–Weil–Bott Theorem (Math G4344, Spring 2012), printed pp.1–7 (standard reference, not scraped)