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The Casselman–Osborne constraint on weights of nilradical cohomology

Statement

Assume the Axiom of Choice. Let $g$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $h$ , positive system $Φ^{+}$ , $n^{+} = ⨁_{α \in Φ^{+}} g_{α}$ , $λ \in Λ^{+}$ dominant integral, and $V = L (λ)$ the finite-dimensional irreducible module of highest weight $λ$ (Highest-weight classification). If $μ \in h^{*}$ is an $h$ -weight of $H^{p} (n^{+}, V)$ for some $p \geq 0$ , then $χ_{μ} = χ_{λ}$ , where $χ_{ν} ⁣ : Z (U (g)) \to C$ is the central character $z \mapsto ν (pr (z))$ attached to the highest weight $ν$ . Equivalently, $μ \in W \cdot λ = {w (λ + ρ) - ρ : w \in W}$ by Central characters are dot-Weyl orbits. This constrains every degree independently of the harmonic computation below.

Facts & Assumptions

Given: The Axiom of Choice; the finite-dimensional $g$ -module $V = L (λ)$ ; a central element $z \in Z (U (g))$ ; and a class $ω \in H^{p} (n^{+}, V)$ of $h$ -weight $μ$ .

[F1]

The module $V$ is finite-dimensional and irreducible with highest weight $λ$ , and every central element acts on a cyclic highest-weight module by a scalar; for the highest vector $v_{λ}$ one has $z v_{λ} = pr (z) (λ) v_{λ}$ (Highest-weight classification, Central elements act by scalars on cyclic highest-weight modules, The Harish-Chandra projection computes the highest-weight scalar, The Harish-Chandra projection).

[F2]

A central character of a module $M$ is a unital algebra homomorphism $χ : Z (U (g)) \to C$ such that every $z$ acts by $χ (z) {id}_{M}$ (Central character of a Lie algebra module).

[F3]

For every $z \in Z (U (g))$ , every $p \geq 0$ and every class $ω \in H^{p} (n^{+}, V)$ , $z \cdot ω = pr (z) \cdot ω$ , where the left side multiplies cochain values by $z$ and the right side is the $h$ -action of the normalizer-action proposition applied to $pr (z) \in S (h)$ (Central actions on nilradical cohomology factor through the Harish–Chandra projection, The normalizer acts on Lie algebra cohomology).

[F4]

The cohomology $H^{p} (n^{+}, V)$ is an $h$ -module, so its $h$ -weight spaces are defined; for a weight vector of weight $μ$ , every $h \in h$ acts by the scalar $μ (h)$ , hence every polynomial $f \in S (h)$ acts by $μ (f)$ (The normalizer acts on Lie algebra cohomology, Weight and weight space, Finite-dimensional modules decompose into weight spaces).

[F5]

The central characters obtained from highest weights are equal exactly on dot-Weyl orbits: $χ_{ν} = χ_{ξ}$ if and only if $ξ \in W \cdot ν$ (Central characters are dot-Weyl orbits, Integral, dominant, and strictly dominant weights, The Harish-Chandra projection is multiplicative on the center).

Proof

technique · compare the central action on the coefficient module with the Cartan action on a cohomology weight vector

1.1F1F2

The central element $z$ acts on the whole finite-dimensional irreducible module $V$ by the scalar $χ_{λ} (z) = pr (z) (λ)$ : by [F1] it acts on the highest vector by that scalar, and cyclicity propagates the scalar to every vector because $z$ commutes with the action of $U (g)$ .

2.1F3F4step 1.1

On the other hand, step 1.1 identifies the left-hand action of $z$ on cochain values with the scalar $χ_{λ} (z)$ , so for the cohomology class $ω$ of weight $μ$ the identity of [F3] gives $χ_{λ} (z) ω = z \cdot ω = pr (z) \cdot ω$ , while the $h$ -action of the polynomial $pr (z) \in S (h)$ on the weight vector $ω$ is multiplication by the scalar $μ (pr (z)) = χ_{μ} (z)$ by [F4].

3.1F5step 2.1∎

Since $ω \neq 0$ , step 2.1 forces $χ_{λ} (z) = χ_{μ} (z)$ for every $z \in Z (U (g))$ , that is $χ_{λ} = χ_{μ}$ . By [F5] equality of central characters is equivalent to $μ \in W \cdot λ$ , which is the displayed reformulation.

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Sources

Faisal Al-Faisal, On the Representation Theory of Semisimple Lie Groups (University of Waterloo MMath thesis, 2010), §3.3 printed pp.71–73 (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)