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Borel-Weil-Bott is compatible with Serre duality

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $λ \in X^{*} (T)$ and put $μ = - λ - 2 ρ$ , so that $L_{λ}^{\lor} \otimes K_{X} ≅ L_{μ}$ by Canonical weight of a flag variety. Then:

(i) $λ + ρ$ is regular if and only if $μ + ρ = - (λ + ρ)$ is regular, and if $λ + ρ$ is regular with Weyl element $w$ (so $w (λ + ρ)$ is dominant) then the Weyl element of $μ$ is $u = w_{0} w$ , with $ℓ (u) = N - ℓ (w)$ and $N = ∣ Φ^{+} ∣$ ;

(ii) for regular $λ$ the Serre pairing $H^{i} (X, L_{λ})^{\lor} ≅ H^{N - i} (X, L_{μ})$ at $i = ℓ (w)$ identifies $H^{ℓ (w)} (X, L_{λ})^{\lor}$ with $H^{N - ℓ (w)} (X, L_{μ}) ≅ L (w \cdot λ)$ , and the Borel-Weil-Bott descriptions $H^{ℓ (w)} (X, L_{λ}) ≅ L (w \cdot λ)^{*}$ and $H^{ℓ (u)} (X, L_{μ}) ≅ L (u \cdot μ)^{*} ≅ L (w \cdot λ)$ match under this pairing;

(iii) if $λ + ρ$ is singular then all cohomology groups of both $L_{λ}$ and $L_{μ}$ vanish.

Facts & Assumptions

Given: The Axiom of Choice, the group $G$ , its Borel $B$ , the flag variety $X = G / B$ of dimension $N = ∣ Φ^{+} ∣$ , a weight $λ$ and $μ = - λ - 2 ρ$ .

[F1]

For a regular weight $ν$ there is a unique $w$ with $w ν$ dominant, and $ℓ (w) = N (ν)$ ; for every $w$ one has $ℓ (w_{0} w) = N - ℓ (w)$ , and regularity is preserved by $W$ (A regular weight has a unique dominant dot translate).

[F2]

Borel-Weil-Bott: if $ν + ρ$ is singular all $H^{i} (X, L_{ν})$ vanish, and if $ν + ρ$ is regular with Weyl element $v$ then $H^{ℓ (v)} (X, L_{ν}) ≅ L (v \cdot ν)^{*}$ and all other cohomology vanishes (The Borel-Weil-Bott theorem).

[F3]

Serre duality: $ω_{X} ≅ L_{- 2 ρ}$ and there is a functorial perfect pairing $H^{i} (X, L_{λ}) \times H^{N - i} (X, L_{λ}^{\lor} \otimes ω_{X}) \to C$ for $0 \leq i \leq N$ ; the canonical identifications give $L_{λ}^{\lor} \otimes ω_{X} ≅ L_{- λ - 2 ρ} = L_{μ}$ (Serre duality for locally free sheaves on a smooth projective variety, Canonical weight of a flag variety, The equivariant line bundle associated to a Borel character).

[F4]

For a dominant integral weight $ν$ , the dual $L (ν)^{*}$ is irreducible of highest weight $- w_{0} ν$ , so $L (- w_{0} ν) ≅ L (ν)^{*}$ ; applying this twice gives $L (ν) ≅ L (- w_{0} ν)^{*}$ for dominant integral $ν$ , the isomorphism class being determined by the highest weight; a particular isomorphism is not unique (Highest weight of the dual representation, Highest-weight classification).

[F5]

The dot action satisfies $u \cdot μ = u (μ + ρ) - ρ$ ; for $u = w_{0} w$ and $μ = - λ - 2 ρ$ one has $u \cdot μ = - w_{0} (w \cdot λ)$ (Dot-Weyl facets and single-wall translation data, Length and longest Weyl-group element).

Proof

1.1F1F5givenalgebra

Part (i). Since $μ + ρ = - (λ + ρ)$ , the pairing of $μ + ρ$ with every coroot is the negative of that of $λ + ρ$ , so $μ + ρ$ is regular exactly when $λ + ρ$ is. If $λ + ρ$ is regular with Weyl element $w$ , then $(w_{0} w) (μ + ρ) = - (w_{0} w) (λ + ρ) = - w_{0} (w (λ + ρ))$ : as $w (λ + ρ)$ is dominant, $w_{0} (w (λ + ρ))$ is antidominant, so its negative is dominant, and by the uniqueness in [F1] the Weyl element of $μ$ is $u = w_{0} w$ . Its length is $ℓ (u) = ℓ (w_{0} w) = N - ℓ (w)$ by [F1].

2.1F2F3F4F5step 1.1algebra

Part (ii). Assume $λ + ρ$ regular. By [F2] applied to $λ$ , $H^{ℓ (w)} (X, L_{λ}) ≅ L (w \cdot λ)^{*}$ . By part (i), $u = w_{0} w$ is the Weyl element of $μ$ , so [F2] applied to $μ$ gives $H^{N - ℓ (w)} (X, L_{μ}) ≅ L (u \cdot μ)^{*}$ . By [F5], $u \cdot μ = - w_{0} (w \cdot λ)$ ; since $w \cdot λ$ is dominant integral, [F4] identifies $L (- w_{0} (w \cdot λ))^{*}$ with $L (w \cdot λ)$ . The Serre pairing of [F3] at $i = ℓ (w)$ is $H^{ℓ (w)} (X, L_{λ})^{\lor} ≅ H^{N - ℓ (w)} (X, L_{μ})$ , and the two Borel-Weil-Bott descriptions identify both sides with $L (w \cdot λ)$ . This identification is $G$ -equivariant: the cup product and contraction are natural for the bundle linearizations, while the canonical trace in [F3] is invariant under automorphisms of $X$ . Thus the Borel-Weil-Bott module descriptions match under the Serre pairing.

3.1F2step 1.1given∎

Part (iii). If $λ + ρ$ is singular, then $μ + ρ = - (λ + ρ)$ is singular as well by part (i), and [F2] gives the vanishing of all cohomology groups of both $L_{λ}$ and $L_{μ}$ .

Depends on

Used by

The top-degree Borel-Weil-Bott case and Serre duality Example

Dependency tree · two levels

60 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

Jacob Lurie, A Proof of the Borel-Weil-Bott Theorem (standard reference, not scraped)
George Boxer and Vincent Pilloni, Notes on Higher Coleman Theory (Montreal 2020) (standard reference, not scraped)