PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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The coadjoint-orbit inclusion is an equivariant moment map

Statement

Assume ${A C}_{ω}$ . Equip the coadjoint orbit $O \subseteq g^{*}$ with its canonical structure and the KKS form $ω$ , and let $Φ : O ↪ g^{*}$ be the inclusion. Then $Φ$ is an equivariant moment map for the coadjoint action on $O$ :

$Φ (h \cdot β) = h \cdot Φ (β), d ⟨ Φ, ξ ⟩ = - ι_{ξ_{O}} ω (ξ \in g) .$

Facts & Assumptions

Given: ${A C}_{ω}$ , a coadjoint orbit $O$ with its KKS form and the coadjoint action on it.

[A1]

${A C}_{ω}$ is countable choice; it is used only through the orbit and fundamental-field suppliers cited in [F1] and [F2].

[F1]

The coadjoint action is $(h \cdot α) (ζ) = α ({Ad}_{h^{- 1}} ζ)$ , so the orbit map $β \mapsto h \cdot β$ is the restriction of the coadjoint action to $O$ . The coadjoint representation, action and orbits.

[F2]

The inclusion satisfies $d ⟨ Φ, ξ ⟩ = - ι_{ξ_{O}} ω$ , and $ω$ is the KKS form with $ω_{β} (ξ_{O} (β), η_{O} (β)) = β ([ξ, η])$ . Coadjoint orbits are symplectic manifolds, The Kirillov--Kostant--Souriau form on a coadjoint orbit.

[F3]

An equivariant moment map is exactly a smooth map satisfying these two conditions. Moment map, component Hamiltonians and infinitesimal moment maps.

Proof

technique · direct

1.1

The inclusion is coadjoint equivariant: for $h \in G$ and $β \in O$ the orbit point $h \cdot β$ is again in $O$ , and $Φ (h \cdot β) = h \cdot β = h \cdot Φ (β)$ because $Φ$ is the identity map on $O$ .

F1given

1.2

The component moment equations hold for $Φ$ by [F2].

2.1

By step 1.1, step 1.2 and the definition of an equivariant moment map, the inclusion $Φ$ is an equivariant moment map for the coadjoint action on $(O, ω)$ .

step 1.1step 1.2F3A1∎

Depends on

Used by

The two-sphere as a coadjoint orbit of SO(3) Example
The shifting trick identifies reduction at a value with a zero reduction Proposition

Dependency tree · two levels

27 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

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)
Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)