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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The differential of the moment map and the orbit-orthogonal identity

Statement

Assume ${A C}_{ω}$ . Let a Hamiltonian action of $G$ on $(M, ω)$ have moment map $μ$ , and let $p \in M$ . Then

$\ker d μ_{p} = (T_{p} (G \cdot p))^{ω}, im d μ_{p} = ann (g_{p}),$

where $(T_{p} (G \cdot p))^{ω}$ is the symplectic orthogonal of the tangent space of the orbit of $p$ and $g_{p}$ is the infinitesimal stabilizer.

Facts & Assumptions

Given: ${A C}_{ω}$ , a Hamiltonian $G$ -space with moment map $μ$ , and a point $p \in M$ .

[A1]

${A C}_{ω}$ is countable choice; it is used only through the orbit and fundamental-field interface of the two suppliers cited in [F3], both of which carry the same assumption.

[F1]

$d μ^{ξ} = - ι_{ξ_{M}} ω$ for every $ξ \in g$ . Moment map, component Hamiltonians and infinitesimal moment maps.

[F3]

The infinitesimal orbit map $ξ \mapsto ξ_{M} (p)$ has image $T_{p} (G \cdot p)$ , the tangent space of the orbit with its canonical structure, and kernel $g_{p}$ . Kernel of the infinitesimal orbit map, Every orbit is an injectively immersed homogeneous space.

[F4]

For a subspace $W$ of a finite-dimensional symplectic vector space, $\dim W + \dim W^{ω} = \dim V$ and $(W^{ω})^{ω} = W$ . Symplectic double-orthogonal and dimension identities, Symplectic orthogonal complement.

Proof

technique · direct

1.1

For $v \in T_{p} M$ and $ξ \in g$ , [F1] gives $⟨ d μ_{p} (v), ξ ⟩ = d μ_{p}^{ξ} (v) = - ω_{p} (ξ_{M} (p), v)$ . Hence $d μ_{p} (v) = 0$ if and only if $ω_{p} (ξ_{M} (p), v) = 0$ for every $ξ$ , that is, if and only if $v$ is symplectically orthogonal to the span of the values $ξ_{M} (p)$ ; by [F3] that span is $T_{p} (G \cdot p)$ . Therefore $\ker d μ_{p} = (T_{p} (G \cdot p))^{ω}$ .

F1F3

2.1

The image is contained in the annihilator: if $ξ \in g_{p}$ , then $ξ_{M} (p) = 0$ by [F3], so for every $v \in T_{p} M$ the same identity gives $⟨ d μ_{p} (v), ξ ⟩ = - ω_{p} (0, v) = 0$ , so $im d μ_{p} \subseteq ann (g_{p})$ .

step 1.1F3

3.1

Dimension count: by step 1.1 and [F4], $\dim \ker d μ_{p} = \dim M - \dim T_{p} (G \cdot p)$ , and by [F3] $\dim T_{p} (G \cdot p) = \dim g - \dim g_{p}$ . Hence $\dim im d μ_{p} = \dim g - \dim g_{p} = \dim ann (g_{p})$ . Since step 2.1 gives containment between spaces of equal dimension, $im d μ_{p} = ann (g_{p})$ .

step 1.1step 2.1F3F4A1∎

Depends on

Used by

Dependency tree · two levels

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Sources

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