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The shifting trick identifies reduction at a value with a zero reduction

Statement

Assume ${A C}_{ω}$ . Let $(M, ω, G, μ)$ be a Hamiltonian $G$ -space, let $α \in g^{*}$ and let $O = G \cdot α$ be its coadjoint orbit with the KKS form $ω_{O}$ ; write $O^{-} = (O, - ω_{O})$ and equip $M \times O^{-}$ with the diagonal $G$ -action, the product form $Ω = {pr}_{M}^{*} ω - {pr}_{O}^{*} ω_{O}$ and

$Ψ (m, β) : = μ (m) - β \in g^{*} .$

Then:

$Ψ$ is a coadjoint-equivariant moment map for the diagonal action.
The zero set $Ψ^{- 1} (0)$ consists of the pairs $(m, μ (m))$ with $μ (m) \in O$ , and $m \mapsto (m, μ (m))$ identifies it $G$ -equivariantly with the saturated level $μ^{- 1} (G \cdot α)$ .
Every $G$ -orbit in $Ψ^{- 1} (0)$ meets the slice $μ^{- 1} (α) \times {α}$ in exactly one $G_{α}$ -orbit, so the inclusion of the slice induces a canonical bijection $μ^{- 1} (α) / G_{α} \to Ψ^{- 1} (0) / G$ . Whenever both orbit spaces carry their free-proper quotient manifold structures, this bijection is a diffeomorphism.
The pullbacks of the reduced form of $M_{α}$ and of the zero-reduced form of $M \times O^{-}$ to $μ^{- 1} (α)$ agree, both being $ι^{*} ω$ . Hence, whenever $0$ is a regular value of $Ψ$ and $G$ acts freely and properly on $Ψ^{- 1} (0)$ , the shift map of item 3 is a symplectomorphism $M_{α} \to Ψ^{- 1} (0) / G$ . Here both the $G_{α}$ - action on $μ^{- 1} (α)$ and the $G$ -action on $Ψ^{- 1} (0)$ are assumed free and proper. Moreover $0$ is a regular value of $Ψ$ if and only if $α$ is a regular value of $μ$ , and the $G$ -action on $Ψ^{- 1} (0)$ is free if and only if the $G_{α}$ -action on $μ^{- 1} (α)$ is free.

Facts & Assumptions

Given: ${A C}_{ω}$ , a Hamiltonian $G$ -space, a covector $α$ , and the orbit $O$ with the opposite KKS form.

[A1]

${A C}_{ω}$ is countable choice; it is used only through the fundamental-field, orbit and reduction suppliers.

[F1]

The orbit inclusion $Φ : O ↪ g^{*}$ is an equivariant moment map for the coadjoint action with the KKS form. The coadjoint-orbit inclusion is an equivariant moment map, Coadjoint orbits are symplectic manifolds.

[F2]

On a product with the diagonal action the moment maps add, and on the opposite symplectic manifold the moment map changes sign; the product form is symplectic. Products and opposites of symplectic moment maps.

[F3]

$μ$ is equivariant with $μ (g \cdot m) = g \cdot μ (m)$ , and the coadjoint action is linear in the second variable: $g \cdot (β_{1} - β_{2}) = g \cdot β_{1} - g \cdot β_{2}$ . Moment map, component Hamiltonians and infinitesimal moment maps, The coadjoint representation, action and orbits.

[F4]

If $α$ is regular for $μ$ and $G_{α}$ acts freely and properly on $μ^{- 1} (α)$ , then the reduction $(M_{α}, ω_{α})$ exists with $π_{α}^{*} ω_{α} = ι^{*} ω$ . Marsden--Weinstein--Meyer symplectic reduction.

[F5]

Regularity of a value for a moment map is equivalent to local freeness of the action along the level. Regularity of a moment map is equivalent to local freeness.

[F6]

The product form restricted to the slice $M \times {α}$ pulls back to $ω$ , because the second factor contributes zero on vectors tangent to the slice. Products and opposites of symplectic moment maps.

Proof

technique · direct

1.1F1F2F3

By [F1] and [F2] the diagonal action on $M \times O^{-}$ has moment map $Ψ (m, β) = μ (m) - β$ , and it is equivariant: $Ψ (g \cdot (m, β)) = μ (g \cdot m) - (g \cdot β) = g \cdot μ (m) - g \cdot β = g \cdot Ψ (m, β)$ by [F3].

2.1step 1.1F3

The zero set is ${(m, β) : β = μ (m)}$ together with the condition $β \in O$ ; the map $m \mapsto (m, μ (m))$ is a $G$ -equivariant bijection $μ^{- 1} (G \cdot α) \to Ψ^{- 1} (0)$ , since $Ψ (m, μ (m)) = 0$ and $μ (m) \in O$ exactly when $μ (m) = g \cdot α$ for some $g$ , i.e. when $m \in g \cdot μ^{- 1} (α) \subseteq μ^{- 1} (G \cdot α)$ .

3.1step 2.1F3F4

Orbit-slice property: given $(m, μ (m)) \in Ψ^{- 1} (0)$ with $μ (m) = g \cdot α$ , the element $g^{- 1}$ moves it to $(g^{- 1} \cdot m, α)$ with $μ (g^{- 1} \cdot m) = α$ , so every orbit meets the slice. Two slice points $(m, α)$ and $(m^{'}, α)$ lie in the same $G$ -orbit exactly when $m^{'} = h \cdot m$ with $h \cdot α = α$ , i.e. $h \in G_{α}$ . Hence the inclusion of the slice induces a canonical bijection $μ^{- 1} (α) / G_{α} \to Ψ^{- 1} (0) / G$ . If both actions are free and proper, the quotient maps are submersions and their local smooth sections make the induced bijection and its inverse smooth.

4.1step 3.1F4F6

Under the stated regularity, freeness, and properness hypotheses, pulling the reduced form of $M_{α}$ back along $μ^{- 1} (α) \to M_{α}$ gives $ι^{*} ω$ by [F4]; pulling the zero-reduced form of $M \times O^{-}$ back along the composite $μ^{- 1} (α) \to Ψ^{- 1} (0) \to Ψ^{- 1} (0) / G$ gives the restriction of $Ω$ to the slice, which is $ι^{*} ω$ by [F6]. Both composite maps are surjective submersions, so the two forms agree under the identification of item 3.

4.2step 3.1F5

Regularity and freeness: for $m \in μ^{- 1} (α)$ , the infinitesimal stabilizers of $m$ for the $G$ -action and for the $G_{α}$ -action coincide, because $g \cdot m = m$ implies $g \cdot α = α$ by equivariance; the stabilizer of the point $(m, α)$ for the $G$ -action on the slice is the same group. Hence, by [F5], $0$ is a regular value of $Ψ$ exactly when $α$ is a regular value of $μ$ , and the $G$ -action on $Ψ^{- 1} (0)$ is free exactly when the $G_{α}$ -action on $μ^{- 1} (α)$ is free.

5.1step 4.1step 4.2A1∎

Combining the items: $Ψ$ is an equivariant moment map (1.1), its zero set is the $G$ -equivariant image of the saturated level (2.1), the orbit-slice bijection identifies the two quotients (3.1), and the forms and hypotheses correspond (4.1, 4.2); when the shifted zero reduction exists, the identification is a symplectomorphism $M_{α} \to Ψ^{- 1} (0) / G$ .

Depends on

Used by

The shifting trick for a nonzero coadjoint orbit Example

Dependency tree · two levels

44 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)