Shifting trick - Imported Knowlpedia Content

Core idea

Let $G$ have a Hamiltonian action on $(M,\omega)$ with equivariant moment map $\mu:M\to\mathfrak g^*$ , let $\alpha\in\mathfrak g^*$ , and let $\mathcal O_\alpha$ be its coadjoint orbit. Equip $\mathcal O_\alpha^{-}$ with the negative of its Kirillov–Kostant–Souriau form. The diagonal action on

(M\times\mathcal O_\alpha^{-},\,\omega\oplus-\omega_{\mathrm{KKS}})

has moment map

\widetilde\mu(m,\xi)=\mu(m)-\xi.

The shifting trick replaces reduction at $\alpha$ by zero-level reduction of this product. Set-theoretically, and symplectically whenever the regular-reduction hypotheses hold,

\mu^{-1}(\alpha)/G_\alpha\cong\widetilde\mu^{-1}(0)/G.

This construction keeps the original group action while moving the chosen momentum level to zero.

Construction of the identification

The zero set consists of pairs $(m,\xi)$ with $\mu(m)=\xi\in\mathcal O_\alpha$ . Every diagonal $G$ -orbit in this set has a representative $(m,\alpha)$ : choose $g$ carrying $\xi$ to $\alpha$ . Two such representatives $(m,\alpha)$ and $(m',\alpha)$ are in the same diagonal orbit exactly when they differ by an element of $G_\alpha$ . This gives the displayed bijection.

Equivalently, both sides identify with the orbit-reduction space

\mu^{-1}(\mathcal O_\alpha)/G.

Under free and proper regularity assumptions, the bijection is a diffeomorphism and the pullback characterizations of the reduced forms show that it is a symplectomorphism Ortega and Ratiu, §4.3.

Why the sign is negative

With the convention that inclusion $\mathcal O_\alpha\hookrightarrow\mathfrak g^*$ is the moment map for the positive KKS form, the moment map on the oppositely symplectic orbit is $-\xi$ . Moment maps add under products, giving $\widetilde\mu(m,\xi)=\mu(m)-\xi$ . Consequently the zero equation is the desired equality $\mu(m)=\xi$ .

Changing the convention for the KKS form or for Hamiltonian vector fields changes the displayed signs together. The invariant content is that the orbit factor is taken with the symplectic sign that makes the product moment map vanish exactly over $\mu^{-1}(\mathcal O_\alpha)$ .

Uses and scope

The construction allows zero-level results, including singular reduction theorems, to be applied at arbitrary coadjoint values. It also makes the coadjoint orbit itself part of the geometry instead of treating $\alpha$ as a fixed parameter.

The shifting trick does not repair a failure of regularity or properness. If the original reduction is singular, the shifted zero-level reduction is singular as well, though the identification remains useful at the level of stratified quotients.

References

Victor Guillemin and Shlomo Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. DOI record. Relevant: Chapter 5, reduction at coadjoint orbits and the shifting construction.
Juan-Pablo Ortega and Tudor S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhäuser, 2004. DOI record. Relevant: §4.3, equivalence of point, orbit, and shifted zero-level reduction.