Symplectic reduction at nonzero momentum

Statement

Let a Lie group $G$ act on $(M,\omega)$ , and suppose the action is Hamiltonian with equivariant moment map $\mu:M\to\mathfrak g^*$ . Fix $\alpha\in\mathfrak g^*$ . If $\alpha$ is a regular value and its coadjoint stabilizer $G_\alpha$ acts freely and properly on $\mu^{-1}(\alpha)$ , then the quotient

M/\!/\!_\alpha G=\mu^{-1}(\alpha)/G_\alpha

is a smooth manifold. It carries a unique symplectic form $\omega_\alpha$ characterized by

\pi^*\omega_\alpha=i^*\omega,

where $i$ is inclusion and $\pi$ is the orbit projection. Its dimension is $\dim M-\dim G-\dim G_\alpha$ .

Why the stabilizer acts

Equivariance gives $\mu(g\cdot m)=\operatorname{Ad}_g^*\mu(m)$ . Therefore $g$ preserves the individual level $\mu^{-1}(\alpha)$ exactly when $g\in G_\alpha$ ; the full group usually moves this level through the family $\mu^{-1}(\operatorname{Ad}_g^*\alpha)$ . This is why quotienting $\mu^{-1}(\alpha)$ by all of $G$ is generally undefined.

For $m\in\mu^{-1}(\alpha)$ , the kernel of $i^*\omega$ equals the tangent space to the $G_\alpha$ -orbit. Thus the restricted form is basic for the free proper action and descends to a nondegenerate two-form. Closedness descends from $d\omega=0$ .

Dimension and special cases

Regularity makes $\mu^{-1}(\alpha)$ have codimension $\dim G$ . Quotienting by the free $G_\alpha$ -action removes another $\dim G_\alpha$ dimensions, giving the formula in the core. If $\alpha$ is fixed by the coadjoint action, then $G_\alpha=G$ and

\dim(M/\!/\!_\alpha G)=\dim M-2\dim G,

the familiar zero-level count.

For an abelian group every coadjoint value is fixed, so nonzero levels are reduced by the full group. For a nonabelian group, $G_\alpha$ can be strictly smaller, and the dimension formula reflects that difference.

Relation to zero-level reduction

The shifting trick converts reduction at $\alpha$ into zero-level reduction of $M$ times the coadjoint orbit through $\alpha$ equipped with the opposite Kirillov–Kostant–Souriau form. This also identifies $\mu^{-1}(\alpha)/G_\alpha$ with $\mu^{-1}(\mathcal O_\alpha)/G$ under the corresponding regularity hypotheses.

References

Juan-Pablo Ortega and Tudor S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhäuser, 2004. DOI record. Relevant: §4.3, point reduction at a coadjoint value.
Jerrold E. Marsden and Tudor S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., Springer, 1999. DOI record. Relevant: §10.3, regular point and orbit reduction.