Marsden–Weinstein reduction

Remarks

The Marsden–Weinstein–Meyer theorem says that $\omega_{\mathrm{red}}$ exists, is unique, and is symplectic, and that $\dim M /\!\!/ G = \dim M - 2 \dim G$. Freeness of the action on $Z$ implies that $0$ is a regular value of $\mu$, which is why $Z$ is a submanifold. Properness holds automatically when $G$ is compact. Reduction at another value $\lambda \in \mathfrak{g}^*$ uses $\mu^{-1}(\lambda)$ divided by the stabilizer of $\lambda$ under the coadjoint action.

Example: let $S^1$ act on $\C^{n+1}$ by $e^{i\theta} \cdot z = e^{i\theta} z$, with the standard symplectic form. A moment map is $\mu(z) = -\frac{1}{2}\abs{z}^2 + \frac{1}{2}$. Then $\mu^{-1}(0)$ is the unit sphere $S^{2n+1}$, the circle acts freely on it, and the reduction is $\mathbb{CP}^n$ with a multiple of the Fubini–Study form.

Non-example: take the same circle action on $\C$ with $\mu(z) = -\frac{1}{2}\abs{z}^2$. The level set $\mu^{-1}(0) = \set{0}$ is a fixed point, so the action on it is not free and the definition does not apply.