Zero-level symplectic reduction and the dimension formula

Given: ${\mathrm{A}\mathrm{C}}_{\omega }$, a Hamiltonian $G$-space, $0$ a regular value of $\mu$, a nonempty level ${\mu }^{-1}(0)$, and $G$ acting freely and properly on that level.

${\mathrm{A}\mathrm{C}}_{\omega }$ is countable choice; it is used only through the reduction and fundamental-field suppliers.

The coadjoint stabilizer of $0$ is all of $G$, because the coadjoint action is linear: $g\cdot 0=0$ for every $g$. The coadjoint representation, action and orbits.

Under these hypotheses the reduction theorem applies with $\alpha =0$ and $G_{\alpha }=G$, producing the unique symplectic form ${\omega }_0$ on $M_0$ with ${\pi }^{\ast }{\omega }_0={\iota }^{\ast }\omega$. Marsden--Weinstein--Meyer symplectic reduction.

Regularity of $0$ means $d{\mu }_p$ is surjective for every $p\in {\mu }^{-1}(0)$; hence $\dim \ker d{\mu }_p=\dim M-\dim \mathfrak{g}$ and $T_p{\mu }^{-1}(0)=\ker d{\mu }_p$. A regular level set is an embedded submanifold, The tangent space of a regular level set is the kernel, Regular and critical points and values.

A smooth free proper action of $G$ on a nonempty manifold of dimension $n$ has a smooth quotient of dimension $n-\dim G$, with quotient projection a surjective submersion. Free proper action quotient manifold.