A moment map $\mu$ is defined when one has a Hamiltonian $G$-action on a symplectic manifold $M$, for some Lie group $G$. My question is, what are the geometric interpretations of the kernel and critical points of the moment map?
For the kernel $\mu^{-1}(0)$, I've read here https://en.m.wikipedia.org/wiki/Moment_map that it is invariant under $G$. If one assumes 0 is a regular point and that $G$ acts freely on the kernel, then $\mu^{-1}(0)/G$ is the symplectic quotient $M//G$. But how do we know that the kernel is a manifold?
I've also read that the critical points of the moment map (which are the points where all its derivatives vanish) correspond to fixed points of the group action when $G$ is a product of circle actions. Is this true for nonabelian $G$?