Given a smooth affine symplectic variety $V$ with an action of a connected algebraic $G$. If $\mu$ is the moment map, the define the affine quotient to be :


$X = \mu^{-1}(0)// G = \text{Spec}\mathbb{C}[\mu^{-1}(0)]^{G}$ 


This is an algebraic Variety (may be singular). 


The Hamiltonian reduction of $V$ is defined to be 


$Y = \mu^{-1}(0)/ G$. 


Q : When are these two notions same ? 



Hamiltonian Reduction and Affine quotient 
We can assume that $G$ is affine, since an abelian variety must act trivially on any affine variety. The closed points of $X$ are exactly the closed $G$-orbits on $\mu^{-1}(0)$. On an affine variety, closed orbits can be distinguished by invariant global functions, and if one orbit is in the closure of another they will go to the same point in $X$. 


Thus, $X$ and $Y$ are the same if all $G$-orbits on $\mu^{-1}(0)$ are closed.


