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Given a smooth affine symplectic variety $V$ with an action of a compact 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 ?

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# Hamiltonian Reduction and Affine quotient

Given a smooth affine symplectic variety $V$ with an action of a compact 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 ?