Consider a $G$-equivariant map $\pi:X\to Y$ for $G$ an affine algebraic group, such that $\pi$ is a good categorical quotient. Is there any relationship between $H^*_G(X)$ and $H^*(Y)$? Is there if $\pi$ is a good geometric quotient, or if the quotient space is smooth?
EDIT: A categorical quotient is an equivariant map $\pi:X\to Y$ that is constant on $G$-orbits. It's good if the topology on $Y$ is induced by $X$ ($\pi$ is a surjective open submersion) and the map from the functions on any affine $V \subset Y$ to $G$-invariant functions on $\pi^{-1}(V)$ is an isomorphism. It's a good geometric quotient if the $G$-orbits closed in $Y$.