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Let $f:X\to Y$ be a flat surjective morphism with reduced fibers between affine varieties over an alg. closed field of char. zero, k. Let G be a reductive group acting on X fiberwise. How do I show that all the G-modules $k[f^{-1}(y)]$ are isomorphic?

Edit: I am adding the original setting of the article: Let S be a reductive monoid with unit group G (that is, S in an affine algebraic variety with a structure of a monoid and G is reductive). Let $G_0$ be the commutator subgroup of G. Since $G\times G$ acts on S, we have $k[S]\hookrightarrow k[G]$ (if it is not obvious, you can just believe me). Consider the $G_0\times G_0$ action on S, and set $A=S//(G_0\times G_0)$ (good quotient). Let $f:S\to A$ (it is surjective). Assume that f is flat with integral fibers. Why are all the $G_0\times G_0$ modules $k[f^{-1}(a)]$ isomorphic?

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    @QiL, Are you sure that all the fibers are isomorphic to $G_0$? This a good quotient, not a geometric quotient..2012-09-01

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