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I think it is true that $G$-equivariant sheaves on $X$ are equal to $G/H$ equivariant sheaves on $X/H$. More precisely I'm interested in the following statement:

Given an algebraic group $G$ with normal subgroup $H$ and an action of $G$ on $X$, such that the quotient $X/H$ exists as a variety and $X\rightarrow X/H$ is an $H$ principal bundle.

Then the category of $G$ equivariant $\mathcal{O}_X$ Modules is equivalent to the category of $G/H$ equivariant modules on the quotient.

The equivalence should be given by pullback along $X\rightarrow X/G$ in one direction and taking invariant sections in the other.

Is there a reference for this?

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    $\begingroup$ I don't know a reference, but it should follow from the following: (1) $G$-equivariant sheaves on X = sheaves on the quotient stack $X//G$; (2) $G/H$-equivariant sheaves on $X/H$ = sheaves on the quotient stack $(X/H)//(G/H)$; (3) $X/H = X//H$ (because of the assumptions on the action of $H$ on $X$); and finally (4) $(X//H)//(G/H) = X//G$. $\endgroup$
    – Sasha
    Jun 30, 2010 at 20:28
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    $\begingroup$ Thanks, this looks good, however I am not yet very experianced with stacks. Have you a reference for some of your claims, especially (3) or (4)? $\endgroup$
    – Jan Weidner
    Jun 30, 2010 at 21:11

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