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I think it is true that $G$-equivariant sheaves on $X$ are equal to $G/H$ equivariant sheaves on $X/H$. … Then the category of $G$ equivariant $\mathcal{O}_X$ Modules is equivalent to the category of $G/H$ equivariant modules on the quotient. …

Then an $\mathcal O_X$-linear map between two $G$-equivariant $\mathcal O_X$-modules commutes with the group actions iff it commutes with the Lie-algebra actions. …

Given a topological group acting on a topological space, Bernstein and Lunts construct the equivariant derived category, which looks like the derived category of the quotient would, if action was free … In their book Bernstein and Lunts give a beautiful description of the equivariant derived category of a point (for a reasonable Lie group). Are there applications of this specific result? …

The first definition goes like this: An equivariant $D_X$ Module is just a $D_X$ module $M$ together with an isomorphism $$\rho^* M\rightarrow \pi^* M$$ of $D_{G\times X}$ -modules. … In addition it requires the action map $$D_X\otimes M \rightarrow M$$ to be equivariant. …

Note that this category is a special case of the category of equivariant objects: http://ncatlab.org/nlab/show/equivariant+object Is this category equivalent to the equivariant derived category? … If not, why is the equivariant derived category a better construction in this case? …

proof the following theorem: Let $\pi:X\rightarrow X/G$ be a principal $G$-bundle (say of varieties, Zariski locally trivial), then $\pi^*$ induces an equivalence between modules on the quotient and equivariant … G\times G\times X \Rrightarrow G\times X \Rightarrow X$$ this should be the category of $G$ - equivariant qc $\mathcal O_X$-modules, where $G$ is an algebraic group and $X$ is an ordinary variety. …

Similary a graded $A$ module is just a $\mathbb{C}^*$ equivariant sheaf. Now I want to know, if there is also a geometric interpretation of filtered rings/modules. …

Is there a natural mixed Hodge structure on its equivariant cohomology? Is it pure if $X$ is smooth projective? … What if we ask the analogous question for $l$-adic equivariant cohomology for varieties over finite fields? …