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Let $Gr$ be the affine Grassmannian of $G=G((t))/G[[t]]$, and let $Perv(Gr)$ be the category of perverse sheaves on $Gr$. We have action of $G((t))$ on the left-hand side of $Perv(Gr)$, also we have action of the tensor category $Rep(G^\vee)$ on the right-hand side, through geometric Satake correspondence. It is clear that we have the action of $G((t))$ on $Perv(Gr)$ as functors.

Follow Gaitsgory's paper "the notion of category over stack", he claimed that an action of algebraic group scheme $H$ on a category $\mathcal{C}$ is actually equivalent to a category $\mathcal{C}$ with the action of the tensor category "Rep(H)".

Go back to the original set-up, that means we also have an action of $G^\vee$ on $Perv(Gr)$. I think, on this $Perv(Gr)$, the actions of $G((t))$ and $G^\vee$ should have different meaning, but why they gave them the same name?

Am I confused?

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up vote 10 down vote accepted

Yes, you are confused. What is claimed by Gaitsgory is that datum of category with action of $H$ is equivalent to datum of of another category with action of $Rep(H)$. You go back and forth between these two categories using constructions of "equivariantization" and "de-equivariantization".

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Thanks. Is the action of $G((t))$ on this category just as functors? Is there any more strucutre, like the action of affine group scheme on the category? An irrelevant question, what is the regular function on $G((t))$? Or maybe we have the notion of categorical Harish-chandra module? – Hong Apr 1 '10 at 19:18
the action of $G((t))$ comes from its action on the affine Grassmannian, so if I understood you correctly, the answer to your first question is yes. Unfortunately I don't know much about your other questions.. – Victor Ostrik Apr 2 '10 at 0:04

The $G((t))$ action and the $Rep(G^\vee)$ [or equivalently of $G^\vee$ itself after deequivariantization, as Victor explains] are of quite different natures -- the former is a "smooth" action, and the latter an "algebraic" or "analytic" actions (the adjectives smooth and analytic come from analogy with p-adic rep theory). i.e. there are many kinds of notion of group action, and they are (to me) most conveniently summarized by describing the corresponding notion of group algebra which acts. An algebraic action of a group on a category is an action of the "quasicoherent group algebra" of G, ie the monoidal category of quasicoherent sheaves wrt convolution. (though I'd feel much safer if we said all this in a derived context, makes me uneasy otherwise). A smooth action is an action of the monoidal category of D-modules on G, the "smooth group algebra" -- analog of smooth functions on a p-adic group. Such an action is the same as an algebraic action, which is infinitesimally trivialized. Such examples are studied in Chapter 7 of Beilinson-Drinfeld's Hecke manuscript and the appendix to the long paper by Gaitsgory-Frenkel, in particular.

PS the "equivariantization" dictionary between categories over BH and categories with H action is a nice simple case of descent --- you describe things over BH as things over a point with descent data, that descent data is given by the map H --> pt, the two maps H x H---> H, and so on. When you assemble this together (most efficiently using the Barr-Beck theorem) you get the desired dictionary. (Of course if you want to consider categories as forming a 2-category you'd need a 2-categorical version of Barr-Beck, but for most practical purposes I know of you can get by with the current Lurie [$(\infty,$]1-categorical version.

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