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Suppose you want to construct a representation of an affine algebraic group $G$, you may start with a $G$-equivariant line bundle $\mathcal{L}$ on a $G$-manifold $X$ and then consider global sections, or cohomologies, for example $H^*(X, \mathcal{L})$ becomes a $G$-module.

Suppose now you want to construct a representation of $G$ on a category $\mathcal{C}$ (the type of representations studied in the appendix of [1] or in chapter 7 of [2]). Then you may consider a $G$-equivariant gerbe $\mathcal{G}$ on $X$ and again take global sections.

In the former case you find out that essentially all (finite dimensional, over $\mathbb{C}$, etc) representations of $G$ arise in a geometric way: there exists a manifold $X$ (ie. the flag variety $G/B$ for a choice of a Borel $B \subset G$) and an equivalence of tensor categories between certain $D_X$-modules and $G$-rep. This is known as Beilinson-Bernstein localization and I'll refer to [3] for the precise statements.

My question is if there's a categorical analogue of this statements along the lines of

Is there an equivalence of $2$-categories, between the category of (categorical) representations of $G$ and some category of equivariant gerbes with flat connections on a space $X$ ?

This space might be infinite dimensional. One thing that comes to mind for example is the fact that $D$-modules on the affine grassmanian $Gr_G$ carries a monoidal action of $Rep (G^{\vee})$, and actions of groups on categories are related to these monoidal actions by de-equivariantization [1]. This might be related to my question, but notice that this is not really what I ask up there.

Besides the folklore that "should be true", is there anything concrete written down?


[1] Frenkel and Gaitsgory. Local geometric Langlands correspondence and affine Kac-Moody algebras

[2] Beilinson and Drinfeld Quantization of Hitchin's integrable system and Hecke eigensheaves

[3] Milicic Localization and Representation Theory of Reductive Lie Groups

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[Edited to reflect Reimundo's comment] The question addresses categorified versions of the Borel-Weil-Bott theorem (and more generally Beilinson-Bernstein localization), which states an equivalence between G-equivariant vector bundles on the flag variety - aka vector bundles on pt/B (modulo an action of the Weyl group - aka double cosets B\G/B - by intertwiners) and algebraic representations of G. There are two pieces of content here: first, that all representations can be realized on G/B, ie representations have highest weights, and second that irreducibles correspond to line bundles, ie their highest weight spaces are one-dimensional. The first has an analog for any representation of the Lie algebra: Beilinson-Bernstein's localization can be rephrased as simply asserting that descent holds from twisted D-modules on the flag variety to representations of the Lie algebra.

I don't know anything about the analog of the second assertion for categorified representations - ie to what extent "indecomposable" representations of some kind are induced from "one-dimensional ones" (ie from gerbes on homogeneous spaces) - except to point out a very nice paper by Ostrik (section 3.4 here) in which analogous results are proved for the case of a finite group.

As for the "descent" (first) part of BWB, it becomes completely trivial once categorified, if we consider so-called algebraic (or quasicoherent) actions of G on categories (equivalently module categories for quasicoherent sheaves on G). In fact the same assertion holds for ANY algebraic subgroup of G, not just a Borel, in sharp distinction to the classical setting: algebraic G-actions on categories are generated by their H-invariants for any H in G! More precisely we have the following theorem:

Passing to H-invariants provides an equivalence of $(\infty,2)$-categories between (dg) categories with a G action and categories with an action of the "Hecke category" QC(H\G/H) of double cosets.

This is a theorem of mine with John Francis and David Nadler in a preprint that's about to appear (copies available).. it's a version of a well known result of Mueger and Ostrik in the finite group case, and is an easy application of Lurie's Barr-Beck theorem. In fact if we use a result of Lurie in DAG XI, that there is no distinction between quasicoherent sheaves of categories on stacks X with affine diagonal and simply module categories over QC(X), we can rephrase the result as follows:

G-equivariant quasicoherent sheaves of (dg-)categories on the flag variety equipped with a "categorified Weyl group action" (module structure for QC(B\G/B) ) are equivalent (as an $(\infty,2)$-category) to (dg)-categories with algebraic G-action.

On the other hand things get much more interesting if we consider "smooth" or "infinitesimally trivialized" G-actions (module categories over D-modules on G) (also discussed in the references you provide). The fundamental example of such a category is indeed Ug-mod, the category of all representations of the Lie algebra, or equivalently (up to some W symmetry) the category D_H(G/N) of all twisted D-modules on the flag variety. In this case my paper with Nadler "Character theory of a complex group" (and other work in progress) precisely studies the full sub(2)category of smooth G actions which ARE generated by their highest weight spaces, ie which do come via a Borel-Weil-Bott type construction (or equivalently, the full subcategory generated by the main example Ug-mod). And not all smooth G-categories are of this form, though one might hope that this is the case in some weaker sense.. in any case it seems that all G-categories of interest in representation theory do fall under this heading. But in any case I don't know a BWB type statement in this setting.

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Hi David and thanks for the reply, I need to absorb the theorem you mention. I wasn't expecting the flag variety to be that space. At least without specifying conditions like "highest weight" or similar, that allow us to induce from the Borel in the usual situation, but I guess this is hidden in your notion of smooth. Also, you're absolutely right, I'm interested in (g,K)-modules, but it gets technical to write what an action of a Lie algebra on a Category is, I guess I could edit my question and add references to Xin-Wen's articles or Baez-Crans and Roytenberg on the 2-vector space case. –  Reimundo Heluani Mar 17 '12 at 8:17
    
Thanks again BZ, this explains it. I didn't know about Victor's paper and it does sound interesting (although no section 3.4 there :) ) I'll wait a couple of days to see if someone here has anything to add about the "highest weight" property and I'll accept it. I'll add a reference to your manuscript when it comes out. –  Reimundo Heluani Mar 19 '12 at 18:23

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