Is there a gerbe Beilinson-Bernstein Localization?

Question

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?

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[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

Answer 1

[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.