Smooth/strong G-categories and Localization/morita Equivalences

These are notes from T. Schedler’s talk (with D. Ben-Zvi) on smooth/strong Gcategories at U. Oregon, Eugene, OR, on Aug 16, 2012. Acknowledgements. This talk was part of the workshop on Categorical Representation Theory at U. Oregon, 13 August–17 August, 2012, organized by N. Proudfoot, led mathematically by D. Ben-Zvi, and funded by an NSF CAREER grant. Thanks especially to D. Ben-Zvi for his many generous explanations and time, as well as the syllabus which included this talk. Thanks to N. Proudfoot for organizing the enriching workshop. Finally, I am very grateful to all the participants for their many helpful questions, comments, and corrections. Please send me more corrections as you find errors! Note: Nothing in this talk is due to me; if I neglected any attributions, I apologize. 1. Motivation: Beilinson-Bernstein localization Possible ref: Ben-Zvi + Nadler (bn.pdf) ; talk 1.1. Recollection: BB localization theorem. Let G be a reductive group and B < G a Borel. Then we have the flag variety G/B. As we saw in a previous talk, there are quasi-inverse equivalences of categories, Γ : D(G/B) ∼ → Ug-mod0,M 7→ Γ(G/B,M); (1.1) ∆ : Ug-mod0 → D(G/B)-mod,M 7→ DG/B ⊗Ug M. (1.2) Here Ug-mod0 denotes Ug-modules (i.e., g-modules) on which the augmentation ideal Z+ ⊆ Z(Ug) of the center acts trivially. The action of Ug on global sections Γ(M) comes from the fact that Γ(D(G/B)) = Ug/Z+ · Ug. Interpretation: This is saying that the variety G/B is D-affine, i.e., that the global sections functor on D-modules on G/B is an equivalence. This is not true for O-modules on G/B, since G/B is not affine!! 1.2. Structure of smooth G-categories. This is an awesome result, but there is additional structure that has been omitted from the above. Namely, both sides of the equivalence canonically carry an action of D(G), the category of D-modules on G. To explain this, note that G obviously acts on G/B by left translation, and hence it should act canonically on the category of D-modules on G/B. Naively, given g ∈ G and a D-module M on G/B, we can translate M by g to g∗M . This alone would give a “weak G-action on D(G/B)” (see below). To upgrade this to a strong action, we need to equip this with a trivialization of the g = LieG action, or more precisely, with the action of the formal group Ĝ1 of G. In this case, we get such a trivialization because the action of g on D(G/B) by translation is inner. More generally, if G acts on a variety X, we have the infinitesimal action of g on D-modules on X via the composition