Categorical Moy-Prasad theory and applications to geometric Langlands

Abstract

For any $G((t))$-category, we define a filtration which provides a categorification of the notion of depth studied by Moy and Prasad in the context of representations of a p-adic group. We also prove a generation statement which is the analogue of their theory of unramified minimal K-types and use it to compute our filtration on a number of categories of interest to geometric Langlands. Finally, joint with Sam Raskin, we apply this theory to prove an affine Beilinson-Bernstein localization statement conjectured by Frenkel and Gaitsgory.