Categorical Moy-Prasad theory

TL;DR

This paper categorifies Moy-Prasad theory by defining a depth filtration on categories with loop group actions, leading to new insights into Whittaker sheaves and Kac-Moody modules, and confirming existing conjectures.

Contribution

It introduces a categorified depth filtration framework and computes it for Whittaker sheaves and Kac-Moody modules, extending and confirming prior conjectures.

Findings

Recovers and extends Raskin's filtration for Whittaker sheaves

Provides new localization statements for Kac-Moody modules

Confirms a conjecture of Chen-Kamgarpour

Abstract

We categorify the theory developed by Moy-Prasad in [MP94]. More precisely, we define a depth filtration on any category with an action of the loop group $G((t))$ and prove a $2$-categorical generation statement inspired by the theory of unrefined minimal K-types. Using our generation theorem, we compute the depth filtration on the category of Whittaker sheaves and on the category of Kac-Moody modules. On the Whittaker side, we show that it recovers and extends the filtration constructed by Raskin in [Ras16]. For KM modules, our computation encodes several new localization statements, as well as confirming a conjecture of Chen-Kamgarpour.