Categorical Moy-Prasad theory

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.