  We show that the nearby cycles functor for the $p$-adic Hecke stack at
Iwahori level is perverse t-exact, by developing a theory of Wakimoto
filtrations, and that it lifts to the $\mathbb{E}_1$-center. We apply these
tools to construct the Arkhipov-Bezrukavnikov functor for $p$-adic affine flag
varieties, and prove that it is an equivalence for type $A$ groups.

Number Theory;Algebraic Geometry;Representation Theory