Bounds on Weights of Nearby Cycles and Wakimoto Sheaves on Affine Flag Manifolds

Abstract

We study certain nearby cycles sheaves on an affine flag manifold which arise naturally in the Beilinson–Gaitsgory deformation of the affine flag manifold to the affine Grassmannian. We study the multiplicity functions we introduced in an earlier paper, which encode the data of the Jordan-Holder series. We prove the multiplicity functions are polynomials in q, and we give a sharp bound for their degrees. Our results apply as well to the nearby cycles in the p-adic deformation of Laumon–Haines–Ngo, and also to Wakimoto sheaves.