Whittaker Patterns in the Geometry of Moduli Spaces of Bundles on Curves

Abstract

Let G be a split connected reductive group over a finite field F_q, and N its maximal unipotent subgroup. V. Drinfeld has introduced a remarkable partial compactification of the moduli stack of N-bundles on a smooth projective curve X over F_q. In this paper we study Drinfeld's moduli space and a certain category of perverse sheaves on it. The definition of this category is motivated by the study of the Whittaker functions on the group G(K), where K=F_q((t)). We prove that our category is semi-simple, and that irreducible objects of this category are "clean", i.e., they are extenstions by 0 of local systems supported on the strata. As an application of these results, we obtain a purely geometric proof of the Casselman-Shalika formula for the Whittaker functions.