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

Let G be a (split connected) reductive group over Fq, and N be its maximal unipotent subgroup. Let BunN be the algebraic stack classifying N–bundles on a smooth projective curve X over Fq. V. Drinfeld has introduced a remarkable partial compactification BunN of BunN . In this paper we study the stack BunN and a certain category of perverse sheaves on it. Let us briefly explain our motivations (for more details the reader is referred to Sect. 1). Consider the group G(K̂) over the local field K̂ = Fq((t)), and its maximal compact subgroupG(Ô), where Ô = Fq[[t]]. The quotient Gr = G(K̂)/G(Ô) can be given a structure of an ind-scheme over Fq, which is called the affine Grassmannian of G. According to the general philosophy of “faisceaux–fonctions” correspondence, the functions on Gr which come from representation theory, such as the spherical and Whittaker functions, should have geometric counterparts as perverse sheaves on Gr. Thus, one is naturally led to the question of constructing and studying certain categories of perverse sheaves on Gr. For example, the geometric counterparts of the Whittaker functions would be the sheaves on Gr, which are N(K̂)–equivariant against a non-degenerate character χ. It is natural to expect that the appropriately defined category of such sheaves should reflect the special properties of the Whittaker functions. Unfortunately, the N(K̂)–orbits in Gr are infinite-dimensional and so there is no obvious way to define this category on Gr. In this paper we argue that an appropriate category can (and perhaps should) be defined on BunN instead. In fact, BunN can be viewed as a suitable “globalization” of the closure of a single N(K̂)–orbit in Gr. Moreover, the former has many advantages over the latter (see Sect. 1.2.2). We define the appropriate “Whittaker category” of perverse sheaves on BunN (more precisely, on its generalization x,∞Bun FT N ) and prove that it does indeed possess all the properties that one could expect from it by analogy with the Whittaker functions. Namely, this category is semi-simple, and each irreducible object of this category is a local system on a single stratum extended by zero (these strata are the analogues of the N(K̂)–orbits in Gr). As an application of the main theorems of this paper, we compute the cohomology of certain sheaves on the intersections of G(Ô)– and N(K̂)–orbits in Gr. In particular, we prove our conjecture from [9] (in the case of GLn this conjecture has been proved by