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

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

1999-07-21

arxiv.org/abs/math/9907133v5

Mathematics

Algebraic Geometry

45 pages, Latex; final version to appear in Annals of Mathematics

Scientific paper

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.

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