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Whittaker patterns in the geometry of moduli spaces of bundles on curves. (English) Zbl 1070.11050
Let $$G$$ be a split and connected reductive group over a finite field $$K$$. Let $$\widehat{K}=K((t))$$ be the field of Laurent series. The purpose of this paper is to give a geometric meaning to the notion of a Whittaker function on $$G(\widehat{K})$$. There are a number of conceptional difficulties here. The authors explain how these may be overcome; the problems are quite technical and the bulk of the paper is devoted to their solution. The main application is to a conjecture formulated by the authors and D. A. Kazhdan in their paper [“Geometric realization of Whittaker functions and the Langlands conjecture” J. Am. Math. Soc. 1, 451–484 (1998; Zbl 1068.11501)]. The conjecture has also been proved by B. C. Ngô, and P. Polo [“Résolutions de Demazure affines et formule de Casselman-Shalika géométrique”, J. Algebr. Geom. 10, 515–547 (2001; Zbl 1041.14002)].

##### MSC:
 11R39 Langlands-Weil conjectures, nonabelian class field theory 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11G45 Geometric class field theory 14D20 Algebraic moduli problems, moduli of vector bundles 22E50 Representations of Lie and linear algebraic groups over local fields
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