Whittaker patterns in the geometry of moduli spaces of bundles on curves. (English) Zbl 1070.11050

Frenkel, E.; Gaitsgory, D.; Vilonen, K.

Ann. Math. (2) 153, No. 3, 699-748 (2001).

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)].

Reviewer: Samuel James Patterson (Göttingen)

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

Keywords:

Whittaker functions; moduli stack; Drinfeld theory; Hecke algebra; reductive group; local field; perverse sheaves

Citations:

Zbl 1041.14002; Zbl 1068.11501