## Abstract

Let X be a smooth projective algebraic curve of genus > 1 over and algebraically closed field k of characteristic p > 0. Denote by Bun_{n} (resp. Loc_{n}) the moduli stack of vector bundles of rank n on X (resp. the moduli stack of vector bundles of rank n endowed with a connection). Let also D_{BUnn} denote the sheaf of crystalline differential operators on Bun_{n} (cf. e.g. [3]). In this paper we construct an equivalence φ_{n} between the bounded derived category D^{b}(M(O_{Locn}^{O})) of quasi-coherent sheaves on some open subset Loc_{n} ^{O} ⊂ Loc_{n} and the bounded derived category D^{b}(M(D_{Bunn}^{O})) of the category of modules over some localization D_{Bunn}^{O} of D_{Bunn}. We show that this equivalence satisfies the Hecke eigen-value property in the manner predicted by the geometric Langlands conjecture. In particular, for any ε ∈ Loc_{n} ^{O} we construct a "Hecke eigen-module" Aut_{ε}. The main tools used in the construction are the Azumaya property of D_{Bunn} (cf. [3]) and the geometry of the Hitchin integrable system. The functor φ_{n} is defined via a twisted version of the Fourier-Mukai transform.

Original language | English |
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Pages (from-to) | 153-179 |

Number of pages | 27 |

Journal | Pure and Applied Mathematics Quarterly |

Volume | 3 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 |

Externally published | Yes |