On a vanishing conjecture appearing in the geometric Langlands correspondence

D. Gaitsgory

paper

On a vanishing conjecture appearing in the geometric Langlands correspondence

arXiv:math/0204081

Abstract

Let $X$ be a smooth complete curve, and let $B u n_{n}$ be the moduli stack of rank $n$ vector bundles on $X$ . Let $E$ be a local system on $X$ . In a recent paper of E.Frenkel, K.Vilonen and the author, it was shown that the vanishing of a certian functor $A v_{E}^{d}$ acting from the category $D (B u n_{n})$ to itself, implies the geometric Langlands conjecture. In this paper we establish the required vanishing result. Our proof works for sheaves with char=0 coefficients, or with torsion coefficients when the parameter $d$ is less than the characteristic.

Revised version