On a vanishing conjecture appearing in the geometric Langlands   correspondence
  Let $X$ be a smooth complete curve, and let $Bun_n$ 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 $Av_E^d$ acting from the category $D(Bun_n)$ 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. 
D. Gaitsgory