Proof of the geometric Langlands conjecture I: construction of the functor

Abstract

We construct the geometric Langlands functor in one direction (from the automorphic to the spectral side) in characteristic zero settings (i.e., de Rham and Betti). We prove that various forms of the conjecture (de Rham vs Betti, restricted vs. non-restricted, tempered vs. non-tempered) are equivalent. We also discuss structural properties of Hecke eigensheaves.