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

TL;DR

This paper constructs a key functor in the geometric Langlands program for characteristic zero settings and establishes the equivalence of various conjectural forms, advancing the theoretical framework.

Contribution

It provides the first construction of the geometric Langlands functor in characteristic zero and proves the equivalence of multiple conjectural forms.

Findings

Construction of the geometric Langlands functor in characteristic zero.

Proof of equivalence among different forms of the conjecture.

Discussion of structural properties of Hecke eigensheaves.

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.