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

Dennis Gaitsgory, Sam Raskin

TL;DR

This work constructs the geometric Langlands functor ${oldsymbol L}_G$ in the de Rham setting by pairing the automorphic category $ ext{D-mod}_{ rac{1}{2}}( ext{Bun}_G)$ with the spectral side $ ext{IndCoh}_{ ext{Nilp}}( ext{LS}_{reve{G}})$ via a coarse functor ${oldsymbol L}_{G, ext{coarse}}$ and a vacuum Poincaré object. A central theorem asserts that coarse sends compact objects to objects bounded below, enabling ${oldsymbol L}_G$ to be defined with the proper left-boundedness and spectral-compatibility; the functor is shown to be ${ m QCoh}( ext{LS}_{reve{G}})$-linear and to admit left/right cohomological amplitude bounds. The Betti version of GLC is developed in parallel, with a Betti vacuum Poincaré object and a Betti Langlands functor ${oldsymbol L}_G^{ ext{Betti}}$, and then linked to the de Rham theory through tempered/restricted variants and Riemann–Hilbert. The paper establishes that the various versions (full/restricted, tempered/non-tempered) are logically equivalent, and it analyzes characteristic cycles of Hecke eigensheaves under the GLC framework. Together, these results lay the groundwork for the full geometric Langlands conjecture and show how de Rham and Betti pictures coherently reflect one another via RH, ultimately enabling a unified proof strategy across settings.

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.

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

TL;DR

This work constructs the geometric Langlands functor in the de Rham setting by pairing the automorphic category with the spectral side via a coarse functor and a vacuum Poincaré object. A central theorem asserts that coarse sends compact objects to objects bounded below, enabling to be defined with the proper left-boundedness and spectral-compatibility; the functor is shown to be -linear and to admit left/right cohomological amplitude bounds. The Betti version of GLC is developed in parallel, with a Betti vacuum Poincaré object and a Betti Langlands functor , and then linked to the de Rham theory through tempered/restricted variants and Riemann–Hilbert. The paper establishes that the various versions (full/restricted, tempered/non-tempered) are logically equivalent, and it analyzes characteristic cycles of Hecke eigensheaves under the GLC framework. Together, these results lay the groundwork for the full geometric Langlands conjecture and show how de Rham and Betti pictures coherently reflect one another via RH, ultimately enabling a unified proof strategy across settings.

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.
Paper Structure (48 sections, 33 theorems, 302 equations)

This paper contains 48 sections, 33 theorems, 302 equations.

Table of Contents

  1. The Langlands functor in the de Rham setting
  2. The Betti setting
  3. Comparison of the different versions of GLC
  4. Characteristic cycles of Hecke eigensheaves
  5. Contents
  6. Notations and conventions
  7. Acknowledgements
  8. Construction of the Langlands functor (de Rham context)
  9. The automorphic category
  10. The spectral action
  11. The vacuum Poincaré sheaf
  12. The coarse version of the functor
  13. The case G=T
  14. Statement of the main result
  15. Compatibility with the spectral action
  16. ...and 33 more sections

Key Result

Theorem 4.1

Suppose that $G$ has connected center, the genus of $X$ is $\geq 2$, and assume the geometric Langlands conjecture. Let $\sigma$ be an irreducible ${\check{G}}$-local system and let ${\mathcal{F}}_{\sigma}$ be the corresponding Hecke eigensheaf. Then the characteristic cycle $\operatorname{CC}({\mat

Theorems & Definitions (66)

  • Remark 3.9
  • Theorem 4.1
  • Remark 1.1.6
  • Theorem 1.2.4
  • Proposition 1.3.3
  • proof
  • Corollary 1.3.4
  • Remark 1.4.3
  • Remark 1.4.4
  • Theorem 1.6.2
  • ...and 56 more