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Hecke operators and analytic Langlands correspondence for curves over local fields

Pavel Etingof, Edward Frenkel, David Kazhdan

Abstract

We construct analogues of the Hecke operators for the moduli space of G-bundles on a curve X over a local field F with parabolic structures at finitely many points. We conjecture that they define commuting compact normal operators on the Hilbert space of half-densities on this moduli space. In the case F=C, we also conjecture that their joint spectrum is in a natural bijection with the set of opers on X for the Langlands dual group with real monodromy. This may be viewed as an analytic version of the Langlands correspondence for complex curves. Furthermore, we conjecture an explicit formula relating the eigenvalues of the Hecke operators and the global differential operators studied in our previous paper arXiv:1908.09677. Assuming the compactness conjecture, this formula follows from a certain system of differential equations satisfied by the Hecke operators, which we prove in this paper for G=PGL(n).

Hecke operators and analytic Langlands correspondence for curves over local fields

Abstract

We construct analogues of the Hecke operators for the moduli space of G-bundles on a curve X over a local field F with parabolic structures at finitely many points. We conjecture that they define commuting compact normal operators on the Hilbert space of half-densities on this moduli space. In the case F=C, we also conjecture that their joint spectrum is in a natural bijection with the set of opers on X for the Langlands dual group with real monodromy. This may be viewed as an analytic version of the Langlands correspondence for complex curves. Furthermore, we conjecture an explicit formula relating the eigenvalues of the Hecke operators and the global differential operators studied in our previous paper arXiv:1908.09677. Assuming the compactness conjecture, this formula follows from a certain system of differential equations satisfied by the Hecke operators, which we prove in this paper for G=PGL(n).

Paper Structure

This paper contains 25 sections, 45 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Main objects
  3. Hecke operators
  4. The case $F={\mathbb C}$
  5. The case $G=PGL_n$
  6. The structure of the paper
  7. Acknowledgments
  8. The abelian case
  9. The global picture
  10. Hecke operators and Hecke functors
  11. Another derivation of the differential equations
  12. Generalities on $D$-modules and integral transforms
  13. Pull-back
  14. Pull-back in the twisted setting
  15. Push-forward
  16. ...and 10 more sections

Key Result

Theorem 1.1

There exists a canonical isomorphism where $\rho$ is the half-sum of positive roots.

Theorems & Definitions (84)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Conjecture 1.2
  • Remark 1.3
  • Corollary 1.3
  • Proposition 1.4: BK
  • proof
  • Remark 1.4
  • Remark 1.5
  • ...and 74 more