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Analytic Langlands correspondence from SoV

Federico Ambrosino, Jörg Teschner

Abstract

The analytic Langlands correspondence proposed by Etingof, Frenkel and Kazhdan describes the solution to the spectral problems naturally arising in the quantisation of the Hitchin integrable systems in terms of real opers, certain second order differential operators on a Riemann surface having real monodromy. We prove this correspondence in the cases associated to the group $\mathrm{PSL}(2,\mathbb{C})$, and Riemann surfaces of genus zero with a number of punctures larger than three. A crucial ingredient is a unitary integral transformation mapping products of solutions to the ordinary differential equation associated to a real oper to eigenfunctions of the quantised Hitchin Hamiltonians. This allows us to construct joint eigenfunctions of Hecke operators and Hitchin Hamiltonians from real opers.

Analytic Langlands correspondence from SoV

Abstract

The analytic Langlands correspondence proposed by Etingof, Frenkel and Kazhdan describes the solution to the spectral problems naturally arising in the quantisation of the Hitchin integrable systems in terms of real opers, certain second order differential operators on a Riemann surface having real monodromy. We prove this correspondence in the cases associated to the group , and Riemann surfaces of genus zero with a number of punctures larger than three. A crucial ingredient is a unitary integral transformation mapping products of solutions to the ordinary differential equation associated to a real oper to eigenfunctions of the quantised Hitchin Hamiltonians. This allows us to construct joint eigenfunctions of Hecke operators and Hitchin Hamiltonians from real opers.

Paper Structure

This paper contains 21 sections, 14 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Analytic Langlands correspondence
  3. Quantisation of the Hitchin system
  4. Baxter operators
  5. Formulation of the analytic Langlands correspondence
  6. Construction of the SoV transformation
  7. Fourier-transformation
  8. Separation of Variables
  9. The inverse transformation
  10. Smoothness
  11. Meromorphic continuation
  12. Outline of the proof of the analytic Langlands correspondence
  13. SoV-transform of eigenstates
  14. Hecke eigenvalue property
  15. An alternative integral representation
  16. ...and 6 more sections

Key Result

Theorem 1

The eigenspaces ${\mathcal{H}}_{\chi}$ are one-dimensional, and the set ${\mathcal{S}}$ is in bijection to the set $\mathrm{Op}_{{\mathbb R}}^{}(C)$ of all real opers on $C$.

Theorems & Definitions (26)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Lemma 3
  • Proposition 2
  • ...and 16 more