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A general framework for the analytic Langlands correspondence

Pavel Etingof, Edward Frenkel, David Kazhdan

Abstract

We discuss a general framework for the analytic Langlands correspondence over an arbitrary local field F introduced and studied in our works arXiv:1908.09677, arXiv:2103.01509 and arXiv:2106.05243, in particular including non-split and twisted settings. Then we specialize to the archimedean cases (F=C and F=R) and give a (mostly conjectural) description of the spectrum of the Hecke operators in various cases in terms of opers satisfying suitable reality conditions, as predicted in part in arXiv:2103.01509, arXiv:2106.05243 and arXiv:2107.01732. We also describe an analogue of the Langlands functoriality principle in the analytic Langlands correspondence over C and show that it is compatible with the results and conjectures of arXiv:2103.01509. Finally, we apply the tools of the analytic Langlands correspondence over archimedean fields in genus zero to the Gaudin model and its generalizations, as well as their q-deformations.

A general framework for the analytic Langlands correspondence

Abstract

We discuss a general framework for the analytic Langlands correspondence over an arbitrary local field F introduced and studied in our works arXiv:1908.09677, arXiv:2103.01509 and arXiv:2106.05243, in particular including non-split and twisted settings. Then we specialize to the archimedean cases (F=C and F=R) and give a (mostly conjectural) description of the spectrum of the Hecke operators in various cases in terms of opers satisfying suitable reality conditions, as predicted in part in arXiv:2103.01509, arXiv:2106.05243 and arXiv:2107.01732. We also describe an analogue of the Langlands functoriality principle in the analytic Langlands correspondence over C and show that it is compatible with the results and conjectures of arXiv:2103.01509. Finally, we apply the tools of the analytic Langlands correspondence over archimedean fields in genus zero to the Gaudin model and its generalizations, as well as their q-deformations.
Paper Structure (64 sections, 65 theorems, 334 equations)

This paper contains 64 sections, 65 theorems, 334 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Summary of the main results
  4. Structure of the paper
  5. Acknowledgments
  6. Analytic Langlands correspondence over a general local field
  7. Varieties over arbitrary fields
  8. Forms of reductive groups
  9. Forms of reductive groups over a local field
  10. Principal $G$-bundles
  11. Moduli of $G$-bundles on a smooth projective curve
  12. The form of $G$ attached to a principal bundle and a point
  13. Principal bundles on curves over a local field
  14. Hecke operators
  15. Hecke operators for effective divisors with coefficients in the coweight lattice
  16. ...and 49 more sections

Key Result

Theorem 2.2

(Kneser, Bruhat-Tits, BT, 4.7) For a simply connected semisimple group $G$ over $F$, one has $H^1(\Gamma_F,G(F_{\rm sep}))=1$.

Theorems & Definitions (153)

  • Example 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • Proposition 2.7
  • proof
  • ...and 143 more