Hecke operators and analytic Langlands correspondence for curves over local fields

TL;DR

This paper constructs Hecke operators for moduli spaces of G-bundles over curves on local fields, proposing a conjectural analytic Langlands correspondence linking their spectra to opers, and proves related differential equations for G=PGL(n).

Contribution

It introduces analogues of Hecke operators in an analytic setting and conjectures their spectral correspondence with opers, advancing the analytic Langlands program for complex curves.

Findings

Conjecture that Hecke operators define commuting compact normal operators.

Propose a bijection between the joint spectrum of Hecke operators and opers with real monodromy.

Prove differential equations satisfied by Hecke operators for G=PGL(n).

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).