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We construct analogues of the Hecke operators for the moduli space ofG-bundles on a curve X over a local field F with parabolic structures atfinitely many points. We conjecture that they define commuting compact normaloperators on the Hilbert space of half-densities on this moduli space. In thecase F=C, we also conjecture that their joint spectrum is in a naturalbijection with the set of opers on X for the Langlands dual group with realmonodromy. This may be viewed as an analytic version of the Langlandscorrespondence for complex curves. Furthermore, we conjecture an explicitformula relating the eigenvalues of the Hecke operators and the globaldifferential operators studied in our previous paper arXiv:1908.09677. Assumingthe compactness conjecture, this formula follows from a certain system ofdifferential equations satisfied by the Hecke operators, which we prove in thispaper for G=PGL(n).

Hecke operators and analytic Langlands correspondence for curves over local fields