  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).
