Hecke operators and analytic Langlands correspondence for curves over local fields

We construct analogues of the Hecke operators for moduli space G-bundles on a curve X over local field F with parabolic structures at finitely many points. conjecture that they define commuting compact normal Hilbert half-densities this space. In case F=C, we also their joint spectrum is in natural bijection set LG-opers real monodromy. This may be viewed as an analytic version Langlands correspondence complex curves. Furthermore, explicit formula relating eigenvalues and global differential operators. Assuming compactness conjecture, follows from certain system equations satisfied by operators, which prove here G= PGLn.