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 $F=\mathbb{C}$, we also conjecture that their joint spectrum is in a natural bijection with the set of ${}^{L}G$-opers on X 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. Assuming the compactness conjecture, this formula follows from a certain system of differential equations satisfied by the Hecke operators, which we prove here for $G={\mathrm{PGL}}_n$.