  The geometric Langlands correspondence for complex algebraic curves differs
from the original Langlands correspondence for number fields in that it is
formulated in terms of sheaves rather than functions (in the intermediate case
of curves over finite fields, both formulations are possible). In a recent
preprint, Robert Langlands made a proposal for developing an analytic theory of
automorphic forms on the moduli space of $G$-bundles on a complex algebraic
curve. Langlands envisioned these forms as eigenfunctions of some analogues of
Hecke operators. In these notes I show that if $G$ is an abelian group then
there are well-defined Hecke operators, and I give a complete description of
their eigenfunctions and eigenvalues. For non-abelian $G$, Hecke operators
involve integration, which presents some difficulties. However, there is an
alternative approach to developing an analytic theory of automorphic forms,
based on the existence of a large commutative algebra of global differential
operators acting on half-densities on the moduli stack of $G$-bundles. This
approach (which implements some ideas of Joerg Teschner) is outlined here, as a
preview of a joint work with Pavel Etingof and David Kazhdan.
