Is there an analytic theory of automorphic functions for complex algebraic curves?

Edward Frenkel

Published 2018-12-19Version 1

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 raised the possibility of developing an analytic theory of automorphic forms on the moduli of G-bundles on a complex algebraic curve. Langlands envisioned these forms as eigenfunctions of certain Hecke operators, which he attempted to define. In these notes I show that Hecke operators are well-defined if G is abelian and give a complete description of their eigenfunctions and eigenvalues in this case. However, for non-abelian G, Hecke operators involve integration, which is problematic in the context of complex curves. Nonetheless, automorphic forms can be defined in a different way -- not as eigenfunctions of Hecke operators, but rather as eigenfunctions of a commutative algebra of global differential operators on the line bundle of half-densities on the moduli of G-bundles.