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

## TL;DR

This paper explores the development of an analytic theory of automorphic functions for complex algebraic curves, focusing on Hecke operators for abelian groups and proposing differential operators for non-abelian cases.

## Contribution

It provides a complete description of Hecke eigenfunctions for abelian groups and proposes an alternative approach using differential operators for non-abelian groups.

## Key findings

- Hecke operators are well-defined for abelian groups and eigenfunctions are characterized.
- Non-abelian groups involve integration in Hecke operators, posing challenges.
- Differential operators offer a promising alternative for non-abelian cases.

## Abstract

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.