Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 16 (2020), 042, 31 pages      arXiv:1812.08160      https://doi.org/10.3842/SIGMA.2020.042
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?

Edward Frenkel
Department of Mathematics, University of California, Berkeley, CA 94720, USA

Received September 30, 2019, in final form April 27, 2020; Published online May 16, 2020

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.

Key words: Langlands Program; automorphic function; complex algebraic curve; principal $G$-bundle; Jacobian variety; differential operator; oper.

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