Langlands reciprocity for algebraic surfaces

Abstract

Twenty-five years ago R. Langlands proposed [L] a "fantastic generalization" of Artin-Hasse reciprocity law in the classical class field theory.He conjectured the existence of a correspondence between automorphic irreducible infinite-dimensional representations of a reductive group G over a global number field on the one hand, and (roughly speaking) finite dimensional representations of the Galois group of the field, on the other hand.Following an earlier idea of A. Weil, Langlands' conjecture was reinterpreted by V. Drinfeld [Dr 3] and G. Laumon [La] in purely geometric terms.In the complex geometry setup the number field gets replaced by a Riemann surface X, the Galois group of the field gets replaced by the fundamental group, π 1 (X), and an automorphic representation gets replaced by a perverse sheaf, cf.[BBD], on the moduli space of algebraic principal G-bundles on X.The purpose of this note is to formulate some results and conjectures related to Langlands reciprocity for algebraic surfaces, as opposed to algebraic curves.That would amount, in number theory, to a non-abelian higher dimensional generalization of the class field theory, the subject that remains a total mystery at the moment.A key step of our geometric approach is a construction of Hecke operators for vector bundles on an algebraic surface.The main point of this paper is that in certain cases the corresponding algebra of Hecke operators turns out to be a homomorphic image of the quantum toroidal algebra.The latter is a quantization, in the spirit of Drinfeld [Dr 1] and Jimbo [Ji], of the universal enveloping algebra of the universal central extension of a "double-loop" Lie algebra.This yields, in particular, a new geometric construction of affine quantum groups of types A (1) , D (1) and E (1) in terms of Hecke operators for an elliptic surface.