Langlands Reciprocity for Algebraic Surfaces

Abstract

This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra generated by the Hecke operators turns out to be a homomorphic image of the {\it quantum toroidal algebra}. The latter is a quantization, in the spirit of Drinfeld-Jimbo, 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, D E in terms of Hecke operators for an elliptic surface.