Langlands Reciprocity for Algebraic Surfaces

Details Langlands Reciprocity for Algebraic Surfaces Langlands Reciprocity for Algebraic Surfaces

1995-02-20

arxiv.org/abs/q-alg/9502013v1

Mathematics

Algebraic Geometry

13 pages, submitted to Mathematical Research Letters

Scientific paper

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.

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