Langlands reciprocity for algebraic surfaces.

*(English)*Zbl 0914.11040From the introduction: 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. A key step of the authors’ 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 and 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 \(A^{(1)}\), \(D^{(1)}\) and \(E^{(1)}\) in terms of Hecke operators for an elliptic surface.

The authors’ approach was motivated in part by a relation between instantons on ALE-spaces and affine Lie algebras discovered by H. Nakajima [Duke Math. J. 76, 365-416 (1994; Zbl 0826.17026)] (which was in its turn motivated by Lusztig’s construction of quantum groups in terms of quivers and the Lagrangian construction of the first author. Nakajima’s results were also used by Vafa and Witten in verifying a special case of their \(S\)-duality conjecture. The authors believe that there is in fact a very close connection between the Langlands’ reciprocity for an algebraic surface and the \(S\)-duality conjecture for the underlying real 4-manifold. Suffice it to say that the interplay between a reductive group \(G\) and the Langlands dual group \(G^\vee\) is quite essential in both cases.

Proofs of the results announced in this paper will appear elsewhere, see http://xxx.lanl.gov/abs/q-alg/9505012.

The authors’ approach was motivated in part by a relation between instantons on ALE-spaces and affine Lie algebras discovered by H. Nakajima [Duke Math. J. 76, 365-416 (1994; Zbl 0826.17026)] (which was in its turn motivated by Lusztig’s construction of quantum groups in terms of quivers and the Lagrangian construction of the first author. Nakajima’s results were also used by Vafa and Witten in verifying a special case of their \(S\)-duality conjecture. The authors believe that there is in fact a very close connection between the Langlands’ reciprocity for an algebraic surface and the \(S\)-duality conjecture for the underlying real 4-manifold. Suffice it to say that the interplay between a reductive group \(G\) and the Langlands dual group \(G^\vee\) is quite essential in both cases.

Proofs of the results announced in this paper will appear elsewhere, see http://xxx.lanl.gov/abs/q-alg/9505012.

##### MSC:

11G45 | Geometric class field theory |

11R39 | Langlands-Weil conjectures, nonabelian class field theory |

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |

19F05 | Generalized class field theory (\(K\)-theoretic aspects) |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |