Langlands Reciprocity for Algebraic Surfaces

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 groupG over a global number field on the one hand, and (roughly speaking) finite dimensionsional 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, π1(X), and an automorphic representation gets replaced by a perverse sheaf, cf. [BBD], on the moduli space of algebraic principal G-bundles on X .