Edward Frenkel

Abstract:   The local Langlands correspondence relates admissible representations of a reductive group G over the field of p-adic numbers to homomorphisms from the Galois group of this field to the Langlands dual group G' of G. We would like to develop a similar picture for representations of affine Kac-Moody algebras, the central extensions of the formal loop algebras g((t)) where g is a simple Lie algebra over the field of complex numbers. In this setting we expect the correspondence to be realized at the level of categories. More precisely, we wish to associate to each principal G'-bundle with a connection on the punctured formal disc (this should be viewed as an analogue of the Galois homomorphism) a certain category of representations of the affine Kac-Moody algebra. These categories may then be used to construct a global Langlands correspondence when the G'-local system extends from the punctured disc to a complex algebraic curve.