logo Local geometric Langlands correspondence and
affine Kac-Moody algebras I

In this talk I will give an introduction to my recent joint work with D.Gaitsgory (part II will be given by Gaitsgory later in the week).
By a local geometric Langlands correspondence for a complex reductive group G we understand a construction which assigns to a local system on the punctured disc for the Langlands dual group of G, a category equipped with an action of the formal loop group G((t)).
We propose a conjectural description of these categories as categories of representations of the corresponding affine Kac-Moody algebra of critical level, and, in some cases, as categories of D-modules on the ind-schemes G((t))/K.
I will give examples of these categories and talk about interrelations between them and supporting evidence for our conjectures.