Local geometric Langlands correspondence and affine KacMoody 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 KacMoody algebra of critical level, and, in some cases, as categories of Dmodules on the indschemes G((t))/K. I will give examples of these categories and talk about interrelations between them and supporting evidence for our conjectures. 