Towards 2-Dimensional Geometric Langlands Duality

In this talk we plan to define and study the spherical Hecke algebra for (untwisted) affine Kac-Moody groups over a local non-archimedian field. We shall prove a generalization of the Satake isomorphism for these algebras, relating it to integrable representations of the Langlands dual affine Kac-Moody group. If time permits, we might discuss how these results can be used to define and study the notion of Hecke eigenfunction in the affine case. This is a joint work with D.Kazhdan.