The spherical Hecke algebra for affine Kac-Moody groups I

Abstract

We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-archimedian field. We prove a generalizationof the Satake isomorphism for this algebra, relating it to integrable repre-sentations of the Langlands dual affine Kac-Moody group. In the nextpublication we shall use these results to define and study the notion ofHecke eigenfunction for the group Gaff.