The spherical Hecke algebra for affine Kac-Moody groups I

Alexander Braverman, David Kazhdan

Published 2008-09-08, updated 2010-09-15Version 3

We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-archimedian field. We prove a generalization of the Satake isomorphism for these algebras, relating it to integrable representations of the Langlands dual affine Kac-Moody group. In the next publication we shall use these results to define and study the notion of Hecke eigenfunction for the group $G_{\aff}$