Iwahori-Hecke algebras for p-adic loop groups

This paper is a continuation of Braverman and Kazhdan (AnnMath (2) 174(3):1603-1642, 2011) inwhich the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik’s double affine Hecke algebra. Furthermore, using these results,we give an explicit description of the affine Satake isomorphism, generalizing Macdonald’s formula for the spherical function in the finite-dimensional case. The results of this paper have been previously announced in Braverman and Kazhdan (European Congress of Mathematics. European Mathematical Society, Zürich, 2014).