The spherical Hecke algebra for affine Kac-Moody groups I

TL;DR

This paper introduces the spherical Hecke algebra for affine Kac-Moody groups over local fields and generalizes the Satake isomorphism to relate it to dual affine Kac-Moody representations.

Contribution

It defines the spherical Hecke algebra for affine Kac-Moody groups and proves a generalized Satake isomorphism linking it to Langlands dual representations.

Findings

Established the structure of the spherical Hecke algebra for affine Kac-Moody groups.

Proved a generalized Satake isomorphism for these algebras.

Set the foundation for defining Hecke eigenfunctions in future work.

Abstract

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 $G_{\aff}$