Representations of affine Kac-Moody groups over local and global fields: a survey of some recent results

TL;DR

This survey reviews recent developments in the representation theory of affine Kac-Moody groups over local and global fields, highlighting analogs of classical theories such as Satake isomorphism and Eisenstein series.

Contribution

It compiles recent results suggesting the existence of an analogous rich representation theory for affine Kac-Moody groups similar to classical reductive groups.

Findings

Affine Satake isomorphism established

Development of affine Iwahori-Hecke algebra

Construction of affine Eisenstein series

Abstract

Let G be a reductive algebraic group over a local field K or a global field F. It is well know that there exists a non-trivial and interesting representation theory of the group G(K) as well as the theory of automorphic forms on the corresponding adelic group. The purpose of this paper is to give a survey of some recent constructions and results, which show that there should exist an analog of the above theories in the case when G is replaced by the corresponding affine Kac-Moody group (which is essentially built from the formal loop group G((t)) of G). Specifically we discuss the following topics : affine (classical and geometric) Satake isomorphism, affine Iwahori-Hecke algebra, affine Eisenstein series and Tamagawa measure.