Holomorphic completions of affine Kac-Moody groups

TL;DR

This paper develops a functional analytic framework for affine Kac-Moody groups by constructing holomorphic loop groups, enabling a comprehensive understanding of their complexifications and symmetric spaces.

Contribution

It introduces the construction of holomorphic loop groups and affine Kac-Moody groups as tame Fréchet manifolds, providing a foundational basis for their complexification and symmetric space theory.

Findings

Holomorphic loop groups are tame Fréchet manifolds.

Complete description of complex Kac-Moody groups.

Solution to the complexification problem of loop groups.

Abstract

We construct holomorphic loop groups and their associated affine Kac-Moody groups and prove that they are tame Fr\'echet manifolds. These results form the functional analytic basis for the theory of affine Kac-Moody symmetric spaces, presented first in the authors thesis. Our approach also solves completely the problem of complexification of loop groups; it allows a complete description of complex Kac-Moody groups and their non-compact real forms.