Kac-Moody Symmetric Spaces: arbitrary symmetrizable complex or almost split real type

TL;DR

This paper introduces a new framework for Kac-Moody symmetric spaces that works for all types of generalized Cartan matrices, including affine and non-invertible cases, and explores their Galois descent properties.

Contribution

It proposes an alternative approach to Kac-Moody symmetric spaces that extends the concept to affine and non-invertible cases, addressing previous limitations.

Findings

Constructed Kac-Moody symmetric spaces for all matrix types.

Extended the theory to include affine Kac-Moody groups.

Analyzed Galois descent for almost split real Kac-Moody symmetric spaces.

Abstract

Kac-Moody symmetric spaces have been introduced by Freyn, Hartnick, Horn and the first-named author for centered Kac-Moody groups, that is, Kac-Moody groups that are generated by their root subgroups. In the case of non-invertible generalized Cartan matrices this leads to complications that -- within the approach proposed originally -- cannot be repaired in the affine case. In the present article we propose an alternative approach to Kac-Moody symmetric spaces which for invertible generalized Cartan matrices provides exactly the same concept, which for the non-affine non-invertible case provides alternative Kac-Moody symmetric spaces, and which finally provides Kac-Moody symmetric spaces for affine Kac-Moody groups. In a nutshell, the original intention by Freyn, Hartnick, Horn and K\"ohl was to construct symmetric spaces that likely lead to primitive actions of the Kac-Moody groups; this, of course, cannot work in the affine case as affine Kac-Moody groups are far from simple. Additionally, we study the Galois descent to almost split real Kac-Moody symmetric spaces based on the theory of almost split Kac-Moody groups developed by R\'emy 2002.