The centralizers of root subgroups in Kac-Moody Steinberg groups

TL;DR

This paper computes the symmetric parts of centralizers of root subgroups in Steinberg groups associated with affine and hyperbolic root systems, revealing complex structures and examples of infinite rank root subsystems.

Contribution

It provides explicit calculations of centralizers in affine and hyperbolic cases, linking affine cases to spherical root systems and uncovering diverse infinite rank root subsystems.

Findings

Centralizers in affine Steinberg groups relate to spherical root systems.

Hyperbolic cases exhibit a diverse 'zoo' of examples.

Many root subsystems of infinite rank are identified.

Abstract

For the affine and hyperbolic root system the symmetric part of the centralizers of root subgroups in the corresponding Steinberg groups are calculated. In the affine case the corresponding root subsystems can be computed in term of the centralizers in the spherical root systems, while in the hyperbolic case there emerges a "zoo" of examples, many of them non-hyperbolic. This also delivers many examples of naturally occuring root subsystems of infinite rank.