The isomorphism problem for almost split Kac-Moody groups

TL;DR

This paper investigates the isomorphism problem for almost split Kac-Moody groups, demonstrating that under certain conditions, isomorphisms preserve key subgroup structures, extending known results from split cases.

Contribution

It establishes that isomorphisms between almost split Kac-Moody groups preserve their twin root data, generalizing prior results from split Kac-Moody groups.

Findings

Isomorphisms preserve twin root data under certain conditions.

Existence of maximal split subgroups in almost split Kac-Moody groups.

Generalization of Borel-Tits result to Kac-Moody groups.

Abstract

We consider the isomorphism problem for almost split Kac--Moody groups, which have been constructed by R\'emy via Galois descent from split Kac-Moody groups as defined by Tits. We show that under certain technical assumptions, any isomorphism between two such groups must preserve the canonical subgroup structure, i.e. the twin root datum associated to these groups, which generalizes results of Caprace in the split case. An important technical tool we use is the existence of maximal split subgroups inside almost split Kac-Moody groups, which generalizes the corresponding result of Borel-Tits for reductive algebraic groups.