Complete Reducibility in Euclidean Twin Buildings

TL;DR

This paper explores the concept of complete reducibility within Euclidean twin buildings, extending ideas from spherical buildings to analyze subgroups of algebraic groups over Laurent polynomial rings and Kac-Moody groups.

Contribution

It introduces the study of completely reducible subcomplexes in Euclidean twin buildings and their applications to algebraic groups over rings and Kac-Moody groups.

Findings

Established a framework for complete reducibility in Euclidean twin buildings

Connected the theory to subgroups of algebraic groups over Laurent polynomial rings

Provided insights into the structure of Kac-Moody groups

Abstract

Completely reducible subcomplexes of spherical buildings was defined by J.P. Serre and are used in studying subgroups of reductive algebraic groups. We begin the study of completely reducible subcomplexes of twin buildings and how they may be used to study subgroups of algebraic groups over a ring of Laurent polynomials and Kac-Moody groups by looking at the Euclidean twin building case.