Abstract involutions of algebraic groups and of Kac-Moody groups

TL;DR

This paper provides a unified framework for understanding involutions in algebraic and Kac-Moody groups using twin buildings and related structures, generalizing previous decompositions and analyzing involution filtrations.

Contribution

It introduces a uniform approach to involutions in algebraic and Kac-Moody groups, extending known decompositions and addressing combinatorial questions.

Findings

Generalized double coset decompositions for algebraic and Kac-Moody groups

Analyzed involution filtrations in twin buildings

Answered a structural question on involution combinatorics

Abstract

Based on the second author's thesis in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac-Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established by Springer and by Helminck-Wang for algebraic groups and by Kac-Wang for certain Kac-Moody groups, we analyze the filtration studied by Devillers-Muhlherr in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised by Bennett-Gramlich-Hoffman-Shpectorov.