A fixed point theorem for Lie groups acting on buildings and applications to Kac-Moody theory

TL;DR

This paper proves a fixed point property for specific groups acting on buildings and applies this to show that topological one-parameter subgroups of Kac-Moody groups are generated by exponentiating certain Lie algebra elements.

Contribution

It introduces a fixed point theorem for groups acting on buildings and applies it to characterize one-parameter subgroups in Kac-Moody groups.

Findings

Fixed point property for groups acting on buildings.

Topological one-parameter subgroups are exponentials of ad-locally finite elements.

Application to Kac-Moody theory confirming subgroup structure.

Abstract

We establish a fixed point property for a certain class of locally compact groups, including almost connected Lie groups and compact groups of finite abelian width, which act by simplicial isometries on finite rank buildings with measurable stabilisers of points. As an application, we deduce amongst other things that all topological one-parameter subgroups of a real or complex Kac-Moody group are obtained by exponentiating ad-locally finite elements of the corresponding Lie algebra.