Abstract simplicity of locally compact Kac-Moody groups

TL;DR

This paper proves that complete Kac-Moody groups over finite fields are abstractly simple, using Mathieu-Rousseau's construction and properties of root subgroups under Weyl group conjugation.

Contribution

It establishes the simplicity of complete Kac-Moody groups over finite fields, leveraging a novel approach involving root subgroup contraction by Weyl group elements.

Findings

Complete Kac-Moody groups over finite fields are abstractly simple.

Root subgroups are contracted by conjugation of Weyl group elements.

The proof relies on Mathieu-Rousseau's construction of these groups.

Abstract

In this paper, we establish that complete Kac-Moody groups over finite fields are abstractly simple. The proof makes an essential use of Mathieu-Rousseau's construction of complete Kac-Moody groups over fields. This construction has the advantage that both real and imaginary root spaces of the Lie algebra lift to root subgroups over arbitrary fields. A key point in our proof is the fact, of independent interest, that both real and imaginary root subgroups are contracted by conjugation of positive powers of suitable Weyl group elements.