Timothée Marquis

Abstract simplicity of locally compact Kac-Moody groups

In this paper, we establish that complete Kac-Moody groups over finite fields are abstractly simple. The proof makes essential use of Mathieu and Rousseaus 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.

Conjugacy classes and straight elements in Coxeter groups

. As a byproduct of our methods, we also obtain a characterisation of straight elements of W, namely of those elements w∈W for which l(wn)=nl(w) for any n∈Z. In particular, we generalise previous characterisations of straight elements within the class of so-called cyclically fully commutative (CFC) elements, and we give a shorter and more transparent proof that Coxeter elements are straight.

Open subgroups of locally compact Kac-Moody groups

Let

Positive energy representations of double extensions of Hilbert loop algebras

G

HALF-LIE GROUPS

be a complete Kac–Moody group over a finite field. It is known that

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

G