Split Kac-Moody groups over a local field, II. Ordered masures

TL;DR

This paper constructs an ordered affine hovel for split Kac-Moody groups over valued fields, generalizing previous constructions for reductive groups and establishing properties and new results about these groups.

Contribution

It introduces a new construction of ordered affine hovels for split Kac-Moody groups over valued fields, extending prior work and proving new properties and simplicity results.

Findings

Constructed ordered affine hovels for Kac-Moody groups

Proved properties of these hovels analogous to previous definitions

Established new simplicity results for maximal Kac-Moody groups

Abstract

For a split Kac-Moody group (in J. Tits' definition) over a field endowed with a real valuation, we build an ordered affine hovel on which the group acts. This construction generalizes the one already done by S. Gaussent and the author when the residue field contains the complex field and the one by F. Bruhat and J. Tits when the group is reductive. We prove that this hovel has all the properties of ordered affine hovels (masures affines ordonn{\'e}es) as defined previously by the author. We use the maximal Kac-Moody group as defined by O. Mathieu and we prove a few new results about it over any field; in particular we prove, in some cases, a simplicity result for this group. At the end an erratum corrects a mistake in the counter-example of 4.12 3 (c).