Almost split Kac–Moody groups over ultrametric fields

Abstract

For a split Kac–Moody group $G$ over an ultrametric field $K$, S. Gaussent and the author defined an ordered affine hovel (for short, a masure) on which the group acts; it generalizes the Bruhat–Tits building which corresponds to the case when $G$ is reductive. This construction was generalized by C. Charignon to the almost split case when $K$ is a local field. We explain here these constructions with more details and prove many new properties, e.g. that the hovel of an almost split Kac–Moody group is an ordered affine hovel, as defined in a previous article.