Almost split Kac-Moody groups over ultrametric fields

TL;DR

This paper explores the geometric structures called hovels associated with almost split Kac-Moody groups over ultrametric fields, extending known constructions from reductive groups and proving new properties of these hovels.

Contribution

It provides detailed explanations and new proofs that the hovels of almost split Kac-Moody groups are ordered affine hovels, generalizing previous work on split cases.

Findings

Hovels of almost split Kac-Moody groups are ordered affine hovels.

Extended the construction of hovels to the almost split case over local fields.

Proved new properties of these hovels in the context of Kac-Moody groups.

Abstract

For a split Kac-Moody group G over an ultrametric field K, S. Gaussent and the author defined an ordered affine hovel 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.