Wonderful compactifications of Bruhat-Tits buildings in the non-split case

TL;DR

This paper extends the understanding of compactifications of Bruhat-Tits buildings for non-split groups over local fields, linking them to wonderful compactifications and analyzing Galois actions.

Contribution

It generalizes the construction of compactifications to non-split semisimple groups over arbitrary fields and explores Galois fixed points in these compactifications.

Findings

Identified the maximal Satake-Berkovich compactification with the embedding into the Berkovich analytification of the wonderful compactification.

Extended the definition of the wonderful compactification to non-split groups using Hilbert schemes.

Analyzed Galois actions on compactifications and characterized Galois-fixed points.

Abstract

Given an adjoint semisimple group $G$ over a local field $k$, we prove that the maximal Satake-Berkovich compactification of the Bruhat-Tits building of $G$ can be identified with the one obtained by embedding the building into the Berkovich analytification of the wonderful compactification of $G$, extending previous results of R\'emy, Thuillier and Werner. In the process, we use the characterisation of the wonderful compactification in terms of Hilbert schemes given by Brion to extend the definition of the wonderful compactification to the case of a non-necessarily split adjoint semisimple group over an arbitrary field and investigate some of its properties pertaining to rational points on the boundary. Lastly, given a finite possibly ramified Galois extension $k'/k$, we take a look at the action of the Galois group on the maximal compactification of the building of $G$ over $k'$ and check that the Galois-fixed points are precisely the limits of sequences of fixed points in the building over $k'$, though they may not lie in the Satake-Berkovich compactification of $G$ over $k$.