Wonderful embedding for group schemes in the Bruhat--Tits theory

TL;DR

This paper constructs a new intrinsic and functorial wonderful embedding for certain group schemes in Bruhat--Tits theory, connecting geometric compactifications with arithmetic group structures.

Contribution

It introduces a novel intrinsic construction of wonderful embeddings for group schemes in Bruhat--Tits theory, avoiding classical ambient space methods and extending to non-quasi-split cases.

Findings

Constructed wonderful embeddings for adjoint, quasi-split groups.

Extended the construction to non-quasi-split groups via étale descent.

Established parallels between these embeddings and classical compactifications.

Abstract

For a reductive group $G$ over a discretely valued Henselian field $k$, using valuations of root datum and concave functions, the Bruhat--Tits theory defines an important class of open bounded subgroups of $G(k)$ which are essential objects in representation theory and arithmetic geometry. Moreover, these subgroups are uniquely determined by smooth affine group schemes whose generic fibers are $G$ over the ring of integers of $k$. To study these group schemes, when $G$ is adjoint and quasi-split, we systematically construct wonderful embedding for these group schemes which are uniquely determined by a big cell structure. The way that we construct our wonderful embedding is different from classical methods in the sense that we avoid embedding a group scheme into an ambient space and taking closure. We use an intrinsic and functorial method which is a variant of Artin--Weil method of birational group laws. Beyond the quasi-split case, our wonderful embedding is constructed by \'etale descent. Moreover our wonderful embedding behaves in a similar way to the classical wonderful compactification of $G$. Our results can serve as a bridge between the theory of wonderful compactifications and the Bruhat--Tits theory.