Bruhat-Tits Theory from Berkovich's Point of View.<br>I - Realizations and Compactifications of Buildings

TL;DR

This paper explores the structure and compactifications of Bruhat-Tits buildings using Berkovich analytic geometry, revealing new insights into boundary strata, stabilizers, and decompositions for reductive groups over non-Archimedean fields.

Contribution

It introduces a novel method to realize and compactify Bruhat-Tits buildings via Berkovich spaces, extending previous results to non-split groups.

Findings

Defined a map from Bruhat-Tits buildings to Berkovich spaces for reductive groups.

Constructed new compactifications of buildings using flag varieties.

Identified boundary strata with buildings of certain parabolics.

Abstract

We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group G over a suitable non-Archimedean field k we define a map from the Bruhat-Tits building B(G,k) to the Berkovich analytic space Gan asscociated with G. Composing this map with the projection of G^an to its flag varieties, we define a family of compactifications of B(G,k). This generalizes results by Berkovich in the case of split groups. Moreover, we show that the boundary strata of the compactified buildings are precisely the Bruhat-Tits buildings associated with a certain class of parabolics. We also investigate the stabilizers of boundary points and prove a mixed Bruhat decomposition theorem for them.