Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings

Bertrand Rémy; Amaury Thuillier; Annette Werner

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 3, page 461-554
  • ISSN: 0012-9593

Abstract

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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 G an associated 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.

How to cite

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Rémy, Bertrand, Thuillier, Amaury, and Werner, Annette. "Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings." Annales scientifiques de l'École Normale Supérieure 43.3 (2010): 461-554. <http://eudml.org/doc/272160>.

@article{Rémy2010,
abstract = {We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group $\mathrm \{G\}$ over a suitable non-Archimedean field $k$ we define a map from the Bruhat-Tits building $\mathcal \{B\}(\mathrm \{G\},k)$ to the Berkovich analytic space $\mathrm \{G\}^\{\rm an\}$ associated with $\mathrm \{G\}$. Composing this map with the projection of $\mathrm \{G\}^\{\rm an\}$ to its flag varieties, we define a family of compactifications of $\mathcal \{B\}(\mathrm \{G\},k)$. 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.},
author = {Rémy, Bertrand, Thuillier, Amaury, Werner, Annette},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {algebraic group; local field; Berkovich geometry; Bruhat-Tits building; compactification},
language = {eng},
number = {3},
pages = {461-554},
publisher = {Société mathématique de France},
title = {Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings},
url = {http://eudml.org/doc/272160},
volume = {43},
year = {2010},
}

TY - JOUR
AU - Rémy, Bertrand
AU - Thuillier, Amaury
AU - Werner, Annette
TI - Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 3
SP - 461
EP - 554
AB - We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group $\mathrm {G}$ over a suitable non-Archimedean field $k$ we define a map from the Bruhat-Tits building $\mathcal {B}(\mathrm {G},k)$ to the Berkovich analytic space $\mathrm {G}^{\rm an}$ associated with $\mathrm {G}$. Composing this map with the projection of $\mathrm {G}^{\rm an}$ to its flag varieties, we define a family of compactifications of $\mathcal {B}(\mathrm {G},k)$. 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.
LA - eng
KW - algebraic group; local field; Berkovich geometry; Bruhat-Tits building; compactification
UR - http://eudml.org/doc/272160
ER -

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