A compactification of the Bruhat-Tits building
type: Item
instance of: scholarly article
title: A compactification of the Bruhat-Tits building
zbMATH Open document ID: 0935.20034
author: Erasmus Landvogt
published in: Lecture Notes in Mathematics
publication date: 1995-12-13T00:00:00Z
review text: The main object of the book under review is the Bruhat-Tits building \(X(G)\) of a connected reductive group \(G\) defined over a local field \(K\). The author regards \(X(G)\) as a \(p\)-adic analogue of a symmetric space, with an eye towards compactifying it. He constructs such a compactification \(\overline X(G)\) being inspired by a polyhedral construction due to Borel-Serre. In this sense, the present construction generalizes that of \textit{P.~Gérardin} [in: Operator algebras and group representations, Vol. I, Monogr. Stud. Math. 17, 208-221 (1984; Zbl 0536.22018)]. The main idea is as follows: first, to compactify an apartment \(A\) of \(X(G)\) (this is done in \S 2 by accomodating the classical approach of \textit{A.~Ash, D.~Mumford, M.~Rapoport}, and \textit{Y.~S.~Tai} [Smooth compactification of locally symmetric varieties (Math. Sci. Press, Brookline, MA) (1975; Zbl 0334.14007)] to the \(p\)-adic case), and then to use the obtained compactification \(\overline A\) as a ``local model'' for compactifying \(X(G)\). Namely, viewing \(X(G)\) as the set of equivalence classes of \(G(K)\times A\) modulo a suitable equivalence relation, one can define \(\overline X(G)\) as the set of equivalence classes of \(G(K)\times\overline A\) modulo a natural extension of this relation. This construction is described in detail in \S 14 (assuming that the residue field of \(K\) is finite) and is illustrated in \S 15 by the example \(G=\text{SL}_n\). One should note that the exposition is almost self-contained. In particular, the author reproduces an essential part of the Bruhat-Tits theory. Among his technical innovations it is worth pointing out systematic use of the Néron-Raynaud integral models of algebraic tori (see Chapter 10 of \textit{S.~Bosch, W.~Lütkebohmert}, and \textit{M.~Raynaud} [Néron models (Springer, Berlin) (1990; Zbl 0705.14001)]). Another interesting approach is used in the construction of an integral model of \(G\): instead of the representation-theoreti…
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 20-02
zbMATH DE Number: 825913
zbMATH Keywords: reductive groups over local fields
zbMATH Keywords: Bruhat-Tits buildings
zbMATH Keywords: compactifications
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
DOI: 10.1007/BFB0094594
Recommended article: Buildings in the representation theory over local number fields
Recommended article: Q3380873
Recommended article: Q3320605
Recommended article: An intrinsic characterization of Bruhat-Tits buildings inside analytic groups
Recommended article: Bruhat–Tits theory from Berkovich's point of view. II Satake compactifications of buildings
Recommended article: Q5142951
Recommended article: Congruences of parahoric group schemes
Recommended article: Compactification of the Bruhat-Tits building of PGL by seminorms.
Recommended article: Reductive groups over a local field
Recommended article: Representation theory and sheaves on the Bruhat-Tits building
description: scientific article; zbMATH DE number 825913