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Bruhat–Tits theory from Berkovich's point of view. II Satake compactifications of buildings
- Source :
- Journal of the Institute of Mathematics of Jussieu. 11:421-465
- Publication Year :
- 2011
- Publisher :
- Cambridge University Press (CUP), 2011.
-
Abstract
- In the paper ‘Bruhat–Tits theory from Berkovich's point of view. I. Realizations and compactifications of buildings’, we investigated various realizations of the Bruhat–Tits building $\mathcal{B}(\mathrm{G},k)$ of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which naturally arise from this point of view. In the present paper, we give a representation theoretic flavour to these compactifications, following Satake's original constructions for Riemannian symmetric spaces.We first prove that Berkovich compactifications of a building coincide with the compactifications, previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding $\mathcal{B}(\mathrm{G},k)$ in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.
- Subjects :
- Linear algebraic group
Pure mathematics
Absolutely irreducible
General Mathematics
010102 general mathematics
General linear group
01 natural sciences
010101 applied mathematics
Analytic geometry
Algebraic group
Embedding
Equivariant map
Compactification (mathematics)
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 14753030 and 14747480
- Volume :
- 11
- Database :
- OpenAIRE
- Journal :
- Journal of the Institute of Mathematics of Jussieu
- Accession number :
- edsair.doi...........665cb3baabdc3cc8c3a7ea2247942997