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Pré-Publication, Document De Travail Année : 2009

Bruhat-Tits theory from Berkovich's point of view. II. Satake compactifications

Bertrand Remy
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Amaury Thuillier
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Annette Werner
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Résumé

In our previous paper "Bruhat-Tits theory from Berkovich's point of view. I — Realizations and compactifications of buildings", we investigated realizations of the Bruhat-Tits building B(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, and we studied in detail the compactifications of the building which arise from this point of view. In this paper, we give a representation theoretic flavor 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 B(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.
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hal-00405113 , version 1 (18-07-2009)

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Bertrand Remy, Amaury Thuillier, Annette Werner. Bruhat-Tits theory from Berkovich's point of view. II. Satake compactifications. 2009. ⟨hal-00405113⟩
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