BRUHAT TITS THEORY FROM BERKOVICH'S POINT OF VIEW II SATAKE COMPACTIFICATIONS OF BUILDINGS

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Niveau: Supérieur, Doctorat, Bac+8
BRUHAT-TITS THEORY FROM BERKOVICH'S POINT OF VIEW. II. SATAKE COMPACTIFICATIONS OF BUILDINGS BERTRAND REMY, AMAURY THUILLIER AND ANNETTE WERNER July 2009 Abstract: In the paper Bruhat-Tits theory from Berkovich's point of view. I — Realizations and compactifi- cations of buildings, we investigated various 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 V. Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which nat- urally arise from this point of view. In the present 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. Keywords: algebraic group, local field, Berkovich geometry, Bruhat-Tits building, compactification.

  • space

  • local field

  • vector space over

  • bruhat- tits building

  • satake map

  • archimedean field

  • berkovich compactifications


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