BRUHAT-TITS THEORY FROM BERKOVICH%26apos;S POINT OF VIEW. II SATAKE COMPACTIFICATIONS OF BUILDINGS

作者：Remy Bertrand^*; Thuillier Amaury; Werner Annette

来源：Journal of the Institute of Mathematics of Jussieu, 2012, 11(2): 421-465.

DOI：10.1017/S1474748011000168

• algebraic group
• local field
• Berkovich geometry
• Bruhat-Tits building
• compactification

In the paper %26apos;Bruhat-Tits theory from Berkovich%26apos;s point of view. I. Realizations and compactifications of buildings%26apos;, 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 Berkovich%26apos;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%26apos;s original constructions for Riemannian symmetric spaces. %26lt;br%26gt;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%26apos;s general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.

• 出版日期2012-4