Bruhat-tits Theory from Berkovich’s Point of View. Ii. Satake Compactifications of Buildings

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 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 naturally 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.