Bruhat-Tits theory from Berkovich’s point of view. I: Realizations and compactifications of buildings. (English) Zbl 1198.51006

In this beautifully written article, the authors investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean local fields.
For a \(k\)-isotropic semisimple algebraic group \(G\) defined over a non-Archimedean valued field \(k\) and for the corresponding Euclidean building \(B(G,k)\) the authors prove that, if \(k\) is complete with respect to its valuation and if \(G\) is almost \(k\)-simple, then for any conjugacy class of proper parabolic \(k\)-subgroups, of type \(t\), there exists a continuous \(G(k)\)-equivariant map \(\theta_t : B(G,K) \to \mathrm{Par}_t(G)^{\mathrm{an}}\) which is a homeomorphism onto its image; the closure of this image is called the Berkovich compactification of type \(t\) of the given Bruhat-Tits building. Cf. [V. G. Berkovich, Spectral theory and analytic geometry over non-archimedean fields. Mathematical Surveys and Monographs, 33. Providence, RI: American Mathematical Society (AMS). (1990; Zbl 0715.14013)] for the notion of a Berkovich \(k\)-analytic space associated to a \(k\)-variety.
The map \(\theta_t\) in fact exists whenever the non-Archimedean valued field \(k\) is such that the Bruhat-Tits building \(B(G,k)\) exists functorially [cf. G. Rousseau, Publ. Math. D’Orsay 77-68, 207 p. (1977; Zbl 0412.22006)].
The paper heavily relies on an intimate knowledge of the theory of algebraic group schemes, and is a very welcome example of how to efficiently work with functoriality properties without losing oneself in abstract category theory.


51E24 Buildings and the geometry of diagrams
20E42 Groups with a \(BN\)-pair; buildings
14L15 Group schemes
Full Text: DOI arXiv Numdam Link