Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings

Abstract

We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group G over a suitable non-Archimedean field k we define a map from the Bruhat-Tits building B(G, k) to the Berkovich analytic space G an asscociated with G. Composing this map with the projection of G an to its flag varieties, we define a family of compactifications of B (G, k). This generalizes results by Berkovich in the case of split groups.Moreover, we show that the boundary strata of the compactified buildings are precisely the Bruhat-Tits buildings associated with a certain class of parabolics. We also investigate the stabilizers of boundary points and prove a mixed Bruhat decomposition theorem for them. AMS classification (2000): 20E42, 51E24, 14L15, 14G22.1. In the mid 60ies, F. Bruhat and J. Tits initiated a theory which led to a deep understanding of reductive algebraic groups over valued fields [BT72], [BT84]. The main tool (and a concise way to express the achievements) of this long-standing work is the notion of a building. Generally speaking, a building is a gluing of (poly)simplicial subcomplexes, all isomorphic to a given tiling naturally acted upon by a Coxeter group [AB08]. The copies of this tiling in the building are called apartments and must satisfy, by definition, strong incidence properties which make the whole space very symmetric. The buildings considered by F. Bruhat and J. Tits are Euclidean ones, meaning that their apartments are Euclidean tilings (in fact, to cover the case of non-discretely valued fields, one has to replace Euclidean tilings by affine spaces acted upon by a Euclidean reflection group with a non-discrete, finite index, translation subgroup [Tit86]). A Euclidean building carries a natural non-positively curved metric, which allows one to classify in a geometric way maximal bounded subgroups in the rational points of a given non-Archimedean semisimple algebraic group. This is only an instance of the strong analogy between the Riemannian symmetric spaces associated with semisimple real Lie groups and Bruhat-Tits buildings [Tit75]. This analogy is our guideline here.Indeed, in this paper we investigate Bruhat-Tits buildings and their compactification by means of analytic geometry over non-Archimedean valued fields, as developed by V. Berkovich -see [Ber98] for a survey. Compactifications of symmetric spaces is now a very classical topic, with well-known applications to group theory (e.g., group cohomology [BS73]) and to number theory (via the study of some relevant moduli spaces modeled on Hermitian symmetric spaces [Del71]). For deeper motivation and a broader scope on compactifications of symmetric spaces, we refer to the recent book [BJ06], in which the case of locally symmetric varieties is also covered. One of our main results is to construct for each semisimple group G over a suitable non-Archimedean valued field k, a family of compactifications of the Bruhat-Tits building B(G, k) of G over k. This fami...

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

“…Theorem 1.4 (see [12,Theorem 4.1]). For any parabolic subgroup Q of G, we use the map ι −1 Q • ϑ t to embed B t (Q ss , k) into Osc t (Q) an ⊂ Par t (G) an .…”

“…Each complete fan F of strictly convex rational polyhedral cones on Λ(S) gives rise to a compactificationΛ(S) F of this vector space, defined by gluing together the compactifications of the cones C ∈ F. More generally, one can compactify in this way any affine space under Λ(S). Proposition 1.6 (see [12,Proposition 3.35]). Let S be a maximal split torus of G. The compactified apartmentĀ t (S, k) is canonically homeomorphic to the compactification of A(S, k)/ Φ associated with the complete fan F t .…”

“…Notations and conventions from [12] are recalled in Section 1. Let us stress one particular working hypothesis: the results in [12] were obtained under a functoriality assumption for buildings with respect to non-Archimedean extension of the ground field (see [12, 1.3.4] for a precise formulation). This assumption, which is fulfilled in particular if k is discretely valued with perfect residue field or if the group under consideration is split, is made throughout the present work.…”

Maximality of hyperspecial compact subgroups avoiding Bruhat–Tits theory

“…The construction of the metric · L is inspired by the embedding of the Bruhat-Tits building in the flag varieties defined by Berkovich and Rémy-Thuillier-Werner [19].…”

Curve Complexes Versus Tits Buildings: Structures and Applications

“…The paper [150] constructs the compactification by using Berkovich analytic geometry over complete non-Archimedean fields, and the paper [151] uses irreducible representations of the algebraic group and is more similar to the Satake compactifications of symmetric spaces. The construction in [171] is also similar to the Satake compactifications of symmetric spaces.…”