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

从Berkovich视角看Bruhat-Tits理论 I——建筑的实现与紧化

Bruhat-Tits Theory from Berkovich's Point of View. I - Realizations and Compactifications of Buildings

该研究利用Berkovich解析几何研究Bruhat-Tits建筑的紧化，构造了从建筑到对应解析空间的映射并得到紧化族，推广了分裂群情形的结果，还刻画了紧化边界层并证明边界点稳定子群的混合Bruhat分解定理。

Bruhat-Tits buildings and analytic geometry

This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry canbe used to compactify buildings. We discuss in detail the example of the special linear group. Moreover, we give an intrinsic description of Bruhat-Tits buildings in the framework of non-Archimedean analytic geometry.

Wonderful compactifications of Bruhat-Tits buildings in the non-split case

Given an adjoint semisimple group $G$ over a local field $k$, we prove that the maximal Satake-Berkovich compactification of the Bruhat-Tits building of $G$ can be identified with the one obtained by embedding the building into the Berkovich analytification of the wonderful compactification of $G$, extending previous results of Rémy, Thuillier and Werner. In the process, we use the characterisation of the wonderful compactification in terms of Hilbert schemes given by Brion to extend the definition of the wonderful compactification to the case of a non-necessarily split adjoint semisimple grou…

Bruhat-Tits 建筑的奇妙紧化

Wonderful compactifications of Bruhat-Tits buildings

本文研究局部域上分裂半单群对应的欧几里得建筑，证明其最大 Satake-Berkovich 紧化可等变等同于通过嵌入奇妙紧化关联的 Berkovich 空间所得紧化，并探讨无穷远结构关系。

Polyhedral compactifications of Bruhat-Tits buildings of quasi-reductive groups

Given a quasi-reductive group $G$ over a local field $k$, using Berkovich geometry, we exhibit a family of $G(k)$-equivariant compactifications of the Bruhat-Tits building $\mathcal B(G, k)$, constructed and investigated by Solleveld and Lourenço. The compactification procedure consists in mapping the building in the analytification $G^{\mathrm{an}}$ of $G$, then composing this map with the projections from $G^{\mathrm{an}}$ to its (in general non-compact) pseudo-flag varieties $(G/P)^{\mathrm{an}}$, for $P$ ranging among the pseudo-parabolic subgroups of $G$. This generalises previous constru…

Maximal connected k-subgroups of maximal rank in connected reductive algebraic k-groups

Let $k$ be any field and let $G$ be a connected reductive algebraic $k$-group. Associated to $G$ is an invariant first studied by Satake and Tits that is called the index of $G$ (a Dynkin diagram along with some additional combinatorial information). Tits showed that the $k$-isogeny class of $G$ is uniquely determined by its index and the $k$-isogeny class of its anisotropic kernel $G_a$. For the cases where $G$ is absolutely simple, Satake and Tits classified all possibilities for the index of $G$. Let $H$ be a connected reductive $k$-subgroup of maximal rank in $G$. We introduce an invariant…