This book gives a comprehensive treatment of the Bruhat-Tits theory over discretely valued Henselian fields. It describes the affine building associated to a connected reductive group defined over a discretely valued Henselian field and the associated structures such as the affine root system and the parahoric group schemes.
Given a connected reductive group \(G\) defined over a discretely valued Henselian field \(k\) with perfect residue field, the theory produces a contractible topological space \(\mathcal{B}(G)\) called the (reduced) Bruhat-Tits building of \(G\). This topological space has the structure of a polysimplicial complex and the topological group \(G(k)\) acts on \(\mathcal{B}(G)\) by automorphisms that preserve the polysimplicial structure.
Chapters 1 and 2 collect much of the background material that is used throughout the book (affine spaces and affine root systems, Tits systems, abstract buildings, bounded open subgroups of reductive groups, Chevalley systems).
Chapter 3 presents the two fundamental examples of quasi-split simply connected groups of rank 1, namely \(\mathrm{SL}_{2}\) and \(\mathrm{SU}_{3}\).
Chapter 4 contains a summary of Bruhat-Tits theory and related material. In the first paragraph, the authors state most major results of the theory in the form of axioms.
Chapter 5 provides the first major application of Bruhat-Tits theory, namely the various decompositions of the topological group \(G(k)\) known under the names of Bruhat, Cartan, and Iwasawa. These decompositions are an essential tool in representation theory and harmonic analysis on \(G(k)\).
In Chapter 6, the development of Bruhat-Tits theory begins. The main purpose of this chapter is to define the apartment associated to a maximal \(k\)-split torus. The fundamental object used in the construction of the apartment is the notion of a valuation of the root datum due to Bruhat-Tits and the equivalence relation on such valuations (equipollence).
Chapter 7 constructs the Bruhat-Tits building of \(G\) and the parahoric subgroups of \(G(k)\), using as an input the apartments constructed in Chapter 6.
Chapter 8 constructs the integral models of \(G\) associated to points of \(\mathcal{B}(G)\) (or more generally pairs of a point and a concave function).
Chapter 9 builds on the fact that for any connected reductive \(k\)-group \(G\), the base change \(G_{K}\) is quasi-split, and hence the Bruhat-Tits building and corresponding integral models for \(G_{K}\) have already been constructed. From these, the arguments of Chapter 9 produce the Bruhat-Tits building and integral models for \(G\). While the case of integral models is essentially trivial, the case of the building itself and its properties is far from trivial, and is the main focus of the chapter.
Chapter 10 is concerned with the additional results that hold when one assumes that the residue field \(\mathfrak{f}\) of \(k\) has dimension at most 1. Among the main results of this chapter are the following: (a) superspecial points of \(\mathcal{B}(G)\) exist only when \(G\) is quasi-split; (b) an anisotropic group \(G\) is automatically of type \(\mathrm{A}_{n}\); (c) anisotropic tori exist when \(G\) is semi-simple; (d) \(\mathrm{H}^{1}(k,G)\) can be described in terms of the building of \(G\).
Chapter 11 introduces the Kottwitz homomorphism and uses it to describe the component group of the special fiber of a given parahoric integral model.
Chapter 12 considers a connected reductive \(K\)-group \(H\) equipped with an action of a finite group \(\Theta\) and studies the relationship between the buildings of \(H\) and \(G = (H^{\Theta})^{0}\), under the assumption that the order of \(\Theta\) is prime to the residue field characteristic.
Chapter 13 introduces the Moy-Prasad filtrations of the group \(G(k)\) (see [A. Moy and the second author, Invent. Math. 116, No. 1–3, 393–408 (1994; Zbl 0804.22008)]), its Lie algebra \(\mathfrak{g}(k)\), and its dual \(\mathfrak{g}^{\ast}(k)\).
Chapter 14 discusses the functorial property of the Bruhat-Tits building.
Chapter 15 gives explicit descriptions of the buildings of classical groups (general and special linear groups, symplectic, orthogonal, and unitary groups) in terms of lattice chains. This description is also given by F. Bruhat and J. Tits in [Bull. Soc. Math. Fr. 112, 259–301 (1984; Zbl 0565.14028); Bull. Soc. Math. Fr. 115, 141–195 (1987; Zbl 0636.20027)].
Chapter 16 presents the work [Mich. Math. J. 54, No. 1, 157–178 (2006; Zbl 1118.22005)] of S. DeBacker on parameterization of \(G(k)\)-conjugacy classes of maximal unramified tori in \(G\) using Bruhat-Tits theory. Chapter 17 gives a similar parameterization of the \(G(k)\)-conjugacy classes of tamely ramified maximal tori.
Chapter 18 presents the formula of the second author [Publ. Math., Inst. Hautes Étud. Sci. 69, 91–117 (1989; Zbl 0695.22005)] for the covolume of S-arithmetic subgroups of an absolutely simple simply connected group over a global field. The derivation of this formula involves a considerable amount of Bruhat-Tits theory, even if \(S\) consists only of Archimedean places.