This appendix discusses the integral models of root groups and their Lie algebras.

The development of Bruhat--Tits theory begins with this chapter. 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 \emph{valuation of the root datum} due to Bruhat--Tits and the equivalence relation on such valuations called \emph{equipollence}. An apartment is by definition an equipollence class of valuations of root data. When the group $G$ is quasi-split we show that there is a canonical such equipollence class for each maximal $k$-split torus of $G$. It is constructed using a Chevalley basis, or, more generally, a Chevalley--Steinberg system. When $G$ is not quasi-split we take the existence of an equipollence class as an abstract input, to be specified later. In addition to the apartment, we discuss in this chapter the affine root system and the affine Weyl group, as well as compatibility with Levi subgroups.

Presents the work of DeBacker on parameterization of $G(k)$-conjugacy classes of maximal unramified tori in $G$ using the building. These results, which give a complete and explicit classification, have become an essential tool in representation theory.

Introduces the Moy--Prasad filtrations of the group $G(k)$, its Lie algebra $\mf{g}(k)$, and its dual $\mf{g}^*(k)$. The Moy--Prasad isomorphism theorem is proved.

Presents the two fundamental examples of quasi-split simply connected groups of rank $1$, namely $\tx{SL}_2$ and $\tx{SU}_3$. For these groups the theory can be described very simply and with minimal notation, serving as an entryway into the general theory.

Establishes additional properties that are valid under the assumption that the residue field has dimension at most 1. These include the vanishing of the first Galois cohomology for simply connected semi-simple groups and the classification of connected reductive groups in terms of affine Dynkin diagrams.

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 the study of representation theory and harmonic analysis on $G(k)$.

Gives explicit descriptions of the buildings of classical groups -- general and special linear groups, symplectic, orthogonal, and unitary groups -- in terms of lattice chains.

Constructs the Bruhat--Tits building of $G$ and the parahoric subgroups of $G(k)$, using as an input the apartments constructed in the previous chapter. The key technical device here is that of a \emph{concave function}. This is a function $f : \hat\Phi \to \R$, where $\Phi$ is the root system of $G$ relative to a maximal $k$-split torus $S \subset G$ and $\hat\Phi=\Phi \cup \{0\}$, satisfying the concavity property $f(a+b) \leq f(a)+f(b)$ for all $a,b \in \hat\Phi$ such that $a+b \in \hat\Phi$. If $\cA$ is the apartment associated to $S$ in the previous chapter and $x \in \cA$, the discussion of the previous chapter allows one to construct an open bounded subgroup $G(k)_{x,\,f}$ of $G(k)$. Using the group-theoretic properties of the groups $G(k)_{x,\,f}$ it is shown in \S\ref{sec:iwahori-tits-system} that parahoric subgroups lead to a Tits system, and hence to a (restricted) Tits building. This is the Bruhat--Tits building $\cB(G/k)$ of $G$.

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 the present chapter produce the Bruhat--Tits building and integral models for $G$. While the case of integral models is essentially trivial, as mentioned in the preceding paragraph, the case of the building itself and its properties is far from trivial, and is the main focus of the chapter. The remainder of this chapter establishes further properties, including the existence of a canonical equipollence class of valuations of the root datum of $G$ relative to maximal split torus $S$, the relationship between the building of $G$ and that of a Levi subgroup of $G$, the relationship between concave function groups for $G(k)$ and those for $G(K)$, the important properties of $G$ being \emph{residually quasi-split} or \emph{residually split} are studied in \S\ref{unr:resqs}. Finally, \S\ref{sec:unr_weil} studies the relationship between the building and the parahoric groups for a group and its Weil-restriction of scalars.

Background material. ,Topics include reviews of Henselian fields, fields of dimension at most 1, tori, reductive groups, Chevalley systems and pinnings, integral models, the dynamic method.Some important definitions, such as of the subgroup $G(k)^0$ of $G(k)$, are also given.

This appendix reviews the various operations that can be performed with integral models. While most material in that appendix is well-known, some parts of it have not been covered in the literature in a way that we have found suitable for our purposes, in particular, the discussions of N\'eron dilatations and the Greenberg functor.