Bruhat–Tits Theory

Introduction

Let $k$ be a non-archimedean local field with ring of integers $\mathcal{O}$ and residue field $\mathbb{F}_q$, and let $\mathbf{G}$ be a connected reductive linear algebraic group over $k$. Bruhat–Tits theory attaches to $\mathbf{G}$ and $k$ a metric simplicial complex $I(\mathbf{G},k)$, the Bruhat–Tits building, on which $G = \mathbf{G}(k)$ acts by isometries with the following properties: the building is a thick affine building of type equal to the affine Weyl group of $\mathbf{G}$, its apartments are the real vector spaces $X_*(S)\otimes\mathbb{R}$ of the maximal $k$-split tori $S$ of $\mathbf{G}$, and its alcoves — the chambers — are the connected components of an apartment after the affine root hyperplanes are removed. The action is transitive on the alcoves and strongly transitive on the pairs (apartment, alcove), the stabilisers of the facets of the building are the parahoric subgroups, and the Bruhat–Tits fixed-point theorem states that every bounded subgroup of $G$ fixes a facet, from which the classification of the maximal bounded (that is, maximal compact) subgroups of $G$ as the stabilisers of the vertices of the building follows.

The theory is the non-archimedean counterpart of the theory of symmetric spaces: where the real reductive group acts on its symmetric space with the maximal compact subgroups as stabilisers, the $p$-adic reductive group acts on its building, the maximal compact subgroups are the vertex stabilisers, and the spherical building at infinity of the building plays the role of the flag manifold at infinity of the symmetric space. The simplest case is $G = SL_2(\mathbb{Q}_p)$, whose building is the Bruhat–Tits tree: the vertices are the homothety classes of $\mathbb{Z}_p$-lattices in $\mathbb{Q}_p^2$, the edges join the classes of nested lattices, the apartments are the bi-infinite geodesic lines, and the affine Weyl group acts by the affine transformations of a line, generated by the reflections in two adjacent vertices. This case is the one in which the whole theory is visible, and it is the source of the classical applications: the amalgam $SL_2(\mathbb{Z}[1/p]) \cong SL_2(\mathbb{Z}) *_{\Gamma_0(p)} SL_2(\mathbb{Z})$ of Bass–Serre Theory, the classification of the maximal compact subgroups of $SL_2(\mathbb{Q}_p)$, and the fixed-point theorem for the actions of finite groups.

The article develops the local-field data and the affine root system, the definition of the building, its metric and its apartments, the detailed example of $SL_n$, the parahoric and Iwahori subgroups, the fixed-point theorems, the buildings at infinity, and the classical applications. The input from above is the theory of the buildings, apartments and Tits systems of Buildings and Tits Systems, the trees and splittings of Bass–Serre Theory, the root systems and Weyl groups of Root Systems and Classification in Part I, and the algebraic groups, arithmetic lattices and congruence subgroups of Arithmetic Groups. The non-archimedean local field, its ring of integers and its residue field, and the affine root system are defined in line, since no article above introduces them; the residue field, the valuation and the completion are standard and are cited to the literature.

The boundary with Part III is the one fixed for this block. What is developed here is the metric and combinatorial structure of the building and the group-theoretic consequences: the affine root data, the apartment, the alcoves, the building, the parahorics, the fixed-point theorems, the buildings at infinity and the amalgam decompositions. What is deferred is the analytic theory: the Hecke algebras $\mathcal{H}(G, K)$ and $\mathcal{H}(G,I)$ of the parahoric and Iwahori subgroups, the spherical functions and the Satake and Macdonald isomorphisms, the harmonic analysis on $G$ and on the building, and the automorphic representations of the adelic groups — all of which belong to Analysis on Groups, where the convolution algebra and the measure are available. The CAT(0) metric structure of the building is stated as a theorem of Bruhat and Tits and quoted from the literature, and the residue field and the valuation are standard; the local fields themselves are not covered here. No physics is invoked.

The Local Data and the Affine Root System

Non-Archimedean Local Fields

Definition. No menu article stands above this one, so the object is defined here and marked as standard (Serre, Local Fields; the entry is in the bibliography). A non-archimedean local field is a field $k$ together with a discrete valuation $v : k^\times \to \mathbb{Z}$ such that $k$ is complete for the distance induced by the absolute value $|x| = q^{-v(x)}$ and the residue field $\mathbb{F}_q = \mathcal{O}/\mathrm{P}$ is finite. The ring of integers is $\mathcal{O} = \{x \in k : v(x)\geq 0\}$, a local ring with maximal ideal $\mathrm{P} = \{x : v(x)\geq1\}$ generated by a uniformiser $\varpi$, and every element of $k^\times$ is $\varpi^n u$ with $n\in\mathbb{Z}$ and $u \in \mathcal{O}^\times$. The standard examples are the $p$-adic fields $\mathbb{Q}_p$ with $\mathcal{O} = \mathbb{Z}_p$ and $q = p$, and the fields $\mathbb{F}_q((t))$ of formal Laurent series with $\mathcal{O} = \mathbb{F}_q[[t]]$ and uniformiser $t$.

Definition. Let $\mathbf{G}$ be a connected reductive linear algebraic group over the local field $k$, in the sense of Arithmetic Groups: a subgroup of $GL_n$ defined by polynomial equations with coefficients in $k$, reductive in the sense of having no nontrivial connected unipotent normal subgroup. A torus $\mathbf{S} \leq \mathbf{G}$ is $k$-split if it is isomorphic over $k$ to a product of copies of the multiplicative group; the $k$-rank of $\mathbf{G}$ is the dimension of a maximal $k$-split torus, and $\mathbf{G}$ is $k$-anisotropic when this rank is zero.

Definition (apartment and affine Weyl group). Let $\mathbf{S}$ be a maximal $k$-split torus of $\mathbf{G}$ and let $X_*(S) = \operatorname{Hom}(\mathbb{G}_m,\mathbf{S})$ be the group of cocharacters. The apartment of $S$ is the real vector space

$$ \mathcal{A}(S) = X_*(S)\otimes_{\mathbb{Z}}\mathbb{R} , $$

of dimension the $k$-rank $r$ of $\mathbf{G}$. The roots of $\mathbf{G}$ with respect to $\mathbf{S}$ are the nontrivial characters $\alpha \in X^*(S)$ occurring in the adjoint action on the Lie algebra, and for each root $\alpha$ and each integer $n \in \mathbb{Z}$ the affine function

$$ \alpha + n : \mathcal{A}(S) \longrightarrow \mathbb{R}, \qquad x \longmapsto \alpha(x)+n , $$

defines an affine root; the affine roots form an affine root system in the sense of Root Systems and Classification, with the finite Weyl group $W$ of the root system of $\mathbf{S}$ acting on $\mathcal{A}(S)$ by linear reflections and the affine Weyl group $W_{\mathrm{aff}} = W\ltimes \Lambda_r$ acting by affine reflections, where $\Lambda_r$ is the coroot lattice. The normaliser of $S$ in $G$ acts on $\mathcal{A}(S)$ by affine isometries, and the quotient of this action is the extended affine Weyl group $\widetilde{W} = W\ltimes\Lambda$ with $\Lambda$ the group of cocharacters modulo the coroot lattice; it is a Coxeter group only in the split case, and in general it is the semidirect product of the affine Weyl group with the group of diagram automorphisms of the Dynkin diagram.

Definition. The walls of the apartment are the hyperplanes $\{x : \alpha(x)+n = 0\}$ of the affine roots; the connected components of the complement of the walls are the alcoves, and the facets are the faces of the alcoves, that is, the connected components of the intersections of the hyperplanes with the complement of the remaining ones. An alcove is an open simplex of dimension $r$, a facet is a simplex of some dimension between $0$ and $r$, and the closure of an alcove is a fundamental domain for the action of the affine Weyl group $W_{\mathrm{aff}}$ on $\mathcal{A}(S)$, with the alcove

$$ C = \{x \in \mathcal{A}(S) : 0 < \alpha(x) < 1 \ \text{ for all positive roots } \alpha\} $$

as the fundamental alcove, whose walls correspond to the nodes of the affine Dynkin diagram. The type of a facet is the subset of the nodes of the affine diagram corresponding to the walls containing it.

Example (the rank-one apartment). For $\mathbf{G} = SL_2$ over $k$, the maximal split torus is the diagonal torus, the $k$-rank is $1$, the apartment is a line $\mathcal{A}(S) \cong \mathbb{R}$, the affine roots are the functions $x \mapsto 2x+n$ and $x\mapsto -2x+n$, the alcoves are the open intervals between consecutive half-integers, and the affine Weyl group is the infinite dihedral group generated by the reflections in $0$ and $\tfrac12$ (or in the walls of the fundamental alcove), acting on the line by translation by $1$ and reflection. The alcoves are the chambers of the tree below, and the apartment is the model for all the others.

The Bruhat–Tits Building

The Definition

Definition. Let $\mathbf{G}$ be a connected reductive group over the non-archimedean local field $k$ and let $S$ be a maximal $k$-split torus. The Bruhat–Tits building $I(\mathbf{G},k)$ is the quotient

$$ I(\mathbf{G},k) = \bigl(G \times \mathcal{A}(S)\bigr)\big/\sim , $$

where $G = \mathbf{G}(k)$ acts on the first factor by left translation and the equivalence relation identifies $(g,x)$ with $(g',x')$ exactly when there is $n \in N_G(S)(k)$ with $x' = nx$ and $g^{-1}g'n$ lying in the subgroup of $G$ fixing $x$ pointwise. The image of $\{g\}\times\mathcal{A}(S)$ is the apartment $g\mathcal{A}(S)$; the images of the facets and the alcoves of $\mathcal{A}(S)$ are the facets and the alcoves of the building; and $G$ acts on $I(\mathbf{G},k)$ by $g\cdot[h,x] = [gh,x]$. Two points of the building lie in a common apartment, and the building is recovered from any apartment together with the action of $G$, so the construction is independent of $S$ up to a canonical isomorphism.

Theorem (Bruhat–Tits). Let $G = \mathbf{G}(k)$ be as above.

(a) The building $I(\mathbf{G},k)$ is a thick affine building of type $\widetilde{W}$; its apartments are the conjugates $g\mathcal{A}(S)$ of the standard apartment, each isometric to $\mathcal{A}(S)$; its residues are the spherical buildings of the parahoric subgroups, and its building at infinity is the spherical building of $\mathbf{G}$ over $k$.

(b) $G$ acts on $I(\mathbf{G},k)$ by isometries and type-preserving simplicial automorphisms, transitively on the alcoves; the stabiliser of an alcove is an Iwahori subgroup, and the stabilisers of the facets are the parahoric subgroups.

(c) The group $G$ acts strongly transitively in the following sense: for every pair consisting of an apartment and an alcove in it, the stabiliser of the alcove acts transitively on the apartments containing it, and the pairs (alcove, apartment) form a single $G$-orbit; consequently $G$ has a Tits system with $B$ an Iwahori subgroup, $N$ the normaliser of $S$, and Weyl group the affine Weyl group, in the split case.

(d) The building is a complete metric space with the metric obtained by gluing the Euclidean metrics of the apartments; it is $\mathrm{CAT}(0)$, and the action of $G$ is by isometries with the apartments as flat subspaces of maximal dimension.

Proof sketch. The construction of the building and the identification of the apartments are direct from the quotient description, the relation being designed so that the pointwise stabiliser of a facet is the corresponding parahoric subgroup and the stabiliser of an alcove is an Iwahori subgroup. The building axioms (that any two alcoves lie in a common apartment and that every panel lies in at least three alcoves) are verified by reducing to the case of the standard apartment with the action of the affine Weyl group and the Bruhat decomposition for the parahoric subgroups. The metric is well defined because the stabiliser of a point acts by isometries of the apartment containing it, and the $\mathrm{CAT}(0)$ property is the theorem of Bruhat and Tits, proved by checking the link conditions. The details are the theorems of Bruhat and Tits and are quoted from the literature, the $\mathrm{CAT}(0)$ statement among them.

The Building of $SL_n$

Example ($SL_n$ over a local field). Let $\mathbf{G} = SL_n$ over $k$ with ring of integers $\mathcal{O}$. The building $I(SL_n,k)$ is the simplicial complex whose vertices are the homothety classes $[L]$ of the $\mathcal{O}$-lattices $L \subseteq k^n$, where $L \sim L'$ when $L' = \varpi^mL$ for some $m \in \mathbb{Z}$, and whose simplices are the sets of classes $\{[L_0],\dots,[L_d]\}$ for which representatives can be chosen with

$$ L_0 \supsetneq L_1 \supsetneq \cdots \supsetneq L_d \supsetneq \varpi L_0 . $$

Equivalently, the vertices are the classes of lattices and a set of $d+1$ vertices is a simplex when the corresponding classes form a chain of lattices modulo homothety and the successive quotients are the $\mathcal{O}$-modules with total length $n$, that is, $\bigoplus_i L_i/L_{i+1}$ has length $n$; the maximal simplices have $n$ vertices, one for each position of a maximal chain, so the building has dimension $n-1$, agreeing with the dimension $n-1$ of the apartments. The apartments are the classes of lattices spanned by the multiples of a fixed $k$-basis of $k^n$, and each apartment is the Coxeter complex of the affine Weyl group $W_{\mathrm{aff}} = S_n\ltimes \mathbb{Z}^{n-1}$, acting on $\mathcal{A} \cong \mathbb{R}^{n}/\mathbb{R}(1,\dots,1)$. The group $SL_n(k)$ acts transitively on the chambers, the stabiliser of a chamber is the Iwahori subgroup of matrices that are upper triangular mod $\mathrm{P}$, and the maximal parahoric subgroups — the stabilisers of the vertices — are the conjugates of $SL_n(\mathcal{O})$, which is the maximal compact subgroup.

Example ($SL_2$ and the tree). For $n = 2$ the building is one-dimensional: the vertices are the homothety classes of $\mathbb{Z}_p$-lattices in $\mathbb{Q}_p^2$, each of which is a vertex of degree $q+1$ because a lattice class has $q+1$ maximal sublattices up to homothety, and the building is a regular tree of degree $q+1$. This is the Bruhat–Tits tree: its apartments are the bi-infinite geodesic lines, its alcoves are the edges, its facets are the vertices and the edges, its Iwahori subgroups are the stabilisers of the edges, and its maximal parahorics are the stabilisers of the vertices, conjugate to $SL_2(\mathbb{Z}_p)$. The tree is the model case of the whole theory examined in Buildings and Tits Systems, and its ends are the points of the projective line $\mathbb{P}^1(k)$.

The Metric Structure

Definition. The Bruhat–Tits metric on $I(\mathbf{G},k)$ is defined by glueing: for two points $x,y$ of the building, choose an apartment containing both, and set $d(x,y)$ equal to the Euclidean distance of the two points in that apartment; the value is independent of the apartment because two apartments intersect in a union of facets and the metrics agree there. The resulting metric is complete, the apartments are isometric copies of $\mathcal{A}(S) \cong \mathbb{R}^r$, and $G$ acts by isometries.

Theorem (metric properties). The building $I(\mathbf{G},k)$ with the Bruhat–Tits metric is a $\mathrm{CAT}(0)$ metric space: it is geodesic, and every geodesic triangle is at least as thin as the Euclidean triangle with the same side lengths. Consequently every bounded subset is contained in a bounded subset of an apartment, the distance to a convex set is realised uniquely, and every finite set of points lies in a common apartment. The latter property implies the fixed-point theorem below.

Proof sketch. The $\mathrm{CAT}(0)$ property is checked on the links of the vertices, where it reduces to the spherical building being $\mathrm{CAT}(1)$; the spherical buildings are flag complexes and the link condition is a combinatorial verification in the Coxeter complex. The consequence for finite sets of points is the standard property of $\mathrm{CAT}(0)$ spaces that finitely many points lie in a flat subspace of the appropriate dimension, combined with the fact that the maximal flat subspaces of the building are exactly the apartments. The statements are Bruhat–Tits's and are quoted from the literature, and the general theory of $\mathrm{CAT}(0)$ spaces is not developed here.

The Group-Theoretic Consequences

Parahoric and Iwahori Subgroups

Definition. A parahoric subgroup of $G = \mathbf{G}(k)$ is the stabiliser of a facet of the building $I(\mathbf{G},k)$, and an Iwahori subgroup is the stabiliser of an alcove. The parahoric subgroups containing a fixed Iwahori subgroup $B$ are the stabilisers of the facets of a fixed apartment containing the alcove of $B$, and they are the standard parahorics of the Tits system of $G$ with respect to $B$; the maximal parahorics are the stabilisers of the vertices of the building.

Theorem (structure of the parahorics). Let $G = \mathbf{G}(k)$ with $k$ non-archimedean and $\mathbf{G}$ connected reductive.

(a) Every parahoric subgroup is open and compact, and every bounded (equivalently, every compact) subgroup of $G$ is contained in a parahoric subgroup, namely the stabiliser of a facet that it fixes.

(b) The maximal bounded subgroups of $G$ are the maximal parahoric subgroups, that is, the stabilisers of the vertices of the building; for a split group the maximal parahoric stabilising the standard vertex of the fundamental alcove is $\mathbf{G}(\mathcal{O})$, and the remaining maximal parahorics are conjugate to the stabilisers of the other vertices of that alcove.

(c) The Iwahori subgroups are conjugate and $G$ has a Bruhat decomposition relative to a parahoric $P$: for $P$ the stabiliser of a facet of type $J$, the double cosets $P w P$ are indexed by the elements $w$ of the quotient of the extended affine Weyl group by the subgroup $W_J$ of the affine Weyl group generated by $J$, and $G$ is their disjoint union.

Proof sketch. (a) The stabiliser of a facet is open and compact because it contains the pointwise stabiliser of the facet, which is a pro-$\mathrm{P}$-group, and is the intersection of $G$ with a compact open subgroup of $GL_n(k)$; conversely a compact subgroup fixes a point of the building by the fixed-point theorem below, and its stabiliser contains the parahoric of the facet generated by the point. (b) The stabiliser of a vertex is maximal among the compact open subgroups because the vertices are the maximal facets of the building and every facet is contained in a vertex. (c) The double coset decomposition is the Bruhat decomposition of the Tits system of $G$ with respect to the extended affine Weyl group, and the parametrisation by the quotient is the affine Weyl group computation. The results are Bruhat–Tits's and are quoted from the literature.

The Fixed-Point Theorems

Theorem (Bruhat–Tits fixed-point theorem). Let $G = \mathbf{G}(k)$ and let $H \leq G$ be a bounded subgroup.

(a) $H$ fixes a facet of the building $I(\mathbf{G},k)$: there is a facet $F$ with $hF = F$ for all $h \in H$, and in fact $H$ fixes $F$ pointwise. The fixed-point set of a compact group of isometries of the building is a non-empty complete subcomplex, and every compact subgroup fixes the smallest facet containing any of its fixed points.

(b) If $H$ is a compact open subgroup, the fixed facet can be taken so that $H$ is contained in its stabiliser, which is a parahoric subgroup, and $H$ is of finite index in the parahoric if it is maximal.

(c) Consequently $\mathbf{G}(k)$ is compact modulo its centre exactly when $\mathbf{G}$ is $k$-anisotropic, that is, when the $k$-rank of $\mathbf{G}$ is zero; in that case the building is a single point and the maximal compact subgroup is $G$ itself.

Proof sketch. (a) The fixed-point set of a compact group of isometries of a $\mathrm{CAT}(0)$ space is non-empty and convex: a minimising sequence for the displacement has a limit in the complete space and the centre of the bounded orbit is fixed. For a compact group acting simplicially on the building, the fixed-point set is a subcomplex, and one shows by descending induction on the dimension that it contains a facet; this is the fixed-point theorem of Bruhat and Tits. (b) The stabiliser of the fixed facet is the parahoric containing the compact open subgroup, and the containment is by definition. (c) If the rank is positive the building is non-compact and a compact subgroup cannot be all of $G$; if the rank is zero the building is a point, the action is trivial and the compactness of $G$ modulo its centre is the standard property of the anisotropic groups. The statements are quoted from the literature.

Corollary (the amalgamation theorem). Let $\mathbf{G}$ be simply connected and semisimple over $k$, let $B$ be an Iwahori subgroup and let $\{P_i\}_{i\in I}$ be the maximal parahorics containing $B$. Then $G = \mathbf{G}(k)$ is generated by the subgroups $P_i$ subject to no relation other than those holding inside the $P_i$ and along their intersections $P_i\cap P_j$, that is, $G$ is the direct limit of the parahorics over the chamber complex of a single alcove; when the $k$-rank is one, so that the building is a tree and the alcove has two vertices, this says that

$$ G = P_1 *_{P_1\cap P_2} P_2 $$

is the amalgam of the two maximal parahorics along their common Iwahori subgroup. Consequently the structure of a simply connected $p$-adic group is determined by the finite data of the parahorics and the affine Dynkin diagram, and in rank one the group is the fundamental group of a graph of groups in the sense of Bass–Serre Theory.

Proof sketch. The group generated by the $P_i$ with the stated relations acts on the building by the given action; the quotient of the building by this action is a single alcove with the vertex groups $P_i$, and the group is the fundamental group of this complex of groups. The rank-one statement is the case in which the alcove is an edge and the complex of groups is a graph of groups with two vertices and one edge, which is exactly the amalgam $P_1*_{P_1\cap P_2}P_2$; the general case is the amalgamation theorem of Bruhat–Tits and Tits, quoted from the literature.

The Building at Infinity and the Classical Applications

The Spherical Building at Infinity

Definition. Let $I(\mathbf{G},k)$ be the Bruhat–Tits building of $k$-rank $r$. The building at infinity $\partial I(\mathbf{G},k)$ is the set of equivalence classes of rays that stay in a bounded neighbourhood of an apartment, with the incidence and the spherical metric induced from the apartments; the boundary of an apartment $\mathcal{A}(S) \cong \mathbb{R}^r$ is the sphere $S^{r-1}$ at infinity, and the building at infinity is a spherical building of type the finite Weyl group of the $k$-root system.

Theorem. The building at infinity $\partial I(\mathbf{G},k)$ is a spherical building of rank $r$ on which $G$ acts by type-preserving automorphisms, with the following properties:

(a) $\partial I(\mathbf{G},k)$ is the building of the Tits system of the maximal parabolic subgroups of $G$, so that its simplices are the cosets of the parabolics and its chambers are the cosets of the minimal parabolics;

(b) the apartments of $\partial I(\mathbf{G},k)$ are the spheres at infinity of the apartments of $I(\mathbf{G},k)$, and the Weyl chambers of the apartment are the alcoves at infinity;

(c) for $\mathbf{G}$ split of rank $r$, the building at infinity is the spherical building of the root system of $\mathbf{G}$ over $k$, so that it is the flag complex of the polar space or the projective space according to the type, as in Buildings and Tits Systems.

Proof sketch. The building at infinity of an affine building is a spherical building whose chambers are the alcoves of the affine building and whose Weyl distance is the finite part of the affine Weyl group; the identification with the cosets of the parabolics is the Tits system of the group with $B$ an Iwahori subgroup and the parabolic subgroups the stabilisers of the facets at infinity. The details are standard and are quoted from the literature.

Trees, Amalgams and the Rank-One Case

Example (the tree and the amalgam $SL_2(\mathbb{Z}[1/p])$). Let $G = SL_2(\mathbb{Q}_p)$ acting on its Bruhat–Tits tree $T$, a regular tree of degree $p+1$. The vertices of $T$ are the homothety classes of $\mathbb{Z}_p$-lattices in $\mathbb{Q}_p^2$, and the group $SL_2(\mathbb{Z})$ — the stabiliser of the vertex of the standard lattice $\mathbb{Z}_p^2$ — fixes a vertex, so the subgroup $SL_2(\mathbb{Z}[1/p]) \leq SL_2(\mathbb{Q}_p)$ acts on $T$ with quotient a segment: the two vertices of the segment are the images of the two vertices of $T$ adjacent along the standard apartment, the two vertex groups are two conjugates of $SL_2(\mathbb{Z})$, and the edge group is their intersection, the congruence subgroup $\Gamma_0(p)$. This is the decomposition

$$ SL_2(\mathbb{Z}[1/p]) \cong SL_2(\mathbb{Z}) *_{\Gamma_0(p)} SL_2(\mathbb{Z}) $$

announced in Bass–Serre Theory: the amalgam is read off from the action of the arithmetic group on the Bruhat–Tits tree, and it is the arithmetic form of the amalgamation theorem above.

Example (the maximal compact subgroups of $GL_2(\mathbb{Q}_p)$). The maximal compact subgroups of $GL_2(\mathbb{Q}_p)$ are the stabilisers of the vertices of the tree, that is, the conjugates of $GL_2(\mathbb{Z}_p)$; the fixed-point theorem says that every compact subgroup lies in one of them, and the Bruhat–Tits tree organises them as the vertices of a tree, with two maximal compact subgroups intersecting in an Iwahori subgroup exactly when the corresponding vertices are adjacent. This recovers the classical classification of the maximal compact subgroups of $GL_n$ over a local field and exhibits the building as the combinatorial object that records the incidences of the compact subgroups.

Fixes and Rigidity

Remark (the role in rigidity). The Bruhat–Tits building is the non-archimedean input into the rigidity theory: the arithmetic groups act on the product of their buildings at the various places, the congruence subgroups are the stabilisers of the facets, and the congruence subgroup problem of Arithmetic Groups is analysed by the action of the group on the building, with the parahorics supplying the congruence quotients and the amalgamation theorem supplying the finite presentation. The spherical building at infinity plays the role of the boundary on which the superrigidity of the lattices operates, and the fixed-point theorem is the mechanism by which compact subgroups are conjugate into the maximal compact subgroups. The rigidity theorems themselves are those of Lattices in Lie Groups and Arithmetic Groups; the buildings of the groups over the various completions are the objects of this article at each place.

The Boundary with Analysis

The theory of this article is metric, combinatorial and group-theoretic; the analysis on the building is Part III.

The Hecke algebras $\mathcal{H}(G,K)$ of a maximal compact subgroup and $\mathcal{H}(G,I)$ of an Iwahori subgroup, the spherical functions and the Satake isomorphism use the convolution algebra of functions on $G$ and therefore the measure and the integration of Part III; the Iwahori–Hecke algebra is a deformation of the group algebra of the affine Weyl group whose structure constants are polynomials in $q$, and its representation theory is Part III.
The harmonic analysis on the building, the spectral theory of the building Laplacian, the Selberg and Ihara type zeta functions of the quotients of the tree and of the higher buildings are Analysis on Groups in Part III.
The automorphic representations of $\mathbf{G}(\mathbb{A}_k)$ and the Langlands correspondence, whose local input is the representation theory of the parahoric and Iwahori subgroups, are Part III; the adeles are standard.
The $\mathrm{CAT}(0)$ geometry of the building and the theory of its isometry groups are stated above as theorems of Bruhat and Tits, quoted from the literature.
What is not deferred: the local field data; the affine root system and the alcoves; the definition of the Bruhat–Tits building; the metric and the $\mathrm{CAT}(0)$ property at the level of statement; the example of $SL_n$ and of the tree; the parahoric and Iwahori subgroups; the fixed-point theorems and their consequences; the amalgamation theorem; the building at infinity; and the arithmetic application to the amalgam of $SL_2(\mathbb{Z}[1/p])$.

Summary

Let $k$ be a non-archimedean local field with ring of integers $\mathcal{O}$, maximal ideal $\mathrm{P}$ and residue field $\mathbb{F}_q$, and let $\mathbf{G}$ be a connected reductive group over $k$ of $k$-rank $r$. The apartment of a maximal $k$-split torus $S$ is the real vector space $\mathcal{A}(S) = X_*(S)\otimes\mathbb{R}$ of dimension $r$, the affine roots $\alpha+n$ cut it into alcoves and facets, and the affine Weyl group $W_{\mathrm{aff}} = W\ltimes\Lambda_r$ acts by affine reflections with the alcoves as a fundamental domain. The Bruhat–Tits building $I(\mathbf{G},k)$ is the quotient of $G\times\mathcal{A}(S)$ by the identification of the apartments along the stabilisers of the facets; it is a thick affine building of type the extended affine Weyl group, its apartments are the conjugates of $\mathcal{A}(S)$, it carries the Bruhat–Tits metric making it a $\mathrm{CAT}(0)$ space, and $G = \mathbf{G}(k)$ acts on it by isometries, transitively on the alcoves and strongly transitively on the pairs (alcove, apartment).

The stabilisers of the facets are the parahoric subgroups and those of the alcoves the Iwahori subgroups; the maximal bounded subgroups of $G$ are the maximal parahorics, the stabilisers of the vertices, of which the standard one is $\mathbf{G}(\mathcal{O})$ for a split group. The Bruhat–Tits fixed-point theorem says that every bounded subgroup of $G$ fixes a facet, whence the classification of the maximal compact subgroups and the compactness of $G$ modulo its centre exactly when $\mathbf{G}$ is $k$-anisotropic. The building for $SL_n$ has as vertices the homothety classes of $\mathcal{O}$-lattices in $k^n$ and as simplices the chains of lattice classes, with apartments the Coxeter complexes of the affine Weyl group $S_n\ltimes\mathbb{Z}^{n-1}$; for $n=2$ it is the regular tree of degree $q+1$. The building at infinity is the spherical building of the maximal parabolic subgroups of $G$, and the action of $SL_2(\mathbb{Z}[1/p])$ on the tree gives the amalgam $SL_2(\mathbb{Z}) *_{\Gamma_0(p)} SL_2(\mathbb{Z})$. The Hecke algebras, the spherical functions and the automorphic representations built on the building belong to Part III.

Summary of Notation

Symbol	Meaning
$k$, $\mathcal{O}$, $\mathrm{P}$, $\varpi$, $\mathbb{F}_q$	Local field, ring of integers, maximal ideal, uniformiser, residue field
$\mathbf{G}$, $G = \mathbf{G}(k)$	Reductive group over $k$; its group of $k$-points
$k$-rank $r$, $k$-anisotropic	Dimension of a maximal $k$-split torus; rank $0$
$\mathbf{S}$, $X_(S)$, $X^(S)$	Maximal $k$-split torus; cocharacters; characters
$\mathcal{A}(S) = X_*(S)\otimes\mathbb{R}$	Apartment, a Euclidean space of dimension $r$
$\alpha + n$	Affine root: the affine function $x\mapsto\alpha(x)+n$ on $\mathcal{A}(S)$
$W$, $W_{\mathrm{aff}} = W\ltimes\Lambda_r$, $\widetilde W$	Weyl group; affine Weyl group; extended affine Weyl group
wall, facet, alcove, type	Hyperplane of an affine root; face of an alcove; chamber; type of a facet
$I(\mathbf{G},k)$	Bruhat–Tits building
$g\mathcal{A}(S)$	Apartment of the building
Bruhat–Tits metric	Metric glued from the Euclidean metrics of the apartments; $\mathrm{CAT}(0)$
parahoric subgroup	Stabiliser of a facet of the building
Iwahori subgroup $B$	Stabiliser of an alcove
Bruhat decomposition for $P$	$G = \bigsqcup_{w} PwP$, $w$ in the quotient of $\widetilde W$
fixed-point theorem	Every bounded subgroup fixes a facet
amalgamation theorem	$G$ is the amalgam of the maximal parahorics along $B$
$\partial I(\mathbf{G},k)$	Building at infinity, a spherical building of rank $r$
$[L]$, homothety class	Vertex of the building of $SL_n$; $\varpi$-multiples identified
$SL_n(\mathcal{O})$	Maximal parahoric of $SL_n(k)$
$\Gamma_0(p)$	Edge group of the amalgam $SL_2(\mathbb{Z}) *_{\Gamma_0(p)} SL_2(\mathbb{Z})$
$\mathcal{H}(G,K)$, $\mathcal{H}(G,I)$	Hecke algebras (Part III)
$\mathrm{CAT}(0)$	Curvature condition of the Bruhat–Tits metric, a theorem of Bruhat and Tits