Compactification

of the Bruhat-Tits Building of PGL

by Lattices of Smaller Rank

Annette Werner

Received: April 19, 2001 Revised: September 3, 2001

Communicated by Peter Schneider

Abstract. In this paper we construct a compactification of the Bruhat-Tits building associated to the group P GL(V) by attaching all the Bruhat-Tits buildings of P GL(W) for non-trivial subspaces W of V as a boundary.

2000 Mathematics Subject Classification: 20E42, 20G25

Keywords and Phrases: Bruhat-Tits buildings, PGL, compactification

1 Introduction

Let K be a non-archimedean local field and V a finite-dimensional vector space over K. In this paper we construct a compactification of the Bruhat-Tits building X associated to the group P GL(V) by attaching all the Bruhat-Tits buildings of P GL(W) for non-trivial subspaces W of V as a boundary. Since the vertices of such a building correspond to homothety classes of lattices of full rank in W, we can also view this process as attaching to X (whose underlying simplicial complex is defined by lattices of full rank in V) all the lattices in V of smaller rank. This compactification differs from both the Borel-Serre compactification and Landvogt’s polyhedral compactification of X. The different features of these three constructions can be illustrated in the case of a three-dimensional V by looking at the compactification of one apartment:

Borel-Serre  Landvogt  Our compactification

In the Borel-Serre compactification, the points on the boundary correspond to the rays, i.e. to the half-lines emanating from the origin, and parallel lines have the same limits. In Landvogt’s compactification, parallel lines have different limit points, whereas the rays in one segment (chamber) all converge to the corner vertex. In our compactification all rays contained in the two chambers around a boundary vertex converge to this vertex, and so do lines which are parallel to the middle axis. The two rays at the boundary of this double chamber (they look shorter in our picture) converge to points on the boundary lines, and their parallels converge to different points on these lines. The idea to attach lattices of smaller rank to compactify X already appeared in Mustafin’s paper [Mu]. The goal of this paper is a generalization of Mumford’s p-adic Schottky uniformization to higher dimensions. Mustafin’s construction and investigation of the compactified building take up about one page. He works with lattices and defines the compactification as the union of X and all the lattices in V of smaller rank, i.e. he only uses a set of vertices as the boundary. His construction remains rather obscure (at least to the author), and does not include proofs. The construction of our compactification is based on the same idea of attaching lattices of smaller rank, but is entirely different. First we compactify one apartment Λ in X (corresponding to a maximal torus T in P GL(V)) by attaching some apartments of lower dimension corresponding to certain tori which are quotients of T. We define a continuous action of the normalizer N of T on this compactification \overline{Λ}. Then we glue all the compactified apartments corresponding to maximal tori in P GL(V) together. To be precise, we take a certain compact subgroup U_{0}^{∧} in G, and we define for each x ∈ \overline{Λ} a subgroup P_{x} of G, which turns out later to be the stabilizer of x. Our compactification \overline{X} is defined as the quotient of U_{0}^{∧} × \overline{Λ} by the following equivalence relation: (g, x) ∼ (h, y) iff there exists some n in N such that nx = y and g^{−1}hn lies in P_{x}. This is similar to the construction of the building X. Then we prove the following results: X is an open, dense subset of \overline{X}, and \overline{X} carries a G-action compatible with the one on X. Besides, \overline{X} is compact and contractible and can be identified with the union of all Bruhat-Tits buildings corresponding to non-zero subspaces W of V. In order to prove these results, we have to investigate in detail the structure of our stabilizer groups P_{x}. In particular, we show a mixed Bruhat decomposition theorem for them.

It is of course a natural question whether a similar compactification also exists for other groups. At present, I can see no generalization of this approach to arbitrary reductive groups, but probably some other cases can be treated individually. In order to facilitate such generalizations we work in a group theoretic set up where possible, and do not use the realization of X as the space of norms up to similarity. Moreover, it would be interesting to see if there exists an analogue of our construction in the archimedean world of symmetric spaces. Acknowledgements: I would like to thank C. Deninger, E. Landvogt and P. Schneider for helpful discussions concerning this paper.

2 The Bruhat-Tits building for P GL

Throughout this paper we denote by K a non-archimedean local field, by R its valuation ring and by k the residue class field. Besides, v is the valuation map, normalized so that it maps a prime element to 1. We adopt the convention that “⊂” always means strict subset, whereas we write “⊆”, if equality is permitted. Let V be an n-dimensional vector space over K. Let us recall the definition of the Bruhat-Tits building for the group G = P GL(V) (see [Br-Ti] and [La]). We fix a maximal K-split torus T and let N = N_{G}T be its normalizer. Note that T is equal to its centralizer in G. We write G = G(K), T = T(K) and N = N(K) for the groups of rational points. By X_{∗}(T) respectively X^{∗}(T) we denote the cocharacter respectively the character group of T. We have a natural perfect pairing

<, >: X_{∗}(T) × X^{∗}(T) ⟶ ℤ
(λ, χ)  ⟼< λ, χ >,

where < λ, χ > is the integer such that χ ∘ λ(t) = t^{<λ,χ>} for all t ∈ 𝔾_{m}. Let Λ be the ℝ-vector space Λ = X_{∗}(T) ⊗_{ℤ} ℝ. We can identify the dual space Λ^{∗} with X^{∗}(T) ⊗_{ℤ} ℝ, and extend <, > to a pairing

<, >: Λ × Λ^{∗} ⟶ ℝ.

Since <, > is perfect, there exists a unique homomorphism ν : T → Λ such that

< ν(z), χ >= −v(χ(z))

for all z ∈ T and χ ∈ X^{∗}(T) (compare [La], Lemma 1.1). Besides, by [La], Proposition 1.8, there exists an affine Λ-space A together with a homomorphism ν : N → Aff(A) extending ν : T → Λ. Here Aff(A) denotes the space of affine bijections A → A. The pair (A, ν) is unique up to unique isomorphism. It is called the empty appartment defined by T. Let g be the Lie algebra of G. We have the root decomposition

g = g^{T} ⊕ ⨁_{a∈Φ} g_{a},

where Φ = Φ(T, G) is the set of roots and where g^{T} = {X ∈ g : Ad(t)X = X for all t ∈ T } and g_{a} = {X ∈ g : Ad(t)X = a(t)X for all t ∈ T } for all a ∈ Φ (see [Bo], 8.17 and 21.1). By [Bo], 21.6, Φ is a root system in Λ^{∗} with Weyl group W = N/T. For all a ∈ Φ there exists a unique closed, connected, unipotent subgroup U_{a} of G which is normalized by T and has Lie algebra g_{a} (see [Bo], 21.9). We denote the K-rational points of U_{a} by U_{a}. In our case G = P GL(V) we can describe these data explicitly. Our torus T is the image of a maximal split torus T^{∼} in GL(V). Hence there exists a basis v_{1}, …, v_{n} of V such that T^{∼} is the group of diagonal matrices in GL(V) with respect to v_{1}, …, v_{n}. From now on we will fix such a basis. Let N^{∼} be the normalizer of T^{∼} in GL(V). Then N is the image of N^{∼} in P GL(V). Hence N is the semidirect product of T and the group of permutation matrices, which is isomorphic to W = N/T. Since W is the Weyl group corresponding to Φ, it acts as a group of reflections on Λ, and we have a natural homomorphism

W ⟶ GL(Λ).

Since Aff(Λ) = Λ ⋊ GL(Λ), we can use this map together with ν : T → Λ to define

ν : N = T ⋊ W ⟶ Λ ⋊ GL(Λ) = Aff(Λ).

Hence (Λ, ν) is an empty apartment, and we write from now on A = Λ. Denote by χ_{i} the character

χ_{i} :  T^{∼}  ⟶ 𝔾_{m}
( \begin{matrix}t_{1} \\ & . & . & . \\ & & & & t_{n} \end{matrix} ) ⟼ t_{i}.

Then for all i and j we have characters a_{ij} := χ_{i} − χ_{j}, and

Φ = {a_{ij} : i ≠ j}.

For a = a_{ij} we define now U_{a}^{∼} as the subgroup of GL(V) such that U_{a}^{∼}(\overline{K}) is the group of matrices U = (u_{kl})_{k,l} such that the diagonal elements u_{kk} are equal to one, u_{ij} is an element in \overline{K} and the rest of the entries u_{kl} is zero. Its image in P GL(V) is isomorphic to U_{a}^{∼} and coincides with the group U_{a}. Define

ψ_{a} : U_{a} ⟶ ℤ ∪ {∞}

by mapping the matrix U = (u_{kl})_{k,l} to v(u_{ij}). Then we put for all l ∈ ℤ

U_{a,l} = {u ∈ U_{a} : ψ_{a}(u) ≥ l}.

We also define U_{a,∞} = {1}, and U_{a,−∞} = U_{a}. An affine function θ : Λ → ℝ of the form θ(x) = a(x) + l for some a ∈ Φ and some l ∈ ℤ is called an affine root. We can define an equivalence relation ∼ on Λ as follows:

x ∼ y iff θ(x) and θ(y) have the same sign

or are both equal to 0 for all affine roots θ.

The equivalence classes with respect to this relation are called the faces of Λ. These faces are simplices which partition Λ (see [Bou], V, 3.9). There exists a W -invariant scalar product on Λ (uniquely determined up to scalar factor), see [Bou], VI, 1.1 and 1.2, and all the reflections at affine hyperplanes are contained in ν(N). For all x ∈ Λ let U_{x} be the group generated by U_{a,−a(x)} = {u ∈ U_{a} : ψ_{a}(u) ≥ −a(x)} for all a ∈ Φ. Besides, put N_{x} = {n ∈ N : ν(n)x = x}, and

P_{x} = U_{x}N_{x} = N_{x}U_{x}.

Now we are ready to define the building X = X(P GL(V)) as

X = G × Λ/ ∼,

where the equivalence relation ∼ is defined as follows (see [La], 13.1):

(g, x) ∼ (h, y)  iff there exists an element n ∈ N
such that ν(n)x = y and g^{−1}hn ∈ P_{x}.

We have a natural action of G on X via left multiplication on the first factor. The G-action on X continues the N-action on Λ, so that we will write nx for our old ν(n)x if x ∈ Λ and n ∈ N. Besides, we can embed the apartment Λ in X, mapping a ∈ Λ to the class of (1, a). (This is injective, see [La], Lemma 13.2.) For x ∈ Λ the group P_{x} is the stabilizer of x. A subset of X of the form gΛ for some g ∈ G is called apartment in X. Similarly, we define the faces in gΛ as the subsets gF, where F is a face in Λ. Then two points (and even two faces) in X are always contained in a common apartment ([La], Proposition 13.12 and [Br-Ti], 7.4.18). Any apartment which contains a point of a face contains the whole face, and even its closure (see [La], 13.10, 13.11, and [Br-Ti], 7.4.13, 7.4.14). We fix once and for all a W -invariant scalar product on Λ, which induces a metric on Λ. Using the G-action it can be continued to a metric d on the whole of X (see [La], 13.14 and [Br-Ti], 7.4.20). Note that if n = 2, then X is an infinite regular tree, with q + 1 edges meeting in every vertex, where q is the cardinality of the residue field. We denote by X^{0} the set of vertices (i.e. 0-dimensional faces) in X. We define a simplex in X^{0} to be a subset {x_{1}, …, x_{k}} of X^{0} such that x_{1}, …, x_{k} are the vertices of a face in X. Let η_{i} : 𝔾_{m} → T be the cocharacter induced by mapping x to the diagonal matrix with diagonal entries d_{1}, …, d_{n} such that d_{k} = 1 for k ≠ i and d_{i} = x. Then η_{1}, …, η_{n−1} is an ℝ-basis of Λ, and the set of vertices in Λ is equal to ⨁_{i=1}^{n−1} ℤη_{i}.

Let L be the set of all homothety classes of R-lattices of full rank in V. We write {M} for the class of a lattice M. Two different lattice classes {M′} and {N′} are called adjacent, if there are representatives M and N of {M′} and {N′} such that

πN ⊂ M ⊂ N.

This relation defines a flag complex, namely the simplicial complex with vertex set L such that the simplices are the sets of pairwise adjacent lattice classes. We have a natural G-action on L preserving the simplicial structure. Moreover, there is a G-equivariant bijection

φ : L ⟶ X^{0}

preserving the simplicial structures. If {N} ∈ L can be written as {N} = g{M} for some g ∈ G and M = π^{k_{1}}Rv_{1} + … + π^{k_{n}}Rv_{n}, then φ({N}) is given by the pair (g, φ{M}) ∈ G × Λ, where

φ({M}) = ∑_{i=1}^{n−1}(k_{n} − k_{i})η_{i}

is a vertex in Λ.

3 Compactification of one apartment

We write \underline{n} for the set \{1, …, n\}. We continue to fix the basis v_{1}, …, v_{n} of V and the maximally split torus T from section 2. Recall that Λ = ⨁_{i=1}^{n−1} ℝη_{i}, and that η_{n} satisfies the relation η_{1}+…+η_{n} = 0. We will often write Λ = ∑_{i=1}^{n} ℝη_{i}, bearing this relation in mind. Let now I be a non-empty subset of \underline{n}, and let V_{I} be the subspace of V generated by the v_{i} for i ∈ I. We write G^{V_{I}} for the subgroup of G = P GL(V) consisting of the elements fixing the subspace V_{I}, and G_{I} for the group P GL(V_{I}). Then we have a natural restriction map

ρ_{I} : G^{V_{I}} ⟶ G_{I}.

The torus T is contained in G^{V_{I}}, and its image under ρ_{I} is a maximal K-split torus T_{I} in G_{I}, namely the torus induced by the diagonal matrices with respect to the base {v_{i} : i ∈ I} of V_{I}. As usual, we write T_{I}, G_{I} and G^{V_{I}} for the groups of K-rational points. We put Λ_{I} = X_{∗}(T_{I}) ⊗_{ℤ} ℝ. Then ρ_{I} induces a surjective homomorphism ρ_{I∗} : X_{∗}(T) → X_{∗}(T_{I}), hence a surjective homomorphism of ℝ-vector spaces

r_{I} : Λ ⟶ Λ_{I}.

For all i ∈ I we write η_{i}^{I} for the cocharacter of T_{I} induced by mapping x to the diagonal matrix with entry x at the i-th place, i.e. η_{i}^{I} = ρ_{I∗}η_{i}.

Then Λ_{I} = ∑_{i∈I} ℝη_{i}^{I}, subject to the relation ∑_{i∈I} η_{i}^{I} = 0. In particular, Λ_{{i}} = 0. Note that r_{I}(∑_{i=1}^{n} x_{i}η_{i}) = ∑_{i∈I} x_{i}η_{i}^{I}. Let ν_{I} : T_{I} → Λ_{I} be the unique homomorphism satisfying < ν_{I}(z), χ >= −v(χ(z)) for all χ ∈ X^{∗}(T_{I}). It is compatible with ν, i.e. the following diagram is commutative:

T ⟶^{ν} Λ
ρ_{I} ↓  ↓^{r_{I}}
T_{I} ⟶^{ν_{I}} Λ_{I}

Now we define

\overline{Λ} = Λ ∪ ⋃_{∅≠I⊂\underline{n}} Λ_{I} = ⋃_{∅≠I⊆\underline{n}} Λ_{I}.

Recall that we write “⊂” for a strict subset, and “⊆” if equality is permitted. Here Λ\underline{n} = Λ and r\underline{n} is the identity. Let us now define a topology on \overline{Λ}. For all I ⊂ \underline{n} we put

D_{I} = ∑_{i∉I} ℝ_{≥0}(−η_{i}).

We think of D_{I} as a “corner” around Λ_{I}. For all open and bounded subsets U ⊂ Λ we define

C_{U}^{I} = (U + D_{I}) ∪ ⋃_{I⊆J⊂\underline{n}} r_{J}(U + D_{I}).

We take as a base of our topology on \overline{Λ} the open subsets of Λ together with these sets C_{U}^{I} for all non-empty I ⊂ \underline{n} and all open bounded subsets U of Λ. Note that every point x ∈ \overline{Λ} has a countable fundamental system of neighborhoods. This is clear for x ∈ Λ. If x is in Λ_{I} for some I ⊂ \underline{n}, then choose some z ∈ Λ with r_{I}z = x, and choose a countable decreasing fundamental system of bounded open neighborhoods (V_{k})_{k≥1} of z in Λ. Put U_{k} = V_{k} + ∑_{i∉I} k(−η_{i}). This is an open neighborhood of z + k ∑_{i∉I}(−η_{i}). Then (C_{U_{k}}^{I})_{k≥1} is a fundamental system of open neighborhoods of x. Hopefully the next result will shed some light on the definition of the topological space \overline{Λ}. Recall from section 2, that we have a G-equivariant bijection φ between equivalence classes of lattices of full rank in V and vertices in the building X. If we restrict φ to lattices which can be diagonalized with respect to v_{1}, …, v_{n}, i.e. which have an R-basis consisting of multiples of these elements, then we get a bijection between these diagonal lattices and vertices in Λ. Applying this to the group G_{I} = P GL(V_{I}), we get a bijection φ_{I} between classes of diagonal lattices in V_{I} with respect to the v_{i} for i ∈ I, and vertices in Λ_{I}.

Proposition 3.1 Let (M_{k})_{k≥1} be a sequence of diagonal lattices in V and let N be a diagonal lattice in V_{I}. The sequence of vertices φ({M_{k}}) converges to the vertex φ_{I}(\{N\}) ∈ Λ_{I} in our topology on \overline{Λ} iff after passing to a subsequence there are lattices M′_{k} equivalent to M_{k} such that M′_{k+1} ⊆ M′_{k} and such that ⋂_{k} M′_{k} is equivalent to N.

Proof: We can write M′_{k} = ⨁_{i=1}^{n} π^{a_{i,k}}Rv_{i} for some integers a_{i,k}. Since M′_{k+1} ⊂ M′_{k}, we have a_{i,k+1} ≥ a_{i,k}. Therefore for all i the sequence a_{i,k} becomes stationary or goes to infinity, so that ⋂_{k} M′_{k} = ⨁_{i∈I′} π^{a_{i}}Rv_{i}, where I′ is the set of all i, such that a_{i,k} becomes stationary, i.e. a_{i,k} = a_{i} for all k big enough. Let us call this intersection module N′. It is a lattice in V_{I′}, and by assumption equivalent to N, so that I = I′. Besides, we have φ({M′_{k}}) = ∑_{i=1}^{n}(−a_{i,k})η_{i}, and φ_{I}({N′}) = ∑_{i∈I}(−a_{i})η_{i}^{I}. If k is big enough, we have φ({M′_{k}}) = ∑_{i∈I}(−a_{i})η_{i} + ∑_{i∉I}(−a_{i,k})η_{i}, with a_{i,k} arbitrarily large. If we take one of the systems of fundamental neighborhoods of φ_{I}({N}) = φ_{I}({N′}) constructed previously, we find that every one of them must contain a point φ({M′_{k}}), so that φ({M_{k}}) converges indeed to φ_{I}({N}). To prove the other direction, assume that φ({M_{k}}) converges to φ_{I}({N}) for M_{k} = ⨁_{i=1}^{n} π^{a_{i,k}}Rv_{i} and N = ⨁_{i∈I} π^{b_{i}}Rv_{i}. Looking at the fundamental neighborhoods as above, we find that for any fixed i_{0} ∈ I the sequence a_{i,k} − a_{i_{0},k} is unbounded for i ∉ I, and goes to b_{i} − b_{i_{0}} for i ∈ I. This implies our claim. □ We will now show that the space \overline{Λ} is compact. Fix some i ∈ \underline{n}. We write D_{i} for D_{\{i\}} = ∑_{j≠i} ℝ_{≥0}(−η_{j}), the “corner in Λ around the point Λ_{{i}}”. Besides, let

E_{i} = D_{i} ∪ ⋃_{i∈J⊂\underline{n}} r_{J}(D_{i}) ⊂ \overline{Λ},

the “closed corner in \overline{Λ} around the point Λ_{\{i\}}”.

Lemma 3.2 i) Each point in \overline{Λ} lies in one of the E_{i}. ii) Each E_{i} is closed in \overline{Λ}.

Proof: i) Let x = ∑_{j=1}^{n} x_{j}(−η_{j}) be a point in Λ. Note that the relation ∑_{j=1}^{n} η_{j} = 0 implies that we can write x = ∑_{j≠i}(x_{j} − x_{i})(−η_{j}) for all i. Now E_{i} ∩ Λ = D_{i} is the set of all x such that all x_{j} − x_{i} are non-negative. In other words, a point x = ∑_{j=1}^{n} x_{j}(−η_{j}) is in D_{i} iff x_{i} is the minimum of all the coefficients x_{j}. This implies that for given x we always find some E_{i} containing it. As similar argument holds if x is contained in a boundary piece Λ_{J}, since if J contains i, the boundary piece E_{i} ∩ Λ_{J} is the set of all x = ∑_{j∈J} x_{j}(−η_{j}^{J}) such that x_{i} is the minimum of all the x_{j} for j ∈ J. (If i is not contained in J, then of course E_{i} ∩ Λ_{J} is empty.) ii) Take some x ∈ \overline{Λ} not contained in E_{i}. Then x is in some Λ_{J} for J ⊆ \underline{n} (possibly \underline{n}). Since the point Λ_{\{i\}} is contained in E_{i}, we know that J ≠ \{i\}. Let us first assume that i is contained in J. We write x = ∑_{j∈J,j≠i} x_{j}(−η_{j}^{J}). Since x is not in E_{i}, our considerations in part i) imply that one of the x_{j} for j ∈ J, say x_{j_{0}}, must be negative. The point z = ∑_{j∈J,j≠i} x_{j}(−η_{j}) in Λ projects to x, i.e. r_{J}(z) = x. This point must also be in the complement of E_{i}. Since E_{i} ∩ Λ = D_{i} is closed, we find a bounded open neighborhood U of z in Λ which is disjoint from E_{i}, and which contains only points y = ∑_{j≠i} y_{j}(−η_{j}) with y_{j_{0}} < 0. Then the open neighborhood C_{U}^{J} of x is also disjoint from E_{i}, which proves our claim. If i ∉ J and x = ∑_{j∈J} x_{j}(−η_{j}^{J}), we choose some j_{0} ∈ J and some x_{i} > x_{j_{0}}. The point z = ∑_{j∈J} x_{j}(−η_{j}) + x_{i}(−η_{i}) ∈ Λ is not contained in E_{i}. Hence we can again find a bounded open neighborhood U of z in Λ which is disjoint from E_{i}, and which contains only points y = ∑_{j≠i} y_{j}(−η_{j}) with y_{j_{0}} < 0. Then C_{U}^{J} is an open neighborhood of x contained in the complement of E_{i}. □ Let R_{≥0,∞} be the compactified half-line ℝ_{≥0}∪{∞} with the topology generated by all intervals [0, a[, ]b, c[ and ]b, ∞] for a > 0 and b, c ≥ 0. The space ℝ_{≥0,∞} is compact and contractible. A contraction map r : ℝ_{≥0,∞} × [0, 1] → ℝ_{≥0,∞} is given by

r(x, t) = \frac{(1 − t)x}{1 + tx} for x ∈ ℝ, and r(∞, t) = \{ \begin{matrix}∞, & if t = 0 \\ \frac{1−t}{t}, & if t ≠ 0 \end{matrix} ,

see [La], 2.1. Let us fix some i ∈ \underline{n}. We will now compare E_{i} to ℝ_{≥0,∞}^{n−1}, which we write as ⨁_{j≠i} ℝ_{≥0,∞}e_{j} for a basis e_{j}. Recall from the proof of Lemma 3.2 that in the case i ∈ I we can describe E_{i} ∩ Λ_{I} as the set of all x = ∑_{j∈I,j≠i} x_{j}(−η_{j}^{I}) with non-negative x_{j}. Hence the following map is a bijection

α_{I} :  E_{i} ∩ Λ_{I}  → {∑_{j≠i} x_{j}e_{j} ∈ ⨁_{j≠i} ℝ_{≥0,∞}e_{j} : x_{j} = ∞ iff j ∉ I}
∑_{j∈I,j≠i} x_{j}(−η_{j}^{I}) ↦ ∑_{j∈I,j≠i} x_{j}e_{j} + ∑_{j∉I} ∞ e_{j}.

For I = \underline{n} the map α\underline{n} : E_{i} ∩ Λ = D_{i} → ⨁_{j≠i} ℝ_{≥0}e_{j} can be continued to a homomorphism of ℝ-vector spaces

α_{Λ} : Λ ⟶ ⨁_{j≠i} ℝe_{j},

which is a homeomorphism. Putting all the maps α_{I} together, we get a bijection

α : E_{i} ⟶ ⨁_{j≠i} ℝ_{≥0,∞}e_{j},

whose restriction to E_{i} ∩ Λ is a homeomorphism. We even have the following fact:

Lemma 3.3 With respect to the topology on E_{i} induced by \overline{Λ}, the map α is a homeomorphism on the whole of E_{i}.

Proof: For all j ≠ i choose an open interval A_{j} in ℝ_{≥0,∞}, which is either of the form A_{j} = [0, a_{j}[ or A_{j} =]b_{j}, c_{j}[ or of the form A_{j} =]b_{j}, ∞]. We claim that the preimage of A = ∑_{j≠i} A_{j}e_{j} is open in E_{i}.

We put A′_{j} =] − 1, a_{j}[, if A_{j} = [0, a_{j}[. In all the other cases we put A′_{j} = A_{j}. Let A′ = ∑_{j≠i} A′_{j}e_{j} and put

W = α_{Λ}^{−1}(A′ ∩ ⨁_{j≠i} ℝe_{j}).

Since α_{Λ} is a homeomorphism, W is open in Λ. Obviously, we have W ∩ E_{i} = α^{−1}(A) ∩ Λ. Now put

I = \{j ∈ \underline{n} : ∞ ∉ A_{j}\} ∪ \{i\}.

We can assume that I ≠ \underline{n}. Choose some positive real number b strictly bigger than all the b_{j} for j ∉ I. Then U = W ∩ {x = ∑_{j≠i} x_{j}(−η_{j}) : x_{j} < b for j ∉ I} is an open bounded subset of Λ. Note that U + D_{I} = W. We claim that α^{−1}(A) = C_{U}^{I} ∩ E_{i}. Indeed, every element u in W = U + D_{I} can be written as u = ∑_{j≠i} x_{j}(−η_{j}) with x_{j} ∈ A′_{j}. Let J be a subset of \underline{n} containing I. If r_{J}(u) = ∑_{j∈J,j≠i} x_{j}(−η_{j}^{J}) is in E_{i}, we have x_{j} ∈ A_{j} for all j ∈ J not equal to i. This implies that α(r_{J}(u)) is contained in A. On the other hand, let y = ∑_{j≠i} y_{j}e_{j} be an element in A, i.e. y_{j} ∈ A_{j}. Put J = {j ≠ i : y_{j} ≠ ∞} ∪ {i}. Then J contains I. We put x_{j} = y_{j}, if j ≠ i is in J. If j ∉ J, we choose an arbitrary element in A_{j} ∩ ℝ and call it x_{j}. Then x = ∑_{j≠i} x_{j}(−η_{j}) is contained in W ∩ D_{i}, so that r_{J}(x) is an element in C_{U}^{I} ∩ E_{i} which satisfies α(r_{J}(x)) = y. Hence we also find α^{−1}(A) ⊂ C_{U}^{I} ∩ E_{i}. Therefore α is continuous. It remains to show that α is open. Let U be an open, bounded subset of Λ and I ⊂ \underline{n} non-empty. We will show that α(C_{U}^{I} ∩E_{i}) is open. Let x be a point in C_{U}^{I} ∩ E_{i} lying in Λ_{J} for some J containing I and i. Hence x = ∑_{j∈J,j≠i} x_{j}(−η_{j}^{J}) with non-negative x_{j}. Since x is contained in C_{U}^{I}, we can find some z = ∑_{j≠i} z_{j}(−η_{j}) in U + D_{I} such that r_{J}(z) = x (hence z_{j} ≥ 0 for j ∈ J) and z_{j} > 0 for j ∉ J. Then z ∈ (U + D_{I}) ∩ E_{i}. Since the restriction of α to E_{i} ∩ Λ is open, we find open intervals A_{j} (of the form [0, a_{j}[ or ]b_{j}, c_{j}[) in ℝ_{≥0} such that A = ∑_{j≠i} A_{j}e_{j} is an open neighborhood of α(z) in ⨁_{j≠i} ℝ_{≥0}e_{j} which is contained in α((U + D_{I}) ∩ E_{i}). We can also assume that for j ∉ J the interval A_{j} does not contain 0. Now put A′_{j} = A_{j} if j ≠ i is contained in J, and put A′_{j} =]b_{j}, ∞] if j is not contained in J, and A_{j} =]b_{j}, c_{j}[. (The interval A_{j} looks indeed like this since we took care to stay away from zero.) It is easy to see that A′ = ∑_{j≠i} A′_{j}e_{j} is contained in α(C_{U}^{I} ∩ E_{i}). Hence we found an open neighborhood A′ around α(x) in α(C_{U}^{I} ∩ E_{i}). □

Theorem 3.4 The topological space \overline{Λ} is compact and contractible, and Λ is an open, dense subset of \overline{Λ}.

Proof: By the previous result, all E_{i} are compact and contractible. Since \overline{Λ} is the union of the E_{i}, it is also compact. A straightforward calculation shows that the contraction maps are compatible, so that \overline{Λ} is contractible. It is clear that Λ is open and dense in \overline{Λ}. □ Our next goal is to extend the action of N on Λ to a continuous action on the compactification \overline{Λ}. Recall that we identified W with the group of permutation matrices in N, so that N = T ⋊ W. For w ∈ W we denote the induced permutation of the set \underline{n} also by w, i.e. we abuse notation so that w(v_{i}) = v_{w(i)}. Let I be a non-empty subset of \underline{n}. We define a map

w : Λ_{I} ⟶ Λ_{w(I)}

by sending η_{i}^{I} to η_{w(i)}^{w(I)}. This gives an action of W on \overline{Λ}. Note that it is compatible with r_{J}, i.e. we have

w ∘ r_{J} = r_{w(J)} ∘ w

on Λ. Besides, we can combine the maps ν_{I} : T_{I} → Λ_{I} with the restriction map ρ_{I} : T → T_{I} to define a map ν_{I} ∘ ρ_{I} : T → Λ_{I}, so that T acts by affine transformations on Λ_{I}. Recall that r_{I}(ν(t)) = ν_{I}(ρ_{I}(t)) for all t ∈ T. It is easy to check that these two actions give rise to an action of N = T ⋊ W on \overline{Λ}, which we denote by ν.

Lemma 3.5 The action ν : N × \overline{Λ} ⟶ \overline{Λ} is continuous and extends the action of N on Λ.

Proof: Let first w be an element of W, and let C_{U}^{I} one of our open basis sets. Then ν(w)(U + D_{I}) = ν(w)(U) + D_{w(I)}, since ν(w) is a linear map on Λ. Besides, we have ν(w)(r_{J}(U + D_{I})) = r_{w(J)}(ν(w)(U) + D_{w(I)}), so that ν(w)(C_{U}^{I}) = C_{ν(w)(U)}^{w(I)}. Now take some element t ∈ T. Then ν(t)(U + D_{I}) = ν(t)(U) + D_{I}, since ν(t) acts by translation. Besides, we have ν(t)(r_{J}(U + D_{I})) = r_{J}(ν(t)(U + D_{I})), so that ν(t)(C_{U}^{I}) = C_{ν(t)(U)}^{I}. Hence for all n ∈ N the action ν(n) on \overline{Λ} is continuous. Since the kernel of the map ν : T → Λ is an open subgroup of N (see [La], Prop. 1.2), which obviously acts trivially on \overline{Λ}, we find that the action is indeed continuous. □

4 Compactification of the building

We can now define the compactification of the building X. For all non-empty subsets Ω of \overline{Λ} and all roots a ∈ Φ we put

f_{Ω}(a) = inf \{t : Ω ⊆ \overline{\{z ∈ Λ : a(z) ≥ −t\}}\}
= − sup\{t : Ω ⊆ \overline{\{z ∈ Λ : a(z) ≥ t\}}\}

Here we put inf ∅ = sup ℝ = ∞ and inf ℝ = sup ∅ = −∞. Moreover, if Ω = {x} consists of one point only, then we write f_{x}(a) = f_{{x}}(a). Note that

f_{x}(a) = −a(x) for all x ∈ Λ,

and that

f_{Ω_{1}}(a) ≤ f_{Ω_{2}}(a), if Ω_{1} ⊆ Ω_{2}.

Recall our generalized valuation map ψ_{a} : U_{a} → ℤ ∪ {∞} from section 2. We can now define a subgroup

U_{a,Ω} = U_{a,f_{Ω}(a)} = {u ∈ U_{a} : ψ_{a}(u) ≥ f_{Ω}(a)}

of U_{a}, where U_{a,∞} = 1 and U_{a,−∞} = U_{a}. By U_{Ω} we denote the subgroup of G generated by all the U_{a,Ω} for roots a ∈ Φ. Note that if Ω = {x} for some point x ∈ Λ, then this coincides with our previous definition of U_{x}. We will now investigate these groups U_{x} for boundary points of \overline{Λ}.

Proposition 4.1 Recall that we denote by a_{ij} the root of T induced by the character χ_{i} − χ_{j}. Put a = a_{ij}, and let x be a point in Λ_{I} for some I ⊆ \underline{n}.

i) If j ∉ I, we have f_{x}(a) = −∞, so that U_{a,x} is the whole group U_{a}. ii) If j ∈ I and i ∉ I, then we have f_{x}(a) = ∞, so that U_{a,x} = 1. iii) If i and j are contained in I, then a is equal to ρ_{I}^{∗}(b) for some root b of the torus T_{I} in G_{I}. In this case we have f_{x}(a) = −b(x). For any z ∈ Λ with r_{I}(z) = x we also have f_{x}(a) = −a(z).

Proof: i) Choose some z ∈ Λ such that r_{I}(z) = x. If i is contained in I, put z_{k} = z +∑_{l∉I} k(−η_{l}). If i is not in I, then we define z_{k} = z +∑_{l∉I,l≠j} k(−η_{l})− 2kη_{j}. In both cases we find that a(z_{k}) equals a(z) + k, hence it goes to infinity. Since the z_{k} converge to x, we find that x lies indeed in the closure of any set of the form {a ≥ s}, which implies our claim. ii) We choose again some z ∈ Λ with r_{I}(z) = x. Let V_{k} be a countable decreasing fundamental system of bounded open neighborhoods of z. This defines a fundamental system of open neighborhoods C_{U_{k}}^{I} around x, where U_{k} = V_{k} + ∑_{i∉I} k(−η_{i}). Now suppose that x is contained in the closure of the set {z ∈ Λ : a(z) ≥ s}. Then we find for all k some y_{k} in C_{U_{k}}^{I} ∩ Λ satisfying a(y_{k}) ≥ s. We can write y_{k} = z_{k}+λ_{k} for some z_{k} ∈ V_{k} and λ_{k} = ∑_{l∉I} λ_{k,l}(−η_{l}) with λ_{k,l} ≥ k. Now a(z_{k}) is bounded, but a(λ_{k}) = −λ_{k,i}, so that a(y_{k}) cannot be bounded from below. Hence we find indeed that f_{x}(a) must be ∞. iii) Recall that G_{I} is the group P GL(V_{I}), and T_{I} is the maximal K-split torus induced by the diagonal matrices with respect to the v_{i} for i ∈ I. Then the root system corresponding to T_{I} and G_{I} is

Φ_{I} = {b_{ij} : i ≠ j in I}

where b_{ij} is the character mapping a diagonal matrix with entries t_{i} for i ∈ I to t_{i}/t_{j}. Hence it is clear that in our case i, j ∈ I the root a = a_{i,j} of T is induced by the root b = b_{ij} of T_{I}. Note that this implies that for all z ∈ Λ we have a(z) = b(r_{I}(z)). It suffices to show that

\overline{\{z ∈ Λ : a(z) ≥ s\}} ∩ Λ_{I} = \{x ∈ Λ_{I} : b(x) ≥ s\}.

Documenta Mathematica 6 (2001) 315–341

Take some x contained in the left hand side, and choose some z ∈ Λ with r_{I}(z) = x. Besides, we take again fundamental neighborhoods V_{k} around z and use them to construct the open neighborhoods C_{U_{k}}^{I} around x. Each C_{U_{k}}^{I} must contain some y_{k} ∈ Λ satisfying a(y_{k}) ≥ s. Note that we can write y_{k} = z_{k} + λ_{k}, where z_{k} is in V_{k} and λ_{k} is a linear combination of η_{i} for i ∉ I. Besides a(y_{k}) = a(z_{k}), so that the sequence of a(y_{k}) converges to a(z). Since all a(y_{k}) are ≥ s, we find b(x) = b(r_{I}z) = a(z) ≥ s. On the other hand, suppose that x is a point in Λ_{I} satisfying b(x) ≥ s. Again, we choose some z ∈ Λ with r_{I}(z) = x, and neighborhoods V_{k} around z. For any k the point z_{k} = z + k ∑_{l∉I}(−η_{l}) lies in C_{U_{k}}^{I}. Besides, a(z_{k}) = a(z) = b(r_{I}(z)) = b(x) is bounded below by s. This implies that x lies indeed in the closure of {z ∈ Λ : a(z) ≥ s}. □

Proposition 4.2 Let x be in Λ_{I} and let a = a_{ij} ∈ Φ be a root.

i) Each U_{a,x} (and hence also U_{x}) leaves the vector space V_{I} invariant. Hence U_{a,x} is contained in G^{V_{I}}. ii) If i and j are not both in I, we have ρ_{I}(U_{a,x}) = 1. If i and j are both in I, and the root a is induced by the root b of T_{I}, then ρ_{I} induces an isomorphism U_{a,x} → U_{b,x}^{I}, where U_{b,x}^{I} is defined with the root group U_{b}^{I} in G_{I} as described in section 2.

Proof: Recall that u ∈ U_{a} maps v_{l} to itself, if l is not equal to j, and it maps v_{j} to v_{j} + ωv_{i}, where ψ_{a}(u) = v(ω). Hence our claim in i) is clear if j is not contained in I or if both i and j are contained in I. In the remaining case we saw in 4.1 that U_{a,x} is trivial, so that i) holds in any case. Let us now prove ii). If j is not contained in I, then each u ∈ U_{a} induces the identity map on V_{I}. If j is in I, but i is not, then U_{a,x} is trivial. Hence in both cases we find that ρ_{I}(U_{a,x}) = 1. Let us assume that both i and j are contained in I, and let u be an element of U_{a,x}. Then ρ_{I}(u) ∈ P GL(V_{I}) is induced by the matrix mapping v_{l} to v_{l} for all l ≠ j in I, and v_{j} to v_{j} + ωv_{i} with some ω having valuation ≥ f_{x}(a). By our description of the groups U_{b}^{I} in section 2 we find that ρ_{I}(u) is contained in U_{b}^{I} and has valuation ψ_{b}(ρ_{I}x) = v(ω) ≥ f_{x}(a). By 4.1, f_{x}(a) = −b(x), so that ρ_{I}(u) lies indeed in U_{b,x}^{I}. The homomorphism ρ_{I} : U_{a,x} → U_{b,x}^{I} is obviously bijective. □ Note that the map x ↦ f_{x}(a) is in general not continuous on \overline{Λ}. Take some z ∈ Λ and define a sequence x_{k} = z + ∑_{i∉I} k(−η_{i}) for some non-empty I such that the complement \underline{n}\I contains at least two elements i and j. Then a = a_{ij} has the property that a(x_{k}) = a(z), so that f_{x_{k}}(a) is constant. But the sequence x_{k} converges to the point r_{I}(z) in Λ_{I}, for which f_{r_{I}(z)}(a) = −∞ holds. Nevertheless, we have the following result:

Lemma 4.3 Let x_{k} be a sequence of points in \overline{Λ}, which converges to x ∈ \overline{Λ}. Let u_{k} ∈ U_{a,x_{k}} be a sequence of elements, converging to some u in the big group U_{a}. Then u lies in fact in U_{a,x}.

Proof: Note first of all, that the statement is clear if f_{x}(a) = −∞, since then U_{a,x} = U_{a}. It is also clear if f_{x_{k}}(a) converges to f_{x}(a), since the map ψ_{a} : U_{a} → ℤ ∪ {∞} is continuous. Assume that f_{x}(a) = ∞. Then any set \{\overline{z : a(z) ≥ s}\} contains only finitely many elements x_{k}. This implies that the sequence f_{x_{k}}(a) goes to ∞ = f_{x}(a), so that in this case our claim holds by continuity. The only case which is left is that f_{x}(a) is real. Assume that x ∈ Λ_{I}. By 4.1, we must have a = a_{ij} with i and j in I. Choose some z ∈ Λ with r_{I}(z) = x, and a decreasing fundamental system of open neighborhoods V_{k} around z. As before, we use them to define neighborhoods C_{U_{k}}^{I} around x. Since x_{k} converges to x, we can assume that x_{k} is contained in C_{U_{k}}^{I}. Then x_{k} ∈ Λ_{J_{k}} for some J_{k} containing I. By definition of C_{U_{k}}^{I} we find some y_{k} ∈ V_{k} and some coefficients α_{l} such that z_{k} = y_{k} + ∑_{l∉I} α_{l}(−η_{l}) satisfies r_{J_{k}}(z_{k}) = x_{k}. By 4.1 we have f_{x_{k}}(a) = −a(z_{k}) = −a(y_{k}) and f_{x}(a) = −a(z). Hence f_{x_{k}}(a) converges to f_{x}(a), and our claim follows again by continuity. □

Proposition 4.4 For x ∈ \overline{Λ}, n ∈ N and a ∈ Φ we have

nU_{a,x}n^{−1} = U_{\overline{n}(a),ν(n)(x)},

where ν denotes the action of N on \overline{Λ}, and n ↦ \overline{n} denotes the quotient map from N to the Weyl group W (which acts on the roots). In particular, we have nU_{x}n^{−1} = U_{ν(n)(x)}.

Proof: Fix some n ∈ N and denote by p the permutation matrix mapping to \overline{n}, i.e. n = tp for some t ∈ T. We denote by p also the corresponding permutation of \underline{n}. If a = a_{ij}, then \overline{n}(a) = a_{p(i)p(j)}. By 4.1 we find that if x ∈ Λ_{I} and j ∉ I both f_{x}(a) and f_{ν(n)(x)}(\overline{n}a) are equal to −∞. Since nU_{a}n^{−1} = U_{\overline{n}a}, our claim holds in this case. Similarly, if j ∈ I and i ∉ I both f_{x}(a) and f_{ν(n)(x)}(\overline{n}a) are equal to ∞, so that both U_{a,x} and U_{\overline{n}a,ν(n)(x)} are trivial. We can therefore assume that x is in some Λ_{I} such that both i and j are contained in I. Recall that we composed the action of N on Λ from the natural action of the Weyl group on Λ and translation by ν(t) for elements in the torus T. Hence for all z ∈ Λ we have ν(n)(z) = ν(n)(0)+\overline{n}(z). We can now calculate

a(ν(n)(z)) = a(ν(n)(0)) + a(\overline{n}(z))
= a(ν(n)(0)) + (\overline{n}^{−1}a)(z),

so that ν(n^{−1})\{z ∈ Λ : (\overline{n}a)(z) ≥ s\} = \{z ∈ Λ : a(z) ≥ a(ν(n^{−1})(0)) + s\}. Since ν(n^{−1}) is a homeomorphism, we also have

ν(n^{−1})\overline{\{z : \overline{n}a(z) ≥ s\}} = \overline{\{z : a(z) ≥ a(ν(n^{−1})(0)) + s\}}.

Hence

f_{ν(n)(x)}(\overline{n}(a)) = inf\{t : ν(n)(x) ∈ \overline{\{\overline{n}(a) ≥ −t\}}\}
= inf\{t : x ∈ \overline{\{z : a(z) ≥ a(ν(n^{−1})(0)) − t\}}\}
= f_{x}(a) + a(ν(n^{−1})(0)).

Since we have nU_{a,s}n^{−1} = U_{\overline{n}(a),s+a(ν(n^{−1})(0))} for all real numbers s (see [La], 11.6), we find that indeed

nU_{a,x}n^{−1} = U_{\overline{n}(a),f_{x}(a)+a(ν(n^{−1})(0))} = U_{\overline{n}(a),f_{ν(n)(x)}(\overline{n}(a))} = U_{\overline{n}(a),ν(n)(x)},

whence our claim. □ Recall that - forgetting about the special nature of our ground field - our root system Φ in Λ^{∗} defines a finite set of hyperplanes in Λ^{∗} and therefore a decomposition of Λ^{∗} into faces (see [Bou],V,1). The maximal faces are called (spherical) chambers. Any chamber defines an order on Λ^{∗} ([Bou], VI, 1.6). We denote the positive roots with respect to this order by Φ^{+} = Φ^{+}(C), and the negative roots by Φ^{−} = Φ^{−}(C). In fact, for any subset Ψ of Φ such that Ψ is additively closed and Φ is the disjoint union of Ψ and −Ψ, there exists a chamber C such that Ψ = Φ^{+}(C) (see [Bou], VI, 1.7). In particular, Φ^{−} is the set of positive roots for a suitable chamber. The following lemma will be useful to reduce claims about the groups U_{a,Ω} for Ω ⊂ \overline{Λ} to claims about subsets of Λ, where we can apply the “usual” theory of the Bruhat-Tits building.

Lemma 4.5 Let Ω ⊂ \overline{Λ} be a non-empty set, and fix some chamber C. Put Φ^{+} = Φ^{+}(C). Assume that for every a ∈ Φ^{+} we have m elements

u_{a,1}, …, u_{a,m} ∈ U_{a,Ω},

such that at least one of all these u_{a,i}’s is non-trivial. Then there exists a nonempty subset Ω′ of Λ such that u_{a,i} ∈ U_{a,Ω′} for all i = 1, …, m and such that Ω ⊂ \overline{Ω}′. In particular, we have U_{a,Ω′} ⊂ U_{a,Ω} for all roots a ∈ Φ.

Proof: We denote by l_{a} the infimum of all ψ_{a}(u_{a,i}) for i = 1, …, m. If all u_{a,i} are trivial, then l_{a} = ∞. This cannot happen for all a ∈ Φ^{+}. Then we put

Ω′_{a} = \{\overline{z ∈ Λ : a(z) ≥ −l_{a}}\}.

(If l_{a} = ∞, then Ω′_{a} = \overline{Λ}.) Besides, put Ω′ = Λ ∩ ⋂_{a∈Φ^{+}} Ω′_{a}. Note that Ω′ contains the intersection of all sets {z ∈ Λ : a(z) ≥ −l_{a}} for a ∈ Φ^{+}. If l is the minimum of all the l_{a}, then this set contains all z ∈ Λ satisfying a(z) ≥ −l for all a ∈ Φ^{+}. By looking at a base of Φ corresponding to Φ^{+}, we see that such z’s exist. Hence Ω′ is non-empty. By construction, f_{Ω′_{a}}(a) = l_{a}, and the inclusion Ω′ ⊂ Ω′_{a} gives us f_{Ω′}(a) ≤ f_{Ω′_{a}}(a). Therefore ψ_{a}(u_{a,i}) ≥ l_{a} ≥ f_{Ω′}(a), which implies that u_{a,i} is indeed contained in U_{a,Ω′} for all i = 1, …, m. It remains to show that Ω ⊂ \overline{Ω}′. Since f_{Ω′}(a) = f\overline{Ω}′(a) for all roots a, this implies that f_{Ω}(a) ≤ f_{Ω′}(a), hence U_{a,Ω′} ⊂ U_{a,Ω} for all a ∈ Φ. We are done if we prove the following claim:

(∗) For any Ψ ⊆ Φ^{+} and real numbers s_{a}we have ⋂_{a∈Ψ}\{\overline{a ≥ s_{a}}\} = \overline{⋂_{a∈Ψ}\{a ≥ s_{a}\}}.

It is clear that the right hand side is contained in the left hand side. So suppose x is an element in ⋂_{a∈Ψ}\{\overline{a ≥ s_{a}}\}. Let I be the subset of \underline{n} such that x ∈ Λ_{I}. We choose a system of open neighborhoods V_{k} of some point in Λ projecting to x and construct C_{U_{k}}^{I}. We are done if we can show that any C_{U_{k}}^{I} intersects ⋂_{a∈Ψ}{a ≥ s_{a}} non-trivially. Let z_{k} be a point in U_{k} = V_{k} + k ∑_{l∉I}(−η_{l}) with r_{I}(z_{k}) = x. Besides, let s_{k} be the maximum of 0 and all the numbers s_{a} −a(z_{k}) for all a ∈ Ψ. Note that Φ^{+} defines a linear ordering of the set \underline{n} = \{1, …, n\}, namely i ≺ j, iff a_{ij} ∈ Φ^{+}. Hence there is a permutation π of \underline{n} satisfying π(1) ≺ π(2) ≺ … ≺ π(n). Put z = z_{k} − ∑_{l∉I}(k + π^{−1}(l)s_{k})η_{l}. This is an element of C_{U_{k}}^{I}. It remains to show that indeed a(z) ≥ s_{a} for all a ∈ Ψ. Let a = a_{ij} be a root in Ψ. If both i and j are in I, we can apply 4.1 to deduce a(z) = a(z_{k}) = −f_{x}(a) ≥ s_{a}. Since x is contained in \{\overline{a ≥ s_{a}}\}, it cannot happen that j is in I, but i is not. If j is not in I, we find that

a(z) = \{ \begin{matrix}a(z_{k}) + k + π^{−1}(j)s_{k} & ≥ s_{a} & ,if i ∈ I \\ a(z_{k}) + (π^{−1}(j) − π^{−1}(i))s_{k} & ≥ s_{a} & ,if i ∉ I \end{matrix} ,

since a ∈ Φ^{+} implies that π^{−1}(i) < π^{−1}(j). Hence we get a(z) ≥ s_{a} for all a ∈ Ψ, which proves (∗). □

Corollary 4.6 Assume that a and b are roots in Φ which are not linear equivalent (i.e. a ≠ ±b), and so that a + b is in Φ. If both f_{Ω}(a) and f_{Ω}(b) are real numbers, then

f_{Ω}(a + b) ≤ f_{Ω}(a) + f_{Ω}(b).

If f_{Ω}(a) = −∞ and f_{Ω}(b) ≠ ∞, then f_{Ω}(a + b) = −∞.

Proof: By (∗) in the proof of the preceding lemma, we have

\{\overline{a ≥ s}\} ∩ \{\overline{b ≥ r}\} = \{\overline{a ≥ s, b ≥ r}\} ⊂ \{\overline{a + b ≥ s + r}\},

which implies our claim. □ Recall that U_{Ω} is the subgroup of G generated by all the U_{a,Ω}. Now we prove a statement about the structure of these groups U_{Ω} which will we crucial for our later results. For Φ^{+} = Φ^{+}(C) we denote by U_{Φ^{+}} the corresponding subgroup of G (see [Bo], 21.9), and by U_{Φ^{+}} the set of K-rational points. Similarly, we have U_{Φ^{−}} and U_{Φ^{−}}. For any non-empty subset Ω of \overline{Λ} define

U_{Ω}^{+} = U_{Φ^{+}} ∩ U_{Ω} and U_{Ω}^{−} = U_{Φ^{−}} ∩ U_{Ω}.

Of course, these groups depend on the choice of some chamber C. We can use them to get some information about U_{Ω}.

Theorem 4.7 i) The multiplication map induces a bijection

∏_{a∈Φ^{±}} U_{a,Ω} ⟶ U_{Ω}^{±},

where the product on the left hand side may be taken in arbitrary order.

ii) U_{a} ∩ U_{Ω} = U_{a,Ω} for all a ∈ Φ.
iii) U_{Ω} = U_{Ω}^{−}U_{Ω}^{+}(N ∩ U_{Ω}).

Proof: Note that by [La], 12.5, our claim holds for all non-empty subsets Ω ⊂ Λ. Now take Ω ⊆ \overline{Λ}, and denote by L_{a} the group generated by U_{a,Ω} and U_{−a,Ω}, and by Y the subgroup of N generated by all N ∩ L_{a} for a ∈ Φ. For all a ∈ Φ^{+} choose an element u_{a} ∈ U_{a,Ω}. By 4.5 we find a subset Ω′ of Λ such that u_{a} ∈ U_{a,Ω′}. Hence by [La], 12.5, the product of the u_{a} in arbitrary order lies in U_{Ω′}^{+} ⊂ U_{Ω}^{+}. A similar argument using 4.5 shows that the image of ∏_{a∈Φ^{+}} U_{a,Ω} under the multiplication map is indeed a subgroup of U_{Ω}^{+}, which is independent of the ordering of the factors. We denote it by H^{+}. Similarly, we define the subgroup H^{−} of U_{Ω}^{−} as the image of ∏_{a∈Φ^{−}} U_{a,Ω} under the multiplication map. Now we can imitate the argument in [La], Proposition 8.9, to prove that the set H^{−}H^{+}Y does not depend on the choice of the chamber defining Φ^{+} and is invariant under multiplication from the left by Y and U_{a,Ω} for arbitrary roots a ∈ Φ. Hence H^{−}H^{+}Y = U_{Ω}. Since U_{Φ^{+}} ∩U_{Φ^{−}} = {1} and N ∩U_{Φ^{+}}U_{Φ^{−}} = {1} (by [BoTi], 5.15), we find U_{Ω}^{−} = (H^{−}H^{+}Y) ∩ U_{Φ^{−}} = H^{−} and U_{Ω}^{+} = H^{+}, which proves i) and ii). Similarly, N ∩ U_{Ω} = Y, whence iii). □ For any subset Ω of \overline{Λ} we write N_{Ω} = \{n ∈ N : ν(n)x = x for all x ∈ Ω\}. Besides, put

P_{Ω} = U_{Ω}N_{Ω} = N_{Ω}U_{Ω},

which is a group since as in 4.4 one can show that N_{Ω} normalizes U_{Ω}. If Ω = {x} we write P_{Ω} = P_{x}. We can now also describe the groups P_{Ω} for any non-empty subset Ω of \overline{Λ}:

Corollary 4.8 Fix some Φ^{+} = Φ^{+}(C) as above, and let Ω be a non-empty subset of \overline{Λ}.

i) P_{Ω} = U_{Ω}^{−}U_{Ω}^{+}N_{Ω} = N_{Ω}U_{Ω}^{+}U_{Ω}^{−}.
ii) P_{Ω} ∩ U_{Φ^{±}} = U_{Ω}^{±} and P_{Ω} ∩ N = N_{Ω}.

Proof: i) The first equality follows from part iii) of the Theorem, if we show that N ∩U_{Ω} ⊂ N_{Ω}. It suffices to show for each root a that N ∩L_{a} ⊂ N_{Ω}. If both f_{Ω}(a) and f_{Ω}(−a) are real numbers, this follows from [La], 12.1. If f_{Ω}(a) = ∞ of if f_{Ω}(−a) = ∞, then our claim is trivial. Note that if f_{Ω}(a) = −∞, then f_{Ω}(−a) is either ∞ (then we are done) or −∞. Hence the only remaining case is f_{Ω}(a) = f_{Ω}(−a) = −∞. If a = a_{ij}, 4.1 implies that Ω ∩ Λ_{J} can then only be non-empty if i and j are not contained in J. Now 4.2 implies our claim. ii) Let us show first that P_{Ω} ∩ U_{Φ^{−}} = U_{Ω}^{−}. Obviously, the right hand side is contained in the left hand side. Take some u ∈ P_{Ω} ∩ U_{Φ^{−}}. Using i), we can write it as u^{−}u^{+}n for u^{±} ∈ U_{Ω}^{±} and n ∈ N_{Ω}. Then n must be in U_{Φ^{+}}U_{Φ^{−}} ∩ N, which is trivial by [BoTi], 5.15. Hence u^{+} is contained in U_{Φ^{−}} ∩ U_{Φ^{+}}, which is also trivial. Therefore u = u^{−} ∈ U_{Ω}^{−}. The corresponding statement for the +-groups follows by taking Φ^{−} as the set of positive roots. It remains to show P_{Ω} ∩ N = N_{Ω}. Take u ∈ P_{Ω} ∩ N. Then we write it again as u = u^{−}u^{+}n. Hence u^{−}u^{+} is contained in U_{Φ^{−}}U_{Φ^{+}} ∩ N, which is trivial, so that u = n ∈ N_{Ω}, as claimed. □ Now we can show a weak version of the mixed Bruhat decomposition for our groups P_{x}. (The weakness lies in the fact that we can not take two arbitrary points in \overline{Λ} in the next statement.)

Theorem 4.9 Let x ∈ \overline{Λ} and y ∈ Λ. Then we have G = P_{x}NP_{y}.

Proof: Let Λ_{I} be the component of \overline{Λ} containing x. Then we can write x = ∑_{i∈I} x_{i}η_{i}^{I} with some real coefficients x_{i}. We define a sequence of points in Λ by

z_{k} = ∑_{i∈I} x_{i}η_{i} − k ∑_{i∉I} η_{i}.

Obviously, z_{k} converges towards x. Now we choose a linear ordering ≺ on the set \underline{n} in such a way that i ∈ I and j ∉ I implies i ≺ j. The set Φ^{+} = \{a_{ij} ∈ Φ : i ≺ j} defines an order corresponding to some chamber. Note that for any root a_{ij} in Φ^{−} we have

a_{ij}(z_{k}) = \{ \begin{matrix}x_{i} − x_{j} & if i, j ∈ I \\ −k − x_{j} & if i ∉ I, j ∈ I \\ 0 & if i, j ∉ I. \end{matrix}

Hence a_{ij}(z_{k}) is bounded from above by a constant c independent of k and of the root a_{ij} ∈ Φ^{−}. Therefore U_{a_{ij},z_{k}} is contained in U_{a_{ij},−c} = {u ∈ U_{a_{ij}} : ψ_{a}(u) ≥ −c}, which is a compact subgroup of U_{a_{ij}}. By 4.7, we find that all U_{z_{k}}^{−} are contained in a compact subset of U_{Φ^{−}}. We have the “usual” mixed Bruhat decomposition for two points in Λ (see [La], 12.10), hence G = P_{z_{k}}NP_{y} for all k. Using 4.8, we can write an element g ∈ G as

g = u_{k}^{−}u_{k}^{+}n_{k}v_{k}

with u_{k}^{±} ∈ U_{z_{k}}^{±}, n_{k} ∈ N and v_{k} ∈ U_{y}. Let us denote the kernel of the map ν : T → Λ by Z ⊂ T. Then the group U_{y}^{∧} = U_{y}Z is compact and open in G by [La], 12.12. Since all v_{k} lie in this compact subset, by switching to a subsequence we can assume that v_{k} converges to some element v ∈ U_{y}^{∧}. Hence the sequence u_{k}^{−}u_{k}^{+}n_{k} is also convergent in G. Since the Weyl group W = N/T is finite, by passing to a subsequence we can assume that n_{k} = t_{k}n for some t_{k} ∈ T and some fixed n ∈ N. Besides, we can assume that u_{k}^{−} converges to some u^{−} ∈ U_{Φ^{−}}, since the u_{k}^{−} are contained in a compact subset of U_{Φ^{−}}. Hence the sequence u_{k}^{+}t_{k} converges in G. Its limit must be contained in the Borel group U_{Φ^{+}}T. Hence u_{k}^{+} converges towards some u^{+} ∈ U_{Φ^{+}}, and t_{k} converges towards some t ∈ T. Using 4.7, u_{k}^{+} is a product of u_{a,k} ∈ U_{a,z_{k}} for all a ∈ Φ^{+}, and, applying 4.3, we deduce that the u_{a,k} converge towards some element u_{a} ∈ U_{a,x}. Hence we see that u^{+} is contained in U_{Φ^{+}} ∩ U_{x} = U_{x}^{+}. Similarly, u^{−} lies in U_{x}^{−}. Therefore g = u_{k}^{−}u_{k}^{+}t_{k}nv_{k} converges towards u^{−}u^{+}tnv, which is contained in U_{x}^{−}U_{x}^{+}NU_{y} ⊆ P_{x}NP_{y}. Hence g lies indeed in P_{x}NP_{y}. □ Recall that Z ⊂ T denotes the kernel of the map ν : T → Λ, and that the group U_{0}^{∧} = U_{0}Z is compact. We define our compactification \overline{X} of the building X as

\overline{X} = U_{0}^{∧} × \overline{Λ}/ ∼,

where the equivalence relation ∼ is defined as follows:

(g, x) ∼ (h, y)  iff there exists an element n ∈ N
such that ν(n)x = y and g^{−1}hn ∈ P_{x}.

(Using 4.4, it is easy to check that ∼ is indeed an equivalence relation.) We equip \overline{X} with the quotient topology. The inclusion U_{0}^{∧} × Λ ↪ G × Λ induces a bijection (U_{0}^{∧} × Λ)/ ∼ → X, which is a homeomorphism if we endow the left hand side with the quotient topology (see [Bo-Se], p.221). Hence X is open and dense in \overline{X}. We have a natural action of U_{0}^{∧} on \overline{X} via left multiplication on the first factor, which can be continued to an action of G in the following way: If g ∈ G and (v, x) ∈ U_{0}^{∧} ×\overline{Λ}, we can use the mixed Bruhat decomposition to write gv = unh for some u ∈ U_{0}^{∧}, n ∈ N and h ∈ P_{x}. Then we define g(v, x) = (u, ν(n)x). Using 4.4, one can show that this induces a well-defined action on \overline{X}. Mapping x to the class of (1, x) defines a map \overline{Λ} → \overline{X}. This is injective, since by 4.8 we have P_{x} ∩ N = N_{x}. The G-action on \overline{X} continues the N-action on \overline{Λ}, so that we will write nx instead of ν(n)x for x ∈ X. The following important fact follows immediately from the definition of \overline{X}:

Lemma 4.10 For all x ∈ \overline{Λ} the group P_{x} is the stabilizer of x in G.

We can use the mixed Bruhat decomposition to prove the following important fact:

Proposition 4.11 For any two points x ∈ \overline{X} and y ∈ X there exists a compactified apartment containing x and y, i.e. there exists some g ∈ G such that x and y both lie in g\overline{Λ}.

Proof: We can assume that y lies in Λ. The point x lies in h\overline{Λ} for some h ∈ G, so x = hx′ for some x′ ∈ \overline{Λ}. By our mixed Bruhat decomposition 4.9 we can write h = qnp for q ∈ P_{y}, n ∈ N and p ∈ P_{x′}. Therefore x = hx′ = qnpx′ = qnx′ ∈ q\overline{Λ}, and y = qy lies also in q\overline{Λ}, whence our claim. □

5 Properties of \overline{X}

In this section we want to check that \overline{X} is compact and we want to identify it with the set ⋃_{W ⊆V} X(P GL(W)).

Hence we see that we can compactify the Bruhat-Tits building for P GL(V) by attaching all the Bruhat-Tits buildings for P GL of the smaller subspaces at infinity. The following lemma is similar to [La], 8.11.

Lemma 5.1 Let z be a point in Λ_{I} and y be a point in Λ_{J} for some I ⊆ J ⊆ \underline{n}. Then we find a chamber such that the corresponding set of positive roots Φ^{+} satisfies U_{y}^{+} ⊆ U_{z}^{+}.

Note that the assumptions are fulfilled if J = \underline{n}, i.e. if y lies in Λ. Proof: Since I ⊆ J, we can define a projection map Λ_{J} → Λ_{I}, which we also denote by r_{I}. To be precise, r_{I} maps a point ∑_{i∈J} x_{i}η_{i}^{J} ∈ Λ_{J} to ∑_{i∈I} x_{i}η_{i}^{I} ∈ Λ_{I}. Put y^{∗} = r_{I}(y). Recall that we denote by Φ_{I} the set of roots of T_{I} in G_{I} = P GL(V_{I}). There exists a chamber in Λ_{I}^{∗} with respect to Φ_{I} such that the corresponding subset Φ_{I}^{+} of positive roots satisfies U_{y^{∗}}^{I+} ⊆ U_{z}^{I+}. Here U_{z}^{I+} is defined exactly as the groups U_{x}^{+} for x ∈ Λ in section 4, just replacing Λ by Λ_{I}. Now Φ_{I} = {b_{ij} : i, j ∈ I}, where b_{ij} is the character mapping a diagonal matrix with entries t_{l} (for l ∈ I) to t_{i}/t_{j}. We define a linear ordering on I by i ≺ j iff b_{ij} ∈ Φ_{I}^{+}. This can be continued to a linear ordering on \underline{n} in such a way that i ≺ j whenever i ∈ I and j ∉ I. Let us put Φ^{+} = {a_{ij} : i ≺ j}. We claim that this satisfies our claim. In fact, take a = a_{ij} ∈ Φ^{+}. If i and j are contained in I, we use 4.2 and the construction of Φ_{I}^{+} to deduce that U_{a,y} ⊆ U_{a,z}. By definition of Φ^{+} it cannot happen that j is in I, but i is not. Hence the only remaining case is that j ∉ I. Then f_{z}(a) = −∞ by 4.1, so that trivially U_{a,y} ⊆ U_{a,z}. □ Recall that P_{Ω} = N_{Ω}U_{Ω} with N_{Ω} = {n ∈ N : nx = x for all x ∈ Ω}.

Theorem 5.2 Fix a nonempty Ω ⊂ \overline{Λ}. We denote by (∗) the following condition:

(∗)  The set \{J ⊆ \underline{n} : Ω ∩ Λ_{J} ≠ ∅\} contains a maximal element

with respect to inclusion.

If (∗) if satisfied, then P_{Ω} = ∩_{x∈Ω}P_{x}. In particular, P_{Ω} is the stabilizer of Ω.

Note that (∗) is satisfied if Ω ∩ Λ is not empty. Proof: To begin with, the inclusion P_{Ω} ⊆ P_{x} for all x ∈ Ω is trivial. Let J be the maximal subset of \underline{n} satisfying Ω ∩ Λ_{J} ≠ ∅, and choose some x_{0} ∈ Ω ∩ Λ_{J}. We will first prove that P_{Ω^{∼}} ∩ P_{x} = P_{Ω^{∼}∪{x}} for all x ∈ Ω and all subsets Ω^{∼} of Ω containing x_{0}. Assume that x lies in in Λ_{I} for some I ⊆ J. By 5.1, we find some set of positive roots Φ^{+} such that U_{x_{0}}^{+} ⊆ U_{x}^{+}, hence also U_{Ω^{∼}}^{+} ⊆ U_{x}^{+}. Let now g be an element in P_{Ω^{∼}} ∩ P_{x}, and write g = nu^{−}u^{+} for some n ∈ N_{Ω^{∼}} and some u^{±} ∈ U_{Ω^{∼}}^{±} (using 4.8). Since g and u^{+} are contained in P_{x}, this also holds for nu^{−}, so that we can write nu^{−} = mv^{+}v^{−} for some m ∈ N_{x} and v^{±} ∈ U_{x}^{±}. So m^{−1}n is contained in N ∩ U_{Φ^{+}}U_{Φ^{−}}, hence trivial by [BoTi], 5.15. We find that m = n and u^{−} = v^{+}v^{−}, which implies v^{+} = 1. Hence n is contained in N_{x} ∩ N_{Ω^{∼}} = N_{Ω^{∼}∪{x}}, and u^{−} is contained in U_{x}^{−} ∩ U_{Ω^{∼}}^{−}. Note that for all a ∈ Φ we have the inclusion U_{a,Ω^{∼}} ∩ U_{a,x} ⊆ U_{a,Ω^{∼}∪{x}}, so that we can use 4.7 to deduce u^{−} ∈ U_{Ω^{∼}∪{x}}^{−}. A similar argument as above gives u^{+} ∈ U_{Ω^{∼}}^{+} ∩ U_{x}^{+} ⊂ U_{Ω^{∼}∪{x}}^{+}. Hence our claim is proven. Therefore any finite subset Ω^{∼} ⊂ Ω containing x_{0} satisfies our claim, i.e. P_{Ω^{∼}} = ∩_{x∈Ω^{∼}}P_{x}. We can write Ω = ⋃_{σ∈Σ} Ω_{σ}, where Ω_{σ} for σ ∈ Σ runs over all finite subsets of Ω containing x_{0}. Let us consider some g ∈ ⋂_{x∈Ω} P_{x} = ⋂_{σ∈Σ} P_{Ω_{σ}}. We fix some set of positive roots Φ^{+} and write g = n_{σ}u_{σ}^{+}u_{σ}^{−} for n_{σ} ∈ N_{Ω_{σ}} and u_{σ}^{±} ∈ U_{Ω_{σ}}^{±} by 4.8. Put T_{x_{0}} = T ∩ N_{x_{0}}. Then N_{x_{0}}/T_{x_{0}} is finite. By the pigeon hole principle, there must be one class m in N_{x_{0}}/T_{x_{0}} such that the set Σ′ of all the σ ∈ Σ so that n_{σ} is equal to m modulo T_{x_{0}} still has the property that ⋃_{σ∈Σ′} Ω_{σ} = Ω. (If not, we could find for any class in N_{x_{0}}/T_{x_{0}} an element in Ω not contained in any Ω_{σ} with the property that n_{σ} lies in our class. Collecting these elements together with x_{0} in some finite set gives a contradiction.) Hence for σ ∈ Σ′ we can write n_{σ} = mt_{σ} for some fixed m ∈ N_{x_{0}} and some t_{σ} ∈ T_{x_{0}}. For σ and τ in Σ′ we get t_{σ}u_{σ}^{+}u_{σ}^{−} = t_{τ} u_{τ}^{+}u_{τ}^{−}. Using the fact that T normalizes U_{Φ^{+}} and U_{Φ^{−}} and that N ∩ U_{Φ^{+}}U_{Φ^{−}} is trivial, we find u_{σ}^{−} = u_{τ}^{−}, u_{σ}^{+} = u_{τ}^{+} and t_{σ} = t_{τ}. Therefore the elements t = t_{σ}, u^{±} = u_{σ}^{±} are independent of the choice of σ ∈ Σ′. Note that by definition of the T -action on \overline{Λ} the element t in T_{x_{0}} stabilizes not only x_{0}, but also every point in the components Λ_{I} for I ⊆ J. Thus t ∈ N_{Ω}. Besides, by 4.7 we deduce u^{+} ∈ ⋂_{σ∈Σ′} U_{Ω_{σ}}^{+} = U_{Ω}^{+}. Similarly, u^{−} ∈ U_{Ω}^{−}. Hence we find indeed that g = mtu^{+}u^{−} is contained in P_{Ω}. □

Corollary 5.3 Let Ω and Ω′ be two non-empty subsets of \overline{Λ} such that Ω, Ω′ and Ω ∪ Ω′ satisfy condition (∗) in the Theorem. Then P_{Ω} ∩ P_{Ω′} = P_{Ω∪Ω′}.

Proof: This is an immediate consequence from 5.2. □ The following result is similar to [La], 9.6.

Proposition 5.4 Let g ∈ G and let J be a subset of \underline{n} so that Λ_{J} ∩ g^{−1}\overline{Λ} is not empty. Then there exists some element n ∈ N such that

gx = nx for all x ∈ g^{−1}\overline{Λ} ∩ ( ⋃_{I⊆J} Λ_{I}).

Proof: Note that the set Ω = g^{−1}\overline{Λ} ∩ (⋃_{I⊆J} Λ_{I}) satisfies condition (∗) from 5.2. Fix some x_{0} ∈ Λ_{J} ∩ g^{−1}\overline{Λ}. For all x ∈ Ω we have g^{−1}N ∩ P_{x} ≠ ∅, since x = g^{−1}y for some y ∈ \overline{Λ}. We will now show that for all finite subsets Δ of Ω containing x_{0} we have g^{−1}N ∩ P_{Δ} ≠ ∅. Let us suppose that this claim holds for some Δ and let us show it for Δ ∪ {x}, where x is some point in Ω. So there is some n_{Δ} ∈ N with g^{−1}n_{Δ} ∈ P_{Δ}. We also find some n_{x} ∈ N satisfying g^{−1}n_{x} ∈ P_{x}. By 5.1, we find a set of positive roots Φ^{+} such that U_{x_{0}}^{+} ⊆ U_{x}^{+}, so that also U_{Δ}^{+} ⊆ U_{x}^{+}. Hence we apply 4.8 to deduce

n_{Δ}^{−1}n_{x} ∈ P_{Δ}P_{x} = N_{Δ}U_{Δ}^{−}U_{Δ}^{+}U_{x}^{+}U_{x}^{−}N_{x} = N_{Δ}U_{Δ}^{−}U_{x}^{−}U_{x}^{+}N_{x} ⊆ N_{Δ}U_{Φ^{−}}U_{Φ^{+}}N_{x}.

Since N ∩ U_{Φ^{−}}U_{Φ^{+}} is trivial, we find n′_{Δ} ∈ N_{Δ} and n′_{x} ∈ N_{x} such that n = n_{x}n′_{x} = n_{Δ}n′_{Δ} satisfies g^{−1}n ∈ P_{Δ} ∩ P_{x}, which is equal to P_{Δ∪{x}} by 5.3. This proves our claim. Now we write as in Theorem 5.2 Ω = ⋃_{σ∈Σ} Ω_{σ}, where Ω_{σ} runs over all finite subsets of Ω containing x_{0}. For all σ we choose some n_{σ} ∈ N such that g^{−1}n_{σ} ∈ P_{Ω_{σ}}. Put n_{σ} = n_{0} if Ω_{σ} is the set {x_{0}}. The same argument as in 5.2 shows that we can find a subset Σ′ ⊆ Σ such that Ω = ⋃_{σ∈Σ′} Ω_{σ} and such that n_{σ}^{−1}n_{0} is equal to some fixed m ∈ N_{x_{0}} modulo T_{x_{0}} for all σ ∈ Σ′. Since any element in T_{x_{0}} leaves the components Λ_{I} for I ⊆ J pointwise invariant, it also stabilizes Ω. Therefore n_{σ}^{−1}n_{0}m^{−1} lies in N_{Ω} ⊂ P_{Ω}. Since g^{−1}n_{σ} is contained in P_{Ω_{σ}}, the same holds for g^{−1}n_{0}m^{−1}, so that g^{−1}n_{0}m^{−1} lies in ⋂_{σ∈Σ′} P_{Ω_{σ}}, which is equal to P_{Ω} by 5.3. Hence n = n_{0}m^{−1} satisfies nx = gx for all x ∈ Ω, as desired. □ Now we can prove

Theorem 5.5 \overline{X} is compact.

By [Bou-T], I.10.4, Proposition 8, we know that since U_{0}^{∧} × \overline{Λ} is compact, the quotient after ∼ is Hausdorff (hence compact), iff the relation ∼ is closed in (U_{0}^{∧} × \overline{Λ}) × (U_{0}^{∧} × \overline{Λ}). Let (u_{k})_{k} and (v_{k})_{k} be sequences in U_{0}^{∧} converging to u respectively v, and let x_{k} and y_{k} be sequences in \overline{Λ} converging to x respectively y, so that (u_{k}, x_{k}) ∼ (v_{k}, y_{k}). We have to show that (u, x) ∼ (v, y). By definition of ∼ we have x_{k} = u_{k}^{−1}v_{k}y_{k}, so that both 0 and x_{k} lie in \overline{Λ} and in u_{k}^{−1}v_{k}\overline{Λ}. Using 5.4, we find some n_{k} ∈ N such that n_{k}z = v_{k}^{−1}u_{k}z for all z ∈ \overline{Λ}∩u_{k}^{−1}v_{k}\overline{Λ}. In particular, n_{k} lies in N_{0} and n_{k}x_{k} = v_{k}^{−1}u_{k}x_{k} = y_{k}. Hence g_{k} = u_{k}^{−1}v_{k}n_{k} lies in P_{x_{k}} ∩ P_{0} = P_{{0,x_{k}}}. Since N_{0} is compact, we can pass to a subsequence and assume that n_{k} converges towards some n ∈ N_{0}, so that g_{k} converges towards some u^{−1}vn. By 4.8, we can write g_{k} = w_{k}^{−}w_{k}^{+}m_{k} for some w_{k}^{±} ∈ U_{{0,x_{k}}}^{±} and m_{k} ∈ N_{{0,x_{k}}}. We can again assume that m_{k} converges towards some m ∈ N_{0}. Since N acts continuously on \overline{Λ}, m lies also in N_{x}. Besides, U_{0}^{−} is compact, so that we can assume that w_{k}^{−} converges towards some w^{−} ∈ U_{0}^{−}. Using 4.7 and 4.3 we find that w^{−} lies in fact in U_{x}^{−}. Now w_{k}^{+} also converges towards some w^{+} which lies in U_{x}^{+} by the same argument. Therefore u^{−1}vn lies in P_{x}. Since N acts continuously on \overline{Λ}, we have nx = y, so that indeed (u, x) ∼ (v, y). □

Theorem 5.6 The space \overline{X} is contractible.

Proof: Recall from 3.4 that \overline{Λ} is contractible. If x = ∑_{i≠j∈J} x_{j}(−η_{j}^{J}) is a point in E_{i} ∩ Λ_{J}, where J contains i, then the contraction map is given by

r(x, t) = \{ \begin{matrix}x, & if t = 0 \\ ∑_{i≠j∈J} \frac{(1−t)x_{j}}{1+tx_{j}} (−η_{j}) + ∑_{j∉I} \frac{1−t}{t}(−η_{j}), & if t ≠ 0. \end{matrix}

Now we define

R : U_{0}^{∧} × \overline{Λ} × [0, 1] ⟶ U_{0}^{∧} × \overline{Λ}
((g, x), t)  ⟼ (g, r(x, t)).

Obviously, R is continuous. In order to show that R is a contraction map for \overline{X}, it suffices to prove that it is compatible with our equivalence relation. Let us first check that nr(x, t) = r(nx, t) for all n ∈ N_{0} and x ∈ \overline{Λ}. Write n = tp for t ∈ T and a permutation matrix p. Then t lies in N_{0}, so that it acts trivially on all points in \overline{Λ}. Besides, a straightforward calculation shows pr(x, t) = r(px, t). Now assume that (g, x) and (h, y) in U_{0}^{∧} × \overline{Λ} are equivalent. Hence there is some n ∈ N with nx = y and g^{−1}hn ∈ P_{x}. Using 5.4, we can assume that n lies in fact in N_{0}. Now fix some t ∈ [0, 1]. We already know that r(y, t) = r(nx, t) = nr(x, t), so that our claim, namely (g, r(x, t)) ∼ (h, r(y, t)) is proven, if we show that g^{−1}hn ∈ P_{r(x,t)}. We already know that g^{−1}hn lies in P_{0} ∩ P_{x}, which is equal to P_{{0,x}} by 5.3. Let us put x_{t} = r(x, t). We are done if we show that P_{{0,x}} ⊂ P_{x_{t}}. We can assume that t > 0. The point x is contained in some E_{i} ∩ Λ_{J}. Recall from 4.8 that P_{{0,x}} = U_{{0,x}}^{−}U_{{0,x}}^{+}N_{{0,x}} and P_{x_{t}} = U_{x_{t}}^{−}U_{x_{t}}^{+}N_{x_{t}} for some fixed Φ^{+}. A straightforward calculation yields

f_{x_{t}}(a_{kl}) ≤ f_{x}(a_{kl}), if k ≠ i and l are in J and f_{x}(a_{kl}) ≥ 0,
or if k ∉ J and l ∈ J
f_{x_{t}}(a_{kl}) ≤ 0, if k ≠ i and l are in J and f_{x}(a_{kl}) < 0,
or if k = i and l ∈ J, or if l ∉ J

Hence for all a ∈ Φ we have U_{a,0} ⊂ U_{a,x_{t}} or U_{a,x} ⊂ U_{a,x_{t}}, which implies that U_{{0,x}}^{+} ⊂ U_{x_{t}}^{+} and U_{{0,x}}^{−} ⊂ U_{x_{t}}^{−}. Besides, we have N_{{0,x}} ⊂ N_{x_{t}}, since r is compatible with the action of N_{0}, so that our claim follows. □ The following result shows that the boundary of our compactification \overline{X} consists of the Bruhat-Tits buildings of all groups P GL(W), where W is a non-trivial subspace of V:

Theorem 5.7 There is a bijection between \overline{X} and the set

⋃_{0≠W ⊆V} P GL(W) = X ∪ ⋃_{0≠W ⊂V} P GL(W).

Proof: Let us first fix some non-empty I ⊂ \underline{n}. We will start by embedding the building corresponding to G_{I} = P GL(V_{I}) in \overline{X}. Recall that we write G^{V_{I}} for the subgroup of G leaving V_{I} invariant, and ρ_{I} for the map G^{V_{I}} → G_{I}. Then the building for G_{I} is defined as G_{I} × Λ_{I}/ ∼, where ∼ is the equivalence relation from section 2 (replacing V by V_{I} everywhere). Let (h, x) be an element of G_{I} × Λ_{I}. We choose an arbitrary lift h^{↑} of h in G^{V_{I}}, and map (h, x) to the point h^{↑}x in \overline{X}. This induces a map

j_{I} : G_{I} × Λ_{I} ⟶ \overline{X}.

We claim that this is independent of the choice of a lift. We have to show that any element g in the kernel of ρ_{I} stabilizes each x ∈ Λ_{I}. Note that G^{V_{I}} is a parabolic subgroup of G. Let us fix a Borel group B ⊃ T contained in G^{V_{I}}. Then there is a set of positive roots Φ^{+} in Φ such that B = U_{Φ^{+}}T. Besides, we can write G^{V_{I}} = BW′B for some subgroup W′ of W. Hence we find that G^{V_{I}} = U_{Φ^{+}}N^{V_{I}} U_{Φ^{+}}, where N^{V_{I}} = N ∩ G^{V_{I}}. Note that since U_{Φ^{+}} is contained in G^{V_{I}}, no root of the form a = a_{ij} such that j ∈ I, but i ∉ I can be contained in Φ^{+}. We write g = u^{+}nv^{+} for some n in N^{V_{I}} and u^{+}, v^{+} in U_{Φ^{+}}. Using ρ_{I}(g) = 1, we find that ρ_{I}(n) = 1 and ρ_{I}(u^{+}v^{+}) = 1. Hence n stabilizes each x ∈ Λ_{I}, so n ∈ P_{x} ∩ N = N_{x}. Let us write [I] for the subset of all roots that are linear combinations of roots a_{ij} with i and j in I. We write u^{+} = u_{1}^{+}u_{2}^{+} and v^{+} = v_{2}^{+}v_{1}^{+} for some u_{1}^{+}, v_{1}^{+} ∈ m(∏_{a∈Φ^{+}\[I]} U_{a}) and u_{2}^{+}, v_{2}^{+} ∈ m(∏_{a∈Φ^{+}∩[I]} U_{a}), where m is the multiplication map. As in 4.2, we find that ρ_{I}(u_{1}^{+}) and ρ_{I}(v_{1}^{+}) are trivial, and that ρ_{I} is injective on all U_{a} for a ∈ Φ^{+} ∩ [I]. Hence we deduce that u_{2}^{+}v_{2}^{+} is trivial. Besides, u_{2}^{+} commutes with n, so that g = u_{1}^{+}u_{2}^{+}nv_{2}^{+}v_{1}^{+} = u_{1}^{+}nv_{1}^{+}. Recall that the roots a_{ij} for i ∉ I and j ∈ I do not lie in Φ^{+}, so that by 4.1 we have U_{a} = U_{a,x} for all a ∈ Φ^{+}\[I] and all x ∈ Λ_{I}. Hence g is indeed contained in P_{x} for all these x. Now we claim that j_{I} induces an injection

j_{I} : X(G_{I}) = G_{I} × Λ_{I}/ ∼ ↪ \overline{X}.

It suffices to show that for all x ∈ Λ_{I} the map ρ_{I} induces a surjection

ρ_{I} : P_{x} ⟶ P_{x}^{I},

where P_{x}^{I} is defined in the same way as P_{x} (see section 2), just replacing V by V_{I}. Note that this also implies P_{x} = ρ_{I}^{−1}(P_{x}^{I}). First of all, let us show that P_{x} is contained in G^{V_{I}}, so that our statement makes sense. By 4.2, we find that U_{x} is contained in G^{V_{I}}. Besides, if n is an element of N_{x}, we find that n = tp for a torus element t and a permutation matrix p. The corresponding permutation must leave I intact, so that indeed n ∈ N^{V_{I}}. Besides, by 4.2 we know that ρ_{I} maps U_{x} to U_{x}^{I}, and it is easy to see that ρ_{I} also maps N_{x} to N_{x}^{I}, so that we get a homomorphism ρ_{I} : P_{x} → P_{x}^{I}. Now take some h = nu in P_{x}^{I} = N_{x}^{I}U_{x}^{I}, and choose a lift n^{↑} of n in N^{V_{I}}. Since n stabilizes x (viewed in the apartment Λ_{I}), the lift n^{↑} stabilizes the point x ∈ \overline{Λ}. Hence n^{↑} lies in P_{x}. By 4.2, the element u has a lift u^{↑} in U_{x}, hence h^{↑} = n^{↑}u^{↑} is an element in P_{x} projecting to h via ρ_{I}. This proves surjectivity.

Let now W be an arbitrary non-trivial subspace of V. Then there is a linear isomorphism

f : V_{I} ⟶ W

for some I ⊂ \underline{n}. Let S be the maximal torus in P GL(W) induced by the diagonal matrices with respect to the basis f(v_{i}) (for all i ∈ I). Conjugation by f induces an isomorphism P GL(W) → P GL(V_{I}) = G_{I}, which maps S to T_{I}, and the normalizer N(S) of S to N(T_{I}), the normalizer of T_{I} in G_{I}. Hence we get an ℝ-linear isomorphism

τ : X_{∗}(S)_{ℝ} ⟶^{∼} X_{∗}(T_{I})_{ℝ} = Λ_{I}.

One can check that for all x ∈ X_{∗}(S)_{ℝ} and n ∈ N(S) we have τ(nx) = (f^{−1}nf)τ(x). Choose some f^{↑} ∈ G whose restriction to V_{I} is given by f. Then we define a map

j_{W} : P GL(W) × X_{∗}(S)_{ℝ} ⟶ G_{I} × Λ_{I} ⟶^{j_{I}} \overline{X} ⟶^{f^{↑}} \overline{X},

where the first map is given by

(g, x) ⟼ (f^{−1}gf, τ(x)).

Since this maps the equivalence relation on P GL(W) × X_{∗}(S)_{ℝ} defining the building X(P GL(W)) to the equivalence relation on G_{I} × Λ_{I} defining the building X(G_{I}), we have an injection

j_{W} : X(P GL(W)) ⟶^{∼} X(G_{I}) ⟶^{j_{I}} \overline{X} ⟶^{f^{↑}} \overline{X}.

Of course, we have to check that this is well-defined. First of all, if f is fixed, then j_{W} does not depend on the choice of a lifting f^{↑}, since two such liftings differ by something in the kernel of ρ_{I}, and this acts trivially on the image of j_{I}, as we have seen above. What happens if we choose another isomorphism g : V_{J} → W for some J ⊂ \underline{n}? First let us consider the case that we construct j_{W} using f′ = s ∘ f for some isomorphism s : W → W. Then we also use a different construction of the building X(P GL(W)), since we use another torus. Following [La], 13.18, we find that there exists a unique P GL(W)–equivariant isometry between these two constructions (which we tacitly use to identify them). A straightforward calculation shows that our map j_{W} is compatible with this isometry. Now assume that we use an isomorphism g : V_{J} → W to construct j_{W}. Then there exists some isomorphism r : V_{I} → V_{J} mapping the basis v_{i} for i ∈ I to the basis v_{j} for j ∈ J in some way. The map r can be lifted to a permutation matrix n ∈ N. It is easy to see that this implies that our construction of j_{W} does indeed not change if we choose g ∘r instead of g. Hence j_{W} is well-defined. We put all these maps j_{W} together and get a map

j : X ∪ ⋃_{0≠W ⊂V} X(P GL(W)) ⟶ \overline{X},

which is obviously surjective. It remains to show that j is injective. Let us first assume that we have some (g, x) ∈ P GL(W)×X_{∗}(S)_{ℝ} and (h, y) ∈ P GL(W′)× X_{∗}(S′)_{ℝ} such that j_{W} (g, x) = j_{W′}(h, y). Hence there are isomorphisms f_{I} : V_{I} → W and f_{J} : V_{J} → W′ and points x_{0} ∈ Λ_{I}, y_{0} ∈ Λ_{J} such that g^{↑}f_{I}^{↑}x_{0} = h^{↑}f_{J}^{↑}y_{0}. In particular, there is some n ∈ N mapping x_{0} to y_{0}. Hence n maps Λ_{I} to Λ_{J}, which implies nV_{I} = V_{J}. We have already seen that P_{x_{0}} ⊂ G^{V_{I}}, so that h^{↑}f_{J}^{↑}nV_{I} = g^{↑}f_{I}^{↑}V_{I}, which implies W = f_{I}(V_{I}) = f_{J}(V_{J}) = W′. Since we already know that j_{W} is injective, our claim follows. □ To conclude this paper, let us show that we can identify the vertices in \overline{X} with the equivalence classes {N} of R-modules of arbitrary rank in V. Together with Proposition 3.1, this is the link to Mustafin’s paper [Mu]. We call a point x in \overline{Λ}, say x ∈ Λ_{I}, a vertex in \overline{Λ}, if it is a vertex in the apartment Λ_{I}. By section 2 this means, that x = ∑_{i∈I} x_{i}η_{i}^{I} with integer coefficients x_{i}. We call a point y in \overline{X} a vertex if y = gx for some g ∈ G and some vertex x ∈ \overline{Λ}, and we denote the set of vertices in \overline{X} by \overline{X}^{0}. We call two R-lattices in V equivalent, if they differ by a factor in K^{×}. Let us denote by \overline{L} the set of all equivalence classes of R-lattices in V of arbitrary positive rank. We write {M} for the class of such a lattice. Our last result shows that the vertices in \overline{X} correspond to elements of \overline{L}, i.e. lattice classes of arbitrary rank, which explains the title of this paper.

Lemma 5.8 The G-equivariant bijection φ : L ⟶ X^{0} can be continued to a G-equivariant bijection

φ : \overline{L} ⟶ \overline{X}^{0}

in the following way: We write \{M\} ∈ \overline{L} as \{M\} = g\{L\}, where L = ∑_{i∈I} π^{k_{i}}Rv_{i} for some non-empty I ⊆ \underline{n}. Then

φ({M}) = g(∑_{i∈I} k_{i}(−η_{i}^{I})).

Proof: We only need to check that φ is injective and well-defined, which amounts to the following claim: For the vertex x = ∑_{i∈I} k_{i}(−η_{i}^{I}) ∈ Λ_{I} let L_{x} = ∑_{i∈I} π^{k_{i}}Rv_{i}. Then P_{x} is the stabilizer of {L_{x}}. Using 4.1, it is easy to see that all U_{a,x} leave {L_{x}} invariant. Besides, N_{x} leaves {L_{x}} invariant, so that P_{x} is contained in S_{{L_{x}}}, the stabilizer of {L_{x}}. Let now g be an element in S_{{L_{x}}}. Note that L_{x} is a lattice of full rank in V_{I}, so that g is contained in G^{V_{I}}, and ρ_{I}(g) ∈ P GL(V_{I}) = G_{I} stabilizes {L_{x}}. In G_{I} we have the (usual) Bruhat decomposition ρ_{I}(g) = pnq with p and q in P_{x}^{I} and n ∈ N(T_{I}) (see [La], 12.10). Here P_{x}^{I} is the stabilizer of x in the building for G_{I}, so that p and q leave the class {L_{x}} in V_{I} invariant. Hence n ∈ G_{I} also stabilizes {L_{x}}. A straightforward calculation now shows that n fixes x ∈ Λ_{I}. Therefore ρ_{I}(g) lies in P_{x}^{I}, so that g is indeed contained in P_{x}, as we have seen in the proof of 5.7. □ Of course, one could also set up a bijection between \overline{L} and ⋃_{0≠W ⊆V} X(P GL(W))^{0} by using analogues of the map φ on every single

Bruhat-Tits Building of PGL 341

building X(P GL(W)). It is easy to see that this is compatible with the map we just described, and the identification of \overline{X} and ⋃_{0≠W ⊆V} X(P GL(W)) given in 5.7.

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[Bou] N. Bourbaki: Groupes et algèbres de Lie. Chapitres 4, 5 et 6. Hermann 1968.

[Bou-T] N. Bourbaki: General Topology I. Hermann / Addison Wesley 1966. [Br-Ti] F. Bruhat, J. Tits: Groupes réductifs sur un corps local. I. Données radi-

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Annette Werner

Mathematisches Institut Universität Münster Einsteinstr. 62 D - 48149 Münster Germany werner@math.uni-muenster.de 342

Documenta Mathematica 6 (2001)