GROUP-THEORETIC COMPACTIFICATION OF BRUHAT–TITS BUILDINGS

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GROUP-THEORETIC COMPACTIFICATION OF BRUHAT–TITS BUILDINGS

B

Y

VES

GUIVARC’H

AND

B

ERTRAND

RÉMY

ABSTRACT. – LetGFdenote the rational points of a semisimple groupGover a non-archimedean local field F, with Bruhat–Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of GF in the Chabauty topology, which enables us to compactify the vertices of X. We obtain a structure theorem showing that the Bruhat–Tits buildings of the Levi factors all lie in the boundary of the compactification. Then we obtain an identification theorem with the polyhedral compactification (previously defined in analogy with the case of symmetric spaces).

We finally prove two parametrization theorems extending the Bruhat–Tits dictionary between maximal compact subgroups and vertices ofX: one is about Zariski connected amenable subgroups and the other is about subgroups with distal adjoint action.

RÉSUMÉ. – SoitGFle groupe des points rationnels d’un groupe semi-simpleG, défini sur un corps local non archimédienF, et d’immeuble de Bruhat–Tits associéX. Ce papier contient cinq résultats principaux.

Nous démontrons un théorème de convergence de suites de sous-groupes parahoriques pour la topologie de Chabauty, ce qui nous permet de compactifier les sommets de X. Nous obtenons un théorème de structure qui prouve que les immeubles de Bruhat–Tits des facteurs de Lévi deGapparaissent tous dans le bord de la compactification. Ensuite nous obtenons un théorème d’identification avec la compactification polyédrique (définie auparavant par analogie avec le cas des espaces symétriques). Nous démontrons enfin deux théorèmes de paramétrage qui étendent le dictionnaire de Bruhat–Tits entre sous-groupes compacts maximaux et sommets deX: l’un porte sur les sous-groupes moyennables d’adhérence de Zariski connexe et l’autre porte sur les sous-groupes dont l’action adjointe est distale.

Introduction

There exist lots of deep motivations to construct compactifications of symmetric spaces and Euclidean buildings. One of them is to determine the cohomological properties of arithmetic groups. When the ambient algebraic group is defined over a global field of characteristic zero, this was done in [18,19]. In positive characteristic, there are still important open questions [17, §VII], [7]. Another related motivation is the compactification of locally symmetric manifolds [49], in particular those carrying a natural complex structure. Some compactifications [6] are useful tools in number theory and also provide nice examples of complex projective varieties of general type [5] or moduli spaces [4]. We refer to [9] for a recent review of compactifications of symmetric spaces and of their quotients by lattices of the isometry group.

In this paper we are interested in compactifying Euclidean buildings by group-theoretic techniques. In return, we obtain group-theoretic results, e.g. geometric parametrizations of

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classes of remarkable closed subgroups in non-archimedean semisimple Lie groups. The analogy with the case of symmetric spaces is of course highly relevant. In fact, our general project is to generalize to Bruhat–Tits buildings all the compactification procedures described in [36] or [35]

in the real case. Furstenberg (i.e. measure-theoretic) and Martin compactifications are included in the project, but will not appear in the present paper. Here, we are interested in the simplest approach: in the real case, it consists in seeing each point of a symmetric space as a maximal compact subgroup of the isometry group (via the Bruhat–Tits fixed-point lemma) and reminding that the spaceS(G)of all closed subgroups of a given locally compact groupGhas a natural compact topology. The latter topology is the Chabauty topology [13] and the compactification under consideration is the closure of the image of the map attaching to each point its isotropy subgroup. Of course, one has to check that this map is a topological embedding onto its image and this is done by proving the Chabauty convergence of a suitable class of sequences in the symmetric space. This compactification is equivariantly homeomorphic to the maximal Satake compactification [50], which itself was identified by C.C. Moore with the maximal Furstenberg compactification [42].

In the non-archimedean case, an additional subtlety is that the Bruhat–Tits building, i.e. the analogue of the symmetric space in this situation, is bigger than the set of maximal compact subgroups (which corresponds to the vertices of the building). The statement below deals with sequences of maximal compact subgroups only, but our most general result takes into account sequences of parahoric subgroups (see Theorem 3 for a precise version).

CONVERGENCE THEOREM. –Let G be a semisimple group over a local field F and let X be its Bruhat–Tits building. Let {vn}n1 be a sequence of vertices in some closed Weyl chamber Q^X. By passing to stabilizers in G_F we obtain a sequence of maximal compact subgroups{K_v_n}n1. We make the following assumption:

for each codimension one sector panel Π of Q^X, the distance dX(vn,Π) has a, possibly infinite, limit asn→+∞.

Then{Kvn}n1 is a convergent sequence in the space of closed subgroupsS(GF)endowed with the compact Chabauty topology. The limit group D is Zariski dense in some parabolic F-subgroupQfixing a face of the chamber at infinity∂_∞Q. MoreoverDcan be written as a semi-direct productKRu(Q)F, whereKis an explicit maximal compact subgroup of some reductive Levi factor ofQandRu(Q)F is the unipotent radical ofQF.

As already mentioned, this convergence is the key fact to define a compact spaceV^gp_X with a naturalGF-action. We call it the group-theoretic compactification of X. The next step then is to understand the geometry ofV^gp_X by means of the structure ofGF. For instance, in the Borel–Serre compactification the boundary reflects the combinatorics of the parabolic subgroups defined over the ground field of the isometry group; a single spherical building is involved. In our case, the group-theoretic compactification of the Euclidean building of each Levi factor of G_F appears, as in Satake’s compactifications of symmetric spaces [50]. The result below sums up Theorem 16.

STRUCTURE THEOREM. –For any proper parabolic F-subgroup Q, the group-theoretic compactification of the Bruhat–Tits building of the semisimple F-group Q/R(Q) naturally sits in the boundary ofV^gp_X. Let P be a minimal parabolic F-subgroup of G. We set D_∅= KRu(P)F, whereKis the maximal compact subgroup of some reductive Levi factor ofP. Then the conjugacy class ofD_∅isGF-equivariantly homeomorphic to the maximal Furstenberg boundaryF and is the only closedGF-orbit inV^gp_X. In fact, for any closed subgroupD∈V^gp_X there is a sequence{gn}n1inGF such thatlim_n→+∞gnDgn−1exists and lies inF.

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There are then two ways to exploit this result. The first one is to use it to compareV^gp_X with previous compactifications of Bruhat–Tits buildings. One compactification was defined in [37] by compactifying apartments first (the convergence there is the same as in flats of maximal Satake’s compactifications) and then by extending the Bruhat–Tits gluing procedure which definesXout ofG_F and the model of an apartment. We call the so-obtained compactification the polyhedral compactification ofXand we denote byV^pol_X the closure of the vertices in the latter space. Note that it is not obvious at all that a space defined as a gluing is Hausdorff. In connection with this technicality for the polyhedral compactification, we fill in a gap of [loc. cit.] pointed to us by G. Prasad and A. Werner (Proposition 19). Then we compareV^gp_X andV^pol_X , see Theorem 20 for a more precise version.

IDENTIFICATION THEOREM. –LetGbe a semisimple simply connected group defined over a non-archimedean local fieldF. LetX be the corresponding Bruhat–Tits building. Then there exists a naturalGF-equivariant homeomorphismV^pol_X V^gp_X.

The second use of the compactification is to parametrize remarkable classes of closed subgroups. By taking stabilizers, we can extend to the non-archimedean case a theorem of C.C. Moore’s [43] which answers a question of H. Furstenberg’s and unifies the classification of maximal compact and minimal parabolic subgroups in the same geometric framework. The proofs of the result below (made more precise in Theorem 33) and of the next one use subtle results due to Ph. Gille on unipotent elements in algebraic groups, in order to cover the case of a local ground field of characteristicp >0.

PARAMETRIZATION THEOREM (AMENABLE CASE). –Any closed, amenable, Zariski con- nected subgroup ofG_F fixes a facet inX^pol. The closed amenable Zariski connected subgroups ofG_F, maximal for these properties, are the vertex fixators for theG_F-action on the polyhedral compactificationX^pol.

To state our last main result, we need to go back to the very definition of the group-theoretic compactification. Since it is the closure of the maximal compact subgroups in the compact space S(GF)of closed subgroups ofGF, it is natural to ask for an intrinsic characterization of the groups appearing after passing to the closure. A satisfactory answer is given by the notion of distality, which formalizes the fact that a group action on a metric space has no contraction.

Here the linear action under consideration is given by the adjoint representation. We refer to Theorem 39 and its preliminaries for a more precise version.

PARAMETRIZATION THEOREM(DISTAL CASE). –Any subgroup of G_F with distal adjoint action is contained in a point of V^gp_X. The maximal distal subgroups of GF are the groups of V^gp_X, i.e. they are the maximal compact subgroups and the limits of sequences of maximal compact subgroups. In particular, they are all closed and Zariski connected.

Let us finish the presentation of our results by mentioning that our proofs may simplify some arguments in the real case. We also plan to compare our approach to some concrete compactifications defined by A. Werner in the SLn case [54,55]. In the present paper, the latter case is presented in the last section as an illustration of the general semisimple case.

Unfortunately, it would have been too long to develop it completely, but we think that checking the details is useful to have a good intuition of the geometry of Euclidean buildings. At last, since algebraic group theory uses a lot of notation, we found useful to collect some of it below.

Notation. – In all this paper, we are considering the following objects:

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• a locally compact non-archimedean local fieldF, with valuation ringOF, uniformizerF

and residue fieldκF =OF/F. The absolute value is denoted by| · |F and the discrete valuation byvF:F→Z. We set:qF=|κF|;

• a simply connected semisimple algebraicF-groupG;

• the Bruhat–Tits buildingX ofG_/F, whose set of vertices is denoted byV_X.

We letF be an algebraic closure of the fieldF. There is a unique valuationv_F:F→Q(resp.

absolute value| · |_F) extendingv_F (resp.| · |F) and we denote byO_Fthe valuation ring ofF. In general, given an algebraic groupH overF, we denote byLie(H)orhits Lie algebra, by HF itsF-rational points and byLie(H)F orh_F theF-rational points of the Lie algebra ofH. We denote byR(H)(resp. byRu(H)) the radical (resp. the unipotent radical) ofH. At last, if His semisimple, thenrk_F(H)denotes itsF-rank.

Structure of the paper

Section 1 fixes notation and recalls basic facts on algebraic groups and Bruhat–Tits theory; it also introduces the class of fundamental sequences in Euclidean buildings. Section 2 is mainly devoted to studying convergence of fundamental sequences of parahoric subgroups for the Chabauty topology on closed subgroups in the semisimple group G_F; this is the main step to define the group-theoretic compactification V^gp_X of the Bruhat–Tits building X. Section 3 describes V^gp_X and in particular shows that, as a G_F-space, V^gp_X contains a single closed G_F-orbit; moreover the compactification of the Bruhat–Tits building of any Levi factor lies in the boundary ofV^gp_X. We also prove the identification with the polyhedral compactificationX^pol. Section 4 deals with compactifications of trees in a slightly more general context than rank-one algebraic groups over local fields; it can be seen both as an illustration of the previous sections and the first step of induction arguments in the next section. Section 5 contains the proofs of the two parametrization theorems in terms of the geometry of the compactificationX^pol; the two parametrized classes of subgroups are that of maximal Zariski connected amenable and of maximal distal subgroups. Section 6 provides examples of arbitrary positive F-rank since it deals with special linear groups; we recall Goldman–Iwahori’s concrete definition of the Bruhat–Tits building ofSLn(F)and we try to illustrate as many previous notions as possible.

At last, Appendix A provides a proof of the continuity of the action ofGF on the polyhedral compactificationX^pol.

1. Bruhat–Tits buildings. Levi factors. Unbounded sequences

We introduce some algebraic subgroups and notation and we recall the geometric meaning of the valuated root datum axioms for the rational pointsGF. We also recall a more technical point:

the Bruhat–Tits building of a Levi factor naturally sits in the Bruhat–Tits building of the ambient groupG_/F. We finally use the Cartan decomposition with respect to a suitable maximal compact subgroup, in order to distinguish a class of sequences in buildings which will become convergent in the group-theoretic compactification of the next section. We setr= rkF(G).

1.1. Bruhat–Tits buildings

We choose once and for all a maximalF-split torusTinG, to which is associated an apartment Ain the Bruhat–Tits buildingX[22, 2.8.11]. We denote byΦ = Φ(T, G)the corresponding root system [12, 8.17]. It is a (possibly non-reduced) root system [20, 5.8] in the sense of [16, VI.1].

In order to avoid confusions and to emphasize the analogy with symmetric spaces, asectorof

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X [17, VI.7] (in French: “quartier” [22, 7.1.4]) is often called aWeyl chamberin this article. We also use the terminologyalcove[22, 1.3.8] so that the word “chamber” alone is meaningless in the present paper.

1.1.1. Let us pick inAa Weyl chamberQwith tip a special vertex which we callo. Let us denote bycthe alcove contained inQwhose closure containso. We refer toA(resp.Q,o,c) as thestandardapartment (resp. Weyl chamber, vertex, alcove) ofX. The fixatorK_o= Fix_G_F(o) is called thestandard maximal compact subgroupand its subgroupB= FixGF(c)is called the standard Iwahori subgroupof GF. The choice ofQ corresponds to the choice of a subset of positive rootsΦ⁺, or equivalently to the choice of a system of simple roots{as}s∈S which we identify with its indexing setS. We set:Φ⁻=−Φ⁺. By definition ofΦ, we have a decomposition of the Lie algebragas aT-module via the adjoint representation:g=g₀⊕

a∈Φg_a, whereg₀is the fixed-point set ofT [12, 21.7]. The subgroupUawith Lie algebrag_a={X∈g: Ad(t).X= a(t)X for eacht∈T} is theroot groupassociated witha. Ifais non-divisible,U_(a) denotes the groupU_a·U_2a with the convention that U_2a={1} if2a /∈Φ[12, 21.7]. We denote by P the minimal parabolic F-subgroup determined byΦ⁺, i.e. such thatp=g₀⊕

a∈Φ⁺g_a. For a subsetI ofS, we denote byΦI the subset of roots which are linear combinations of simple roots indexed byI. We also set:Φ^±_I = Φ^±∩ΦI andΦ^I,±= Φ^±−ΦI. We also introduce the followingF-subgroups [20, §4]:

• the standard parabolic subgroup P_I of type I, defined by p_I =g₀ ⊕

a∈ΦIg_a ⊕

a∈Φ^I,+g_a;

• thestandard reductive Levi factorMI ofPI, defined bym_I=g₀⊕

a∈ΦIg_a;

• thestandard semisimple Levi factorGI= [MI, MI]ofPI;

• thestandard unipotent radicalU^I=Ru(PI)of typeI, also defined byu^I=

a∈Φ^I,+g_a. The parabolic F-subgroup opposite to P_I with respect to T [12, 14.20] is the connected F-subgroup with Lie algebram_I⊕

a∈Φ^I,−g_a; it intersectsPI along the reductive groupMI. WhenI=∅and when no confusion is possible, we simply omit the index_∅, e.g. we denote M =M_∅. In the classification theory of semisimple F-groups, the (GF-conjugacy class of the) semisimple Levi factor [M, M] is called the anisotropic kernel of G_/F [51, 16.2.1].

The connected center Z(M) is a torus defined over F and contains T as a maximal split subtorus defined overF. We denote byZ(M)anthe maximal anisotropic subtorus defined over F in Z(M). Then, by [12, 8.15] and [20, 2.3], the multiplication map provides an isogeny [M, M]×Z(M)an×T →M. The subgroup [M, M]·Z(M)an is an anisotropic reductive F-subgroup of M, so its F-points form a compact group [48, 5.2.3]. The unique maximal compact subgroup ofT_F, which we denote byT_cpt, is topologically isomorphic to(O_F^×)^r. We denote byZG(T)cptthe group[M, M]F·(Z(M)an)F·Tcpt. It is the unique maximal compact subgroup ofMF=ZG(T)F [24, 5.2.7] and it fixes (pointwise) the apartmentA.

1.1.2. A substantial part of Borel–Tits theory (i.e. the theory of reductive groups over arbitrary ground fields [20]) can be summed up in combinatorial terms [12, 21.15]. The most refined version of this approach is provided by the notion of agenerating root datum [22, 6.1]. This is relevant to the case of an arbitrary ground field. The combinatorics becomes richer when the ground field is a local field F as here. One of the main results of Bruhat–Tits theory is the existence of avaluationon the generating root datum ofGF associated with the choice of the maximalF-split torusT [24, 5.1.20]. Each groupU_(a)is unipotent, abelian or metabelian [21, 4.10], and (roughly speaking) the latter notion corresponds to the existence of a filtration on each group(U_(a))_F. Such a filtration comes from the filtration of the additive group(F,+)given by the preimages of vF. Further compatibilities (e.g. with respect to the action of the normalizer

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NG(T)F ofTF, to taking some commutators etc.) are required, but we are only interested in the geometric interpretation of this valuation [53, 1.4].

Letb∈Φbe non-divisible and let{U(b),m}m∈Zbe the decreasing filtration of(U_(b))F given by Bruhat–Tits theory, i.e. by [24, 5.1.20] and [22, 6.2]. To the pair of opposite roots{±b} is attached a parallelism class∂b of affine hyperplanes inA and a family of affine hyperplanes {H_∂b,m}m∈Z in this class which are called thewallsdirected by∂b. The family{H_∂b,m}m∈Z

provides a useful exhaustion ofAby the fixed-point sets of the groupsU_(b),m [53, 2.1]. More precisely, ifb∈Φ⁺we denote byDb,mthe half-space ofAbounded byH∂b,mwhich contains a translate of the Weyl chamberQ; otherwise, we choose the other half-space to beDb,m. The family{H∂b,m}m∈Z is characterized by the fact that the fixed-point set ofU_(b),minAis equal to the half-spaceDb,m. We have an increasing exhaustionA=

m∈ZDb,m. Geometrically, the biggerm∈Zis, the smallerU_(b),mis, and the bigger the closureDb,m=A^U^(b),mis. The reader may illustrate this by having a look at the third paragraph of 6.2.2, dealing with the example of SL_n(F).

1.1.3. As a combinatorial Euclidean building,X can be endowed with a distance dof non- positive curvature, unique up to homothety on each irreducible factor [22, 2.5]. More precisely, the distance d makes X a CAT(0)-space [8, II]; we fix, once and for all, such a metric d.

The boundary at infinity ∂_∞X, i.e. the space of geodesic rays modulo the relation of being at finite Hausdorff distance from one another [8, II.8], is a geometric realization of the spherical building of parabolic subgroups in G_F [17, VI.9E]. We can also define the F-points of the parabolic subgroups ofG/F to be the stabilizers of the facets at infinity in∂_∞X. For instance, the standard sectorQdefines a chamber at infinity∂_∞Q, and we have:PF= StabGF(∂_∞Q) = FixGF(∂_∞Q). The inclusion∂_∞Q⊂∂_∞Acorresponds to the Levi decompositionP=MU: the groupM is characterized by the fact thatMF is the fixator of the union of the facet∂_∞Q and its opposite in∂_∞A. There is a similar interpretation for each standard Levi decomposition PI=MIU^I.

Recall that aFurstenberg boundaryfor a topological groupGis a compact metrizableG-space Y whose continuousG-action isminimal(i.e. any orbit inY is dense) andstrongly proximal(i.e.

there is a Dirac mass in the closure of anyG-orbit in the spaceM¹(Y)of probability measures onY) [41, VI.1.5]. It is a classical fact that the family of Furstenberg boundaries of a given semisimple Lie group coincides with the family of its flag varieties; in the non-archimedean case, this is checked for instance in [10, 5.1]. Moreover, ifQdenotes a parabolicF-subgroup of G, the F-rational points of G/Q form a homogeneous space under GF since we have:

(G/Q)F =GF/QF [20, 4.13]. In this paper the quotient spacesGF/QF, for Qa parabolic F-subgroup ofG_/F, are indifferently calledFurstenberg boundariesorflag varieties.

DEFINITION. – We denote byF the (maximal)Furstenberg boundaryGF/PF of GF. For each subsetIof simple roots ofS, we denote byF^I the Furstenberg boundaryGF/(PI)F.

There is an obviousGF-equivariant mapπI:F →F^I between Furstenberg boundaries. We denote byω(resp.ωI) the class of the identity inF (resp. inF^I). The preimage ofωI byπI

is(PI)F.ω. It is a copy inF of the (maximal) Furstenberg boundary of the Levi factor(MI)F; we denote it byFI. We denote byU^I,−the unipotent radical of the parabolic subgroup opposite P_I with respect toT; we simply writeU⁻ forU^∅,−. Note that there is a uniqueG_F-invariant class of measures onF [40, §1] and that the(U⁻)_F-orbit ofωis conegligible for this class. In the algebraic terminology,(U⁻)_F.ω is called thebig cellofF. Let us finally recall that there is a natural way to glueG_F-equivariantly∂_∞X toX [8, II.8]. The so-obtained space is called thegeometric compactificationofX; we denote it byX^geom. The partition of the boundary of

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X^geomunder its naturalKo-action is connected to flag varieties by the fact that eachKo-orbit is isomorphic, as aKo-space, to a suitable flag variety ofGF.

1.2. Levi factors

We recall that the Bruhat–Tits building X of G_/F contains inessential realizations of the buildings of the Levi factors inG[22, 7.6]. We also introduce some remarkable closed subgroups ofGF. They will turn out to form the boundary of our compactification ofX, respectively to be the stabilizers of the points in this boundary.

1.2.1. We simply recall here (with our notation) some facts of [22, 7.6] we use later (3.2).

Recall that the apartmentAis the vector spaceX_∗(T)F⊗ZRendowed with a suitable simplicial structure and a natural action ofNG(T)F (whereX_∗(T)F is the group ofF-cocharacters ofT) [53, 1.2]. LetIbe a proper subset of simple roots inS. LetLI denote the affine subspace ofA obtained as the intersection of the walls passing through the vertexoand directed by the simple roots in I. The semisimple Levi factorGI is simply connected [51, 8.4.6, Exercise 6] and we denote by AI the standard apartment of its Bruhat–Tits buildingXI. We are interested in the subset(GI)F.Aof the(GI)F-transforms of the points inA, which we want to compare toXI. Intuitively, the idea is that the directionLI is not relevant to the combinatorics of the semisimple Levi factorGI: it corresponds to the cocharacters ofTwhich centralizeGI. But after shrinking (GI)F.AalongLI, we obtain a realization ofXI. This is formalized by [22, Proposition 7.6.4]

which provides a unique extension p˜I: (GI)F.A→XI of the natural affine mappI:A→AI

between apartments, with the following properties:

(i) the mapp˜I is(GI)F-equivariant;

(ii) the preimage ofAI byp˜I isA and in fact the preimage of any apartment, wall, half- apartment inXIis an apartment, a wall or a half-apartment inX, respectively;

(iii) there is anLI-action on(GI)F.Aextending that onAwith the following compatibility with the(GI)F-action:g.(x+v) =g.x+vfor anyg∈(GI)F,x∈(GI)F.Aandv∈LI; (iv) the factor map^(G^I_L⁾^F^.A

I →XI is a(GI)F-equivariant bijection.

The choice of positive roots inΦcorresponding to the Weyl chamberQ induces a choice of positive roots in the subroot system ofGIand the corresponding Weyl chamber ofAI ispI(Q).

1.2.2. We denote byQ^X the (non-compact) closure of the Weyl chamberQin the building X. It is a simplicial cone. Any of its codimension one faces is equal to the intersection ofQ^X with the wall directed by some simple roots∈S and passing througho. We denote byΠ^sthe latter cone and we call it asector panelofQ^X. For any non-empty subsetI of simple roots, we denote:Q^I=

s∈IΠ^s; for instanceQ^S={o}. We set:Q^∅=Q, so thatQ^X=

I⊂SQ^I. Note also thatLIabove is the affine subspace generated byQ^I.

The points in Q^X are parametrized by the distances to the sector panelsΠ^swhensranges over S. Given a family d={ds}s∈S of non-negative real numbers, we denote by xd the corresponding point in Q^X. Similarly, in the building X_I we parametrize the closed Weyl chamber pI(Q)^X^I by the set of finite sequences d={ds}s∈I of non-negative real numbers (corresponding to the distances to the sector panels pI(Π^s),s∈I). The point defined by the parametersdis denoted byxI,d. The preimagep⁻_I¹(xI,d)is an affine subspace ofAparallel to Q^I, which we denote byLI,d. Each spaceLI,dhas dimensionr− |I|= dim(X)− |I|.

DEFINITION. – We define KI,d to be pointwise fixator of the affine subspace LI,d in the reductive Levi factor(MI)F, i.e.KI,d= Fix(MI)F(LI,d).

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Denoting byσthe facet ofAI containingxI,d, we have:KI,d= Fix_(M_I₎_F(p⁻¹_I (σ)), i.e. the groupKI,donly depends on the facet ofpI(Q^X)containingxd. The groupMI (resp.GI) is generated by the root groupsU_(a)withais a non-divisible root such that∂a⊃ QÎ(resp. and M) [12, 21.11]. The groupK_I,d is the parahoric subgroup of(M_I)_F generated byZ_G(T)_cpt and the groupsU_(a),m withaas before and m∈Zsuch that the closed half-apartment D_a,m containsL_I,d[22, 6.4]. Note thatp_I(QÎ)is a special vertex inA_I, which we choose as origin inA_I. We simply writeK_I whenL_I,d=QÎ=L_I. The groupK_I(resp.K_I∩G_I) is a special maximal compact subgroup of the standard reductive Levi factor(M_I)_F (resp. of the semisimple Levi factor(G_I)_F).

1.2.3. Let us setT^I= (

a∈ΦIKer(a))^◦and let us denote byTI the subtorus defined as the identity component(GI∩T)^◦for the Zariski topology. We have:MI=ZG(TÎ)[12, 21.11] and TI is a maximalF-split torus ofGI. Moreover(TÎ)F stabilizes the affine spanQÎ, fixes the facet at infinity∂_∞QÎand acts on each affine subspaceLI,dinAas a discrete cocompact group of translations. Denoting byZ(MI)the connected center ofMI and byZ(MI)anits maximal anisotropic subtorus defined overF, we have:MI=GI·TÎ·Z(MI)an, which provides:KI,d= (GI∩KI,d)·Fix_Z(M_I₎_F(LI,d), whereGI∩KI,dis a parahoric subgroup of(GI)F. The group Z(MI)F acts onLI,dthrough(TÎ)F and we have:Fix_Z(M_I₎_F(LI,d) = (Z(MI)an)F·(TÎ)cpt

where(TÎ)cptis the unique maximal compact subgroup of(TÎ)F, topologically isomorphic to (O_F^×)^r^−|Î^|.

The groupsDI,dandRI,dbelow play an important role in the definition and the description of the group-theoretic compactification ofX.

DEFINITION. – LetIbe a proper subset of simple roots and letdbe a sequence of non-negative real numbers indexed byI.

(i) We defineD_I,dto be the semidirect productK_I,d(U^I)_F.

(ii) We defineR_I,dto be the semidirect product(K_I,d·(TÎ)_F)(UÎ)_F. It follows from the above definitions thatRI,d=DI,d·(TÎ)F.

1.3. Unbounded sequences

We define some classes of sequences in the Euclidean buildingX. These sequences will turn out to be convergent in the later defined compactifications.

1.3.1. Recall that a sequence{xn}n1 in a topological spacegoes to infinityif it eventually leaves any compact subset of this space.

DEFINITION. – Let{x_n}n1be a sequence of points in the Euclidean buildingX. LetQ^Xbe the closure of the Weyl chamberQ. LetIbe a subset of the corresponding setSof simple roots.

(i) We say that{x_n}n1isI-canonical if the following three conditions are satisfied:

(i-a) for eachn1, we have:xn∈Q^X;

(i-b) for eachs∈S\I, we have:limn→+∞distX(xn,Π^s) = +∞;

(i-c) there exists a facetσinAI∩pI(Q^X)such thatpI(xn)∈σforn1.

(ii) We say that{xn}n1isI-fundamental if there exists a converging sequence{kn}n1in Kosuch that{kn.xn}n1is anI-canonical sequence.

(iii) We simply say that{xn}n1is fundamental if it isI-fundamental for someI⊂S.

Note that anI-fundamental sequence is bounded if, and only if, we have:I=S. Note also that the condition for beingI-canonical depends on the choice of the Weyl chamberQwhile that

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for being fundamental does not. The condition of beingI-canonical here is slightly more general than in the real case [36] since we do not impose that the points be in a face ofQ^X. We have to adopt this definition becauseT_F does not act transitively on the intersections of walls in a given parallelism class.

1.3.2. Moreover in the context of symmetric spaces, the definition of I-fundamental se- quences is not exactly the same [36, Definition 3.35]. The plain analogue of the latter definition would be obtained by replacing condition (i-c) of 1.3.1 by the condition:

(i-c) for eachs∈I, the distancedist_X(x_n,Π^s)converges asn→+∞.

On the one hand, let us pick two pointsxandyin the alcovec, such that for anys∈S we have:distX(x,Π^s)= distX(y,Π^s). Then any sequence{xn}n1 taking infinitely many times each valuexandyisS-fundamental in our sense, while it is not for the above plain translation from the case of symmetric spaces. On the other hand, an injective sequence of points{xn}n1

in the alcovec converging to the tipoisS-fundamental for the modified definition, but is not for the definition we will use (1.3.1). The sequences{x_n}n1and{x_n}n1show that the two definitions of being I-fundamental are different forI=S. To see the same phenomenon for IS, it suffices to pick a non-trivial elementt∈T^I and to replace{xn}n1(resp.{x_n}n1) by{tⁿ.xn}n1(resp.{tⁿ.x_n}n1).

The reason why we use the definition of 1.3.1 is that we want the mapx→Stab_G_F(x) =K_x to be continuous for a certain topology on closed subgroups of G_F (2.1). In the sequence {x_n}_n1, all the elements belong to the alcove c, so the associated sequence of parahoric subgroups{Kx_n}n1is constant equal to the Iwahori subgroupB. But the parahoric subgroup attached too= limn→+∞x_nis the maximal compact subgroupKo. Of course, this phenomenon occurs also for unbounded fundamental sequences; this is reformulated in terms of convergence of parahoric subgroups in 2.3.3.

1.3.3. It is clear that a sequence is unbounded if, and only if, it has a subsequence going to infinity. In our case, the existence of a Cartan decomposition ofGF with respect toQimplies a more precise result. The following lemma eventually says that any sequence in the buildingX has a convergent subsequence in suitable embeddings ofX(Theorem 3).

LEMMA 1. – Any sequence{x_n}n1in the building X has anI-fundamental subsequence for someI⊂S. Moreover we can chooseIto be proper inSwhenever{xn}n1is unbounded.

Proof. –By Cartan decomposition with respect to Ko [22, 4.4.3(2)], the closed Weyl chamber Q^X is a fundamental domain for the K_o-action on X: for each n 1, there exist kn ∈ Ko and qn ∈ Q^X such that xn = kn.qn. Since Ko stabilizes each sphere centered at o, the sequence {xn}n1 is bounded if, and only if, so is {qn}n1. In this case, there exists a subsequence of {qn}n1 converging to some q∈Q^X; since q lies in finitely many closures of facets, we are done. From now on, {x_n}n1 is assumed to be unbounded, so we may—and shall—assume that{xn}_n1goes to infinity. We set:J₁={s∈S:

lim sup_n→+∞distX(qn,Π^s) = +∞}. We have: J1 =∅. We pick s1∈J1 and choose an increasing map ψ1:N →N such that limn→+∞distX(q_ψ₁_(n),Π^s¹) = +∞. Then we set:

J2={s∈S\ {s1}: lim supn→+∞distX(qψ1(n),Π^s) = +∞}. IfJ2=∅, we setI=S\ {s1} andψ=ψ₁; otherwise, we picks₂∈S\ {s₁}and choose an increasing mapψ₂:N→Nsuch that lim_n→+∞dist_X(q_ψ₁_◦_ψ₂_(n),Π^s²) = +∞. After a finite number of iterations, we obtain a subset J ofS and an increasing mapψ:N→N such thatlimn→+∞distX(q_ψ(n),Π^s) = +∞

for s∈J andlim sup_n→+∞distX(q_ψ(n),Π^s)<+∞otherwise. It remains to set I=S\J and to pass to convergent subsequences for distances toΠ^s,s∈I, to obtain an increasing map ϕ:N→Nsuch thatdistX(qϕ(n),Π^s)converges fors∈Iand diverges otherwise. 2

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2. Group-theoretic compactification

We use the fact that for a locally compact groupG, the setS(G)of closed subgroups ofG carries a natural compact topology with several equivalent descriptions [13, VIII §5]. The starting point is to see the vertices of the buildingX as the set of maximal compact subgroups ofG_F, hence as a subset ofS(G_F). In the context of semisimple real Lie groups, the idea is originally due to the first author.

2.1. Chabauty topology and geometric convergence

LetGbe a locally compact group. We denote byC(G)the set of closed subsets ofG.

2.1.1. The spaceC(G)can be endowed with a separated uniform structure [14, II §1] defined as follows [13, VIII §5 6]. For any compact subsetCinGand any neighborhoodV of1GinG, we defineP(C, V)to be the set of couples(X, Y)inC(G)×C(G)such that:

X∩C⊂V ·Y and Y ∩C⊂V ·X.

The setsP(C, V)form a fundamental system ofentouragesof a uniform structure onC(G).

The so-obtained topology onC(G)is called theChabauty topology. The spaceS(G)of closed subgroups is a compact subset ofC(G)for the Chabauty topology [13, VIII §5 3, Théorème 1].

We henceforth assume thatGis metrizable; then so is the Chabauty topology. Moreover we can define the topology ofgeometric convergenceonC(G)[26], in which a sequence{Fn}n1

of closed subsets converge toF∈C(G)if, and only if, the two conditions below are satisfied:

(i) Letϕ:N→N be an increasing map and let{xϕ(n)}n1 be a sequence inGsuch that x_ϕ(n)∈F_ϕ(n)for anyn1. If{xϕ(n)}n1converges to somexinG, thenx∈F. (ii) Any point inFis the limit of a sequence{xn}n1withxn∈Fnfor eachn1.

In fact, both topologies coincide:

LEMMA 2. – Let {F_n}n1 be a sequence of closed subsets inG. Then we have geometric convergencelim_n→+∞F_n=Fif, and only if,F_nconverges toF in the Chabauty topology.

Proof. –For the sake of completeness, we recall the proof of this probably well-known lemma.

2.1.2. Geometric convergence implies Chabauty convergence. Let us assume that we have geometric convergence:limn→+∞Fn =F. LetC be a compact subset in Gand letΩbe an open, relatively compact, neighborhood of1GinG.

Let us first prove that here is an indexM 1 such that F_n∩C⊂Ω·F for any nM. If not, there exist an increasing mapϕ:N→Nand a sequence{x_ϕ(n)}n1 such thatx_ϕ(n)∈ (F_ϕ(n)∩C)\Ω·F for eachn1. By compactness ofCand up to extracting again, we may—

and shall—assume that{x_ϕ(n)}n1 converges, say tox, in C. But by condition (i), we have:

x∈F, so forn1we will have:x_ϕ(n)∈Ω.x, a contradiction.

It remains to prove that there is an indexM 1such thatF∩C⊂Ω·Fn fornM. Let W be an open, symmetric, neighborhood of1G inGsuch thatW ·W ⊂Ω. By compactness of F ∩C, we can write: F ∩C=_l

i=1W xⁱ for x¹, x², . . . , x^l inF ∩C. By condition (ii), for each i∈ {1; 2;. . .;l} we can write: xⁱ = lim_n→+∞xⁱ_n, with xⁱ_n ∈F_n for each n1.

For each i∈ {1; 2;. . .;l}, there is an indexN_i such thatxⁱ_n∈W xⁱ for anynN_i. We set:

M = max_1ilN_i. For anynM and anyi∈ {1; 2;. . .;l}we have:xⁱ∈W xⁱ_n. Letx∈F. Then there isi∈ {1; 2;. . .;l}such thatx∈W xⁱ, so that for anynM we have:x∈Ω·Fn.

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2.1.3. Chabauty convergence implies geometric convergence. Let us assume that Fn converges toF for the Chabauty topology.

Let ϕ:N→N be an increasing map and let {xϕ(n)}n1 be a sequence in G such that xϕ(n)∈Fϕ(n) for anyn1, converging to x∈G. We choose a compact neighborhoodC of xinG. There is an indexN₀1such thatx_ϕ(n)∈Cfor anynN₀. Let us choose{Ω_j}j1a decreasing sequence of compact symmetric neighborhoods of1_GinGsuch that

j1Ω_j={1}. By definition of Chabauty convergence, there existsN₁N₀such that(F, F_ϕ(n))∈P(C,Ω₁) for any nN₁. By induction, we find an increasing sequence of indices {Nj}_j1 such that (F, F_ϕ(n))∈P(C,Ωj) for any nNj. Let n∈N. There is a unique j 1 such that n∈[Nj;Nj+1], and we can write:x_ϕ(n)=ωn.yn withωn∈Ωj andyn∈F. SinceΩj shrinks to{1}asj→+∞, we have:limn→+∞ω⁻¹_n .x_ϕ(n)=x. SinceF is closed, this impliesx∈F and finally condition (i) for geometric convergence.

Letx∈Fand letCand{Ωj}j1be as in the previous paragraph. As before, we can find an increasing sequence of indices{Nj}j1such that(F, Fn)∈P(C,Ωj)for anynNj. Since x∈F∩C, for anyn∈[Nj;Nj+1]we can write:x=ωn.yn, withωn∈Ωj andyn∈Fn. Then sinceΩ_jshrinks to{1}asj→+∞, we have:lim_n→+∞y_n=x, which proves condition (ii) for geometric convergence. 2

2.2. Convergence of parahoric subgroups

In this subsection, we prove that after taking stabilizers the fundamental sequences of points in the buildingX (1.3.1) lead to convergent sequences of parahoric subgroups in the Chabauty topology.

2.2.1. The statement of the convergence result below uses the fact that the Euclidean buildings of the Levi factors of GF appear in the Bruhat–Tits buildingX of GF (1.2). Recall that an I-canonical sequence{xn}n1defines a facet in the Bruhat–Tits buildingXIof the Levi factor GI /F (1.3.1).

THEOREM 3. – LetIbe a proper subset ofS. Let{x_n}n1be anI-canonical sequence of points in the closed Weyl chamberQ^X ofX. Letσbe the facet in the Bruhat–Tits buildingX_I defined by{xn}n1. Letd={ds}s∈I be any family of real non-negative numbers defining a point ofσ. Then the sequence of parahoric subgroups{Kxn}n1converges inS(GF)to the closed subgroupDI,d.

SinceQ^X is a fundamental domain for theK_o-action onX [22, 4.4.3(2)], we readily deduce the following consequence.

COROLLARY 4. – Let{xn}n1 be a fundamental sequence in the Bruhat–Tits buildingX. Then the corresponding sequence of parahoric subgroups{Kxn}n1converges inS(GF)to a Ko-conjugate of some subgroupDI,d.

The rest of the subsection is devoted to proving Theorem 3. Letd={d_s}s∈I be a family of non-negative parameters whose associated point in the Weyl chamberA_I∩p_I(Q^X)lies inside the facetσ. By compactness of the Chabauty topology onS(G_F), it is enough to show thatD_I,d is the only cluster value of{K_x_n}n1. Let{K_x_ψ(n)}n1be a subsequence converging to some closed subgroupDofG_F.

2.2.2. Let us first prove some measure-theoretic results. Recall that ifY is a compact and metrizable topological space, then so is the weak-∗topology on the space of probability measures M¹(Y)by the Banach–Alaoglu–Bourbaki theorem. We apply this to the case whenY is a flag