Buildings and Groups III

(School of Mathematics and Statistics, Henan University, Kaifeng 475004, China)

Abstract: This is the third part of a pedagogical introduction to the theory of buildings of Jacques Tits. We describe the construction and properties of the Bruhat-Tits building of a reductive group over a local field.

Keywords: Reductive groups; Bruhat-Tits buildings

§1. Introduction

This is the third chapter of the paper: Buildings and groups. Chapter I is [62] and chapter II is [34].

In this part we give a description of some parts of Bruhat-Tits theory of buildings for reductive groups over non-archimedean discrete valued fields. The account here follows [24], [25],[28], [101], [64], [88], [70]. Our goal is to help students in China who are interested to learn this theory but are not familiar with the literature. We present the definitions of the structures that appear in or related to Bruhat-Tits buildings; we hope that with these in mind the students would find it easier to get an orientation when reading original papers. We are mainly interested in those parts of the theory which often appear in applications in representation theory - this will be illustrated here by a section on Hecke algebras. Examples for results in this chapter have already appeared in chapter II.

We assume that the readers are familiar with the theory of linear algebraic groups (see for example [8]; see [6], [93], [59] for a quick summary). We shall refer to [43] as SGA3.

It is a pleasure to thank Donald Cartwright, Bill Casselman, Chao Kuok Fai, Paul Gérardin,Robert Howlett, George Seligman, Takuro Shintani, Donald Taylor, Jacques Tits, Harm Voskuil for discussions on buildings over the years. I would like to thank Professor Shuxia Feng, Director of the School of Mathematics and Statistics of Henan University, Professor Xiaosen Han and the Editors of Chinese Quarterly Journal of Mathematics for their support of this project. We would like to thank the referee for a careful reading and useful suggestions.

Notation

For an algebraic group G defined over a fieldFwe writeG=G(F). We shall often identify theF-rational points of an algebraic group with the group.

§2. Root data

We give definitions of structures associated with roots. Such structures play an important role in the theory of reductive groups.

[1] We modify the definition of a root system given earlier for Lie algebras over algebraically closed fields (see [62] Chapter I§2.2).

By aroot systemwe mean a subset Φ of a real vector spaceVsatisfying the following conditions:

(RS0) Φ is finite, spansVand

(RS1) For eachα∈Φ, there is aα∨in the dual spaceV ?ofVsuch thatα∨(α)=2.

(RS2) Forα,β ∈Φ, definesα(β):=β ?α∨(β)α. Thensα(Φ)=Φ for allα∈Φ.

(RS3) Ifα∈Φ, thenα∨(Φ)?Z.

( [19] Ch. VI,§1; [94]§7.4.1 p.124.)

[2] Aroot datum(called ‘donnée radicielle’ in SGA3 XXI) is a quadruple Ψ=(X,Φ,X∨,Φ∨)where:XandX∨are free abelian groups of finite type, in duality by a pairingΦ and Φ∨are finite subsets ofXandX∨resp. and there is a bijectionof Φ onto Φ∨.In addition the following two axioms are imposed:

(RD1) For allα∈Φ we have

(RD2) For allα∈Φ we havesα(Φ)=Φ,sα∨(Φ∨)=Φ∨where

LetQ(resp.Q∨) be the subgroup ofX(resp.X∨) generated by Φ (resp. Φ∨). We say Ψ is adjoint ifX=Qand is simply connected ifX∨=Q∨(SGA3 6.2.6).

[3] LetVbe an Euclidean space and Φ?V ?be a root system. Choosing a basis Υ of Φ determines the positive roots Φ+. Aroot group datumin a groupGassociated to a root system Φ is (Z,(Uα,Mα)α∈Φ) which satisfies the following conditions:

(DR1)Zandare subgroups ofG, for allα∈Φ. WriteU?α:=Uα{1}.

(DR2)For allα,β ∈Φ the commutator subgroup(Uα,Uβ)is contained in the group generated by theUrα+sβforr,s∈N withrα+sβ ∈Φ.

(DR3) Ifα,2α∈Φ then we haveU2α ?Uα.

(DR4) For eachα∈Φ the setMα ∈ZG(a right coset) andU??α ?UαMαUα.

(DR5) For allα,β ∈Φ andn∈Mαwe havenUβn?1=Usα(β).

(DR6) IfU+(resp.U?) denotes the group generated by allUαwithα∈Φ+(resp.α∈Φ+),then we haveZU+∩U?={1}.

We say the above root group datum is of type Φ and we say it is generating ifGis generated byZandUα, forα∈Φ. (See [24] 6.1.1).

[4] A family?=(?α)α∈Φof maps?α:Uα →R∪{∞}(α∈Φ) is called avaluation on the root group datum(Z,(Uα,Mα)α∈Φ) on a groupGif the following conditions are satisfied:

(V0) For eachα∈Φ the image of?αcontains at least three elements.

(V1) For eachα∈Φ andr ∈R∪{∞}the setis a subgroup ofUαand we setUα,∞={1}.

(V2)For allα∈Φ andm∈Mαthe functionis constant.

(V3) Letα,β ∈Φ andr,s∈R; ifthen the commutator group (Uα,r,Uβ,s) is contained in the subgroup of G generated by theUnα+mβ,nr+msforn,m∈N withnα+mβ ∈Φ.

(V4) Ifα,2α∈Φ, then?2αis the restriction of 2?αtoU2α.

(V5) Letα∈Φ,u∈Uα, andIf, then we have(See [24] 6.2.1).

Lemma 2.1.Let ? be a valuation on the root group datum(Z,(Uα,Mα)α∈Φ)in a group G associated to a root systemΦin V ?. For z ∈Z, put z·?α(u)=?α(z?1uz). Then for each z ∈Z there exists a unique vector ν(z)=?+ν(z). Moreover this defines a group homomorphism

(See [24] Prop. (6.2.10), Proof of (i); [64] Prop 10.17).

§3. Reductive group

LetFbe a field and G be a connected reductive group overF( [8] IV§11 p.158). We shall often identify theF-rational points of an algebraic group with the group and we writeGfor G(F).

IfE/Fis a field extension we write GEfor G×F E.

Assume that G has a torus defined overFand split overF. Let S be a maximalF-split torus.X?(S) (resp.X?(S)) denote group of algebraic characters (resp. cocharacters) of S.There is a perfect pairing of abelian groups ( [8] III 8.11):

whereis the integer such that

3.1. Roots

The Lie algebra of G is denoted g. The group G operates on itself by inner automorphisms Intwherex,g ∈G. The differential of Intgat identity is denoted Adg.Thenis aF-morphism of algebraic groups which is called theadjoint representationof G.

Since S is a torus, AdSis diagonalizable, i.e.

where Φ=Φ(G,S) is a subset of the nontrivial charactersX?(S){1}and

Then ( [8] V 21.1; [94] p.115, thm 7.1.9, p.257, lem 15.3.7)

(1) Φ is a root system inX?(S)?R.

(2) The Weyl group generated by the reflections{sα;α∈Φ}is isomorphic to the group(NG(S)/ZG(S))(F) which acts via conjugation - every element of (NG(S)/ZG(S))(F) has a representative inNG(S)(F), forn∈NG(S)(F),χ∈X?(S), put (n·χ)(s)=χ(nsn?1).

Forα,β ∈Φ, let (α,β)={pα+qβ:p,q ∈N>0}∩Φ. A subset Ψ?Φ is calledclosed, if (α,β)?Ψ for allα,β ∈Ψ. If in addition Ψ lies in an open half-space ofX?(S)?R, then Ψ is calledpositively closed.

Let Φndorbe thereduced rootsystem consisting of non-divisible roots in Φ.

3.2. Root data

There is a maximal torus T of G defined overFcontaining S;T splits over a finite separable extension ofF.

The Weyl groupW(G,T)=(NG(T)/ZG(T))(F) acts on the character groupX?(T) and thus onV:=X?(T)?R. We can choose a positive definite symmetric bilinear form (,) onVwhich isW-invariant. Use this form to identifyVwith its dual spaceV ?.

The cocharacter groupX?(S) is a subgroup ofX?(T) viaLetX?(S)⊥:={χ∈X?(T):χ?φ=1,?φ∈X?(S)}. Then we have an exact sequence

and thus an exact sequence

This leads to the orthogonal direct sum

So (,) is a positive definite symmetric bilinear form onX?(S)?R and forχ∈X?(S) we can define

Moreover the orthogonal projectionis the map given by the restriction map

and Φ(G,S)=πΦ(G,T){0}.

Put Φ(G,S)∨:={α∨:α∈Φ(G,S)}.

Theorem 3.1.( [94] 15.3.8)(X?(S),Φ(G,S),X?(S),Φ(G,S)∨)is a root datum.

3.3. Root group data

(1) To each rootα∈Φ there is a unique closed connected unipotent subgroupUα ?Gnormalized byZGSand has Lie algebra gα+g2α; this is called theroot subgroupofα. ( [8],21.9, 14.5; [94]§15.4.)

(2) If Ψ?Φ is positively closed, then

(i)there exists a unique closed,connected,unipotentF-subgroupUΨofGwhich is normalized byZGSand has Lie algebraSetU?={1}.

(3) Suppose thatα,β ∈Φ are linear independent. Then (α,β) is positively closed and the commutator group (Uα,Uβ) is contained inU(α,β).

(4) For an order on Φ, the groupsUΦ+andUΦ?will also be denoted byU+andU?respectively. ThenU+U?∩NG(S)={1}.

(5) Forα∈Φ and eachu∈Uαwe have a one element set

Forthe elementmα(u) induces the reflectionsαinX?(S) and inX?(S). ( [64] 0.19; [14]§5; [101] 1.4; [24] 6.1.2 (2) p. 108.)

3.4. Pinning

[1] A reductiveF-group is said tosplitoverFif there is a maximal torus which is defined overFand splits overF(see [8] 18.7).

LetGbe a reductive group defined overFand split overFandTbe aF-torus ofGsplit overF. An épinglage (or pinning) ofGas defined in SGA3 XXIII 1.1 is defined as a pair(Υ,(eα)α∈Υ) with a choice Υ as a basis of the roots Φ=Φ(G,T) andSuch aneαcorresponds exactly with isomorphismxα:Ga→Uα, namely, givenxαtakeeα=dxα(1). Thus we can also define anépinglageofG(with respect toT) as (Υ,(xα:Ga→Uα)α∈Υ).

Forα∈Φ(G,T), we sayF-isomorphismsxα:Ga→Uαandx?α:Ga→U?αareassociatedif there is aF-group monomorphismsuch that for allu∈Ga(F)=F, the following conditions hold:

Let

AChevalley systemofG(with respect toT) is a setof isomorphisms such that:

a)xαandx?αare associated?α∈Φ.

b)?α,β ∈Φ,?ε=±such that?u∈Ga(F) we have:

You can see such conditions in [97].

[2] A reductiveF-group is calledquasi-splitoverFif it has a Borel subgroup ( [8]§11.1) defined overF. In this case the centralizer of a maximalF-split torus is a maximal torus (see [8] 20.5 and 20.6 (iii)).

Let G be a quasi-split reductiveF-group, S be a maximalF-split torus of G and Z be the centralizer of S in G. Then Z is a maximal torus of G defined overF.

Let ΥEbe a basis of the root system Φ(GE,ZE,E). LetE/Fbe a finite separable Galois extension splitting Z and Θ denotes the Galois group Gal(E/F). Then the action of Θ on conjugacy classes of maximal parabolic subgroups leads to a?-action of Θ on ΥEand the fact thatGis quasi-split implies that the image Υ of the restriction mapon ΥEis a basis of the root system Φ(G,S,F) and each fibre ofπis a single orbit in ΥEfor the?-action of Θ(we shall often not mention the?of this action). (See [99] end of§2.3; [14] 6.4 (2), 6.8).

Θ acts on the set of root subgroups{Ua:a∈Φ(GE,ZE,E)}andσUa=Uσaforσ ∈Θ. Let Θabe the stabilizer ofain Θ and letEa=EΘa. ThenUais defined overEa.

GEsplits overE. AChevalley-Steinberg systemofGis a Chevalley systema∈Φ(GE,ZE,E)) such that:

(i) ifa∈Φ(GE,ZE,E) anda|S ∈Φ(G,S,F) is a non-divisible root, thenxσa=σ?xa ?σ?1for allσ ∈Θ.

(ii) ifa∈Φ(GE,ZE,E) anda|S ∈Φ(G,S,F) is a divisible root and ifsuch that, then, for allσ ∈Θ, there exists ansuch that the following condition holds for allu∈E:

Compare: [95], [55], [56].

Proposition 3.1.( [64] Prop. 4.4) A quasi-split group has a Chevalley-Steinberg system.

§4. Apartments

4.1. Affine space

SupposeEis a set,Vis a vector space over a field and the map

defines an action of the additive group ofVonEsuch that there exists an elementa∈EwithV+a=E. (Note the symbol ‘+’ denotes the action and not the addition of vectors inV.) Then we sayEis an affine space underVand an element t∈Vis a translation onE(Bourbaki,Algebra, Chap II§9.1.).

SupposeE(resp.is an affine space underV(resp.). Then a mapis called an affine map if there is a linear mapsuch that

for allx∈Eand t∈V(Bourbaki, Algebra, Chap II§9.4). We callvthevector partofu. The group of all affine automorphisms ofEis denoted by Aff(E).

If we fix an elemento ∈E,then any elementy ∈Ecan be written as(y?o)+owith(y?o)∈Vand thus we can write an affine mapuwith vector partvas

4.2. Affine apartment

Fa nonarchimedean locally compact field,the ring of integers inF,πa fixed prime element inthe residue field,ω:F×→Z the discrete valuation normalized byv(π)=1.

LetGbe a connected reductive group overF.

Fix a maximalFsplit torusSin G.

is the perfect pairing of abelian groups defined by evaluation ( [8] III 8.11).

WriteV1=X?(S)?ZR. IdentifyandX?(S)?ZR. Extendto

We obtain a unique homomorphism

such that for anyF-algebraic characterχofZG(S) we have

for anyg ∈ZG(S). ( [64] Chap I, Lem 1.1).

LetCdenote the connected center ofG. LetX?(C)denote the group of algebraic cocharacters ofC. LetV0:={v ∈V1:α(v)=0,?α∈Φ}. ThenV0=X?(C)?R. PutV=V1/V0. Then

Let Φ=Φ(G,S,F) be the root system inV ?ofGwith respect toS( [8] V 21.1).

Eachα∈Φ defines a reflectionsαonV. The Weyl group of Φ is the group generated by{sα:α∈Φ}. Hence there is a group homomorphismj:W →GL(V).Wis isomorphic toNG(S)/ZG(S) ( [8] 21.1).

4.3. Affine extension

Denote the kernel ofν1byZb. As the ground fieldFis locally compactZbis the maximal compact subgroup ofZG(S) ( [64] Chap I, Prop 1.2). Then Λ=ZG(S)/Zbis a free abelian group with rank equals to dimV1. Letνbe the compositionThis map induces homomorphismν:Λ→V.

The extended Weyl group is defined to beThere is a group homomorphismwhich makes the diagram

commutative ( [64] Chap I, Prop 1.6).

(1) The above commutative diagram says that there is an affine spaceAunderV, (thusso that there exists a group homomorphismν:NG(S)→Aff(A)extendingν:ZG(S)→V.

(2)Up to a unique isomorphism,Ais unique([64]Chap I,Prop 1.8;[101]§1.2). All possible other such extensions are given bywherex0∈Ais a fixed but arbitrary point.

(3) Ifw ∈Wis the image ofn∈NG(S), thenis equal to the vector part of the affine transformationThe projection ofν(mα(u)) toWis the reflectionsαand soν(mα(u)) is an affine reflection whose vector part issα( [101] 1.4).

(4) The translation part of the affine transformationis given byfor some real number tα(u), i.e.

4.4. Affine roots

We may viewmα(u) as the reflection with respect to the affine hyperplane{x∈A:α(x)+tα(u)=0}( [101] 1.4). Put

This is a discrete subset in R and Γ?α=?Γα( [24] 6.2.16). In the case of Chevalley groups we have seen in chapter II that Γα=Z. The affine functions (α,t)=α(·)+t on A forα∈Φredand t∈Γαare calledaffine roots.Write Φafffor the set of affine roots. We identify an elementα∈Φ with the affine function (α,0); then Φ?Φaff. The action ofWon Φ extends to an action ofon Φaff. Explicitly, ifis the composition of the translation byλ∈Λ withw ∈W, then the action ofon the affine root (α,tα(u)) withu∈Uαis

wherevis the valuation ofF. Check thatThe map Φaff→is surjective and equivariant with respect to the projection map

The setsa?1(0) fora∈Φaffare called thewallsofA. The connected components of the complements inAof the union of all walls are calledalcoves(or chambers). Two points x and y inAare called equivalent if each affine root is either positive or zero or negative at both points;the corresponding equivalence classes are calledfacets. The chambers are the open facets.

A pointxofAis called anspecial pointif for any wallLofAthere exists a wallsuch thatandis a translation ofL. Any special point is an extremal point of the closure of a chamber. For any chamber there exists at least one special point in its closure. ( [19], chap V,§3, no 10, cor of prop 11)

A pointxofAis said to behyperspecialif:

(1) there exists an unramified Galois extensionF1/FandGsplits overF1.

(2) there exists a maximalF1-split torus ofGdefined overFand containingS.

(3) letA1=A(G,S1,F1) be the apartment ofS1and let Φaff1=ΦaffG,S1,F1) be the corresponding affine root system, andxis special for Φaff1( [101]§1.10.2). Hyperspecial points does not always exist.

We choose once and for all a special vertexx0inAand a chambersuch thatWe use the pointx0to identify the affine spaceAwith the real vector space

We callAtheapartment coming from the torus S.

Φ is a root system inV ?, by ( [19] VI§1.1 Prop3)V ?has aW-invariant scalar product which is uniquely determined on every irreducible component ofV ?up to a scalar factor ( [19]VI§1.3 Prop 7). Using the canonical pairing:V ×V ?→R we obtain aW-invariant scalar product onVwhich now use to define a metricdon the apartmentA.

The finite Weyl groupWof the root system Φ is generated by the set{sα:α∈Υ}of reflections with respect to the root hyperplanes of the simple roots.Whas a length functiondefined by counting the number of elements in a word. We extendtoin such a way that the length ofis the cardinality of( [65] 1).

For any affine root (α,t) we have inthe reflection at the affine hyperplaneα(·)=?t given bys(α,t)= image inWofmα(u) whereu∈Uαsuch that t=tα(u). The affine Weyl group is defined as the subgroup

Choose of Υ allows to define, forα,β ∈Φ,α≤βifβ ?αis a sum of simple roots with non-negative integer coefficients. Let Φminbe the set ofα∈Φ such thatαis minimal for≤. Put

Then (1) (Waff,Saff) is a Coxeter system ( [5] Satz 2.2.16) and

§5. Building of a reductive group

5.1. Quasi-split groups

We shall deal with a special case of quasi-split groups.

LetFbe a complete discrete valuation field which is strictly Henselian,ω:F×→Z is the valuation onF, and O=OFthe ring of integers inF.

Let G be a quasi-split reductiveF-group, S be a maximalF-split torus of G and Z be the centralizer of S in G.

LetE/Fbe a splitting Z and Θ denotes the Galois group Gal(E/F). We have root systems Φ=Φ(G,S,F)and Φ(GE,ZE,E). Fix a Chevalley-Steinberg systemonG.

(2) ForR, put

(3) Put

( [64] 4.8.)

The case 2α∈Φ: we havesuch thatWe construct group isomorphisms(SU3parametrization)

Then

(1) Define?α:Uα →R∪{∞}bywhenxα(a,b)=u. Define?2α:U2α →R∪{∞}by?2α(u):=ω(b) whenxα(0,b)=u.

(2)ForR, put

( [64] 4.14.)

Just as in the case of Chevalley groups where we can construct a group scheme over Z whose generic fibre is the given Chevalley group, we can in the situation here, for a bounded subset ?of the apartment construct-group schemewhose generic fibre isG. This is done in 4 steps:

1. Lift the maximal torus Z to a-group schemewhose generic fibre isZ. This is done by using Neron models in ( [64] Prop 3.2).

3. Construct a ration group law in the big cell

5.2. Filtration on root groups

Fa nonarchimedean locally compact field,the ring of integers inF,πa fixed prime element inbe the residue field,ω:F×→Z the discrete valuation normalized byω(π)=1.

Let G be a connected reductive overF. For an extensionE/Fwe put GE=G×F E, and we writeGfor G(F),GEfor GE(E). So for a root subgroupUαis Uα(F).

Fis locally compact impliesκis finite and thus perfect, henceGis quasi-split over the strict HenselizationFshofF( [64] Prop 10.1).

Fix a maximalFsplit torusSin G. Denote the root system of G with respect to S by Φ.Write; letbe a maximalFsh-split torus ofsuch thatandbe the root system ofwith respect to. Moreovercan be chosen to be defined overFand containsS( [64] 10.12 (ii)). So we can assume thatis of the formTFshfor aF-torusT.

We choose abasisΥ of the root system Φ and fix a pointo ∈Awith the property that for allα∈Υ, there exists an elementwithν(mα(u))(o)=o.

Let ? be a non-empty bounded subset ofA. Then ? is a Gal(Fsh/F)-invariant subset ofSinceis quasi-split overFsh, up to a unique isomorphism there is a unique smooth affine-group schemewith generic fibre( [64] Thm 6.1). Moreover thisdescends ( [18]6.2), we have - up to a unique isomorphism there is a unique smooth affine-group schemewith generic fibreGsuch that( [64] Cor 10.10).

Forα∈Φ=Φ(G,S) (resp.denote the root subgroups ofG(resp.) byUα(resp.Ua).

(i) Forlet

PutWe make the usual conventions for the case that

(ii) Forα∈Φ, define?α:Uα →R∪{∞}by

(iii) Let Γα={?α(u):u∈Uα{1}}?R.

Thegives an exhaustive and separated discrete filtration onUαby subgroups ( [24]6.2.12 b).

Forα∈Φred, we also have

Forα∈Φ(G,S,F),Uαis the root subgroup, put

Lemma 5.1.( [64] Lem 10.20, 11.4)is a valuation on the root groupdatum(ZG(S),(Uα,Mα)α∈Φ)on the group G.

Take a nonempty bounded subset ? ofAandα∈Φ, choosesuch thata|S=α, putf?(α):=?sup{a(x):x∈?}.

DefineU?to be the subgroup ofGgenerated by allUα,f?(α)forα∈Φ. When ?={x}we writeUx; same applies to other notations; in particular forwe can write (Uα,)x(which we will shortened as

PutN?={n∈NG(S):ν(n)x=x,?x∈?}.

LetP?be the subgroup ofGgenerated byN?andU?.

For any decomposition Φ=Φ+∪Φ?into positive and negative roots letU±denote the subgroup ofGgenerated by allUαforα∈Φ±.Then ( [64] Chap IV Prop 12.5, 12.6; [24] 6.4.9)

1.n∈NG(S)?nU?n?1=Uν(n)?;

2.U?∩NG(S)?N?.

3.P?={g ∈G:gx=x,?x∈?}

4.U?∩Uα=Uα,f?(α).

5.U?=(U?∩U?)(U?∩U+)(U?∩NG(S)).

6. The bijection induced by the product map is independent of the choice of ordering of the factors on the left hand side.

7. For, we haveP?=∩x∈?Px.

8. Forwe have

5.3. Construction of the building

LetGbe a connected reductive over a nonarchimedean locally compact fieldF.

Recall that the real vector spaceA=(X?(S)/X?(C))?R is the apartment coming from a maximal split torusS. Introduce an equivalence relation～on the setG×Aby (g,x)～(h,y)if there is ann∈NG(S) such thatnx=yandg?1hn∈Ux. The set of equivalence classes is denoted by

( [24] 7.4.2) We sometimes writeX(G) forX,for the equivalence class of (g,x). The formula

defines an action ofGonXand that the map

is injective andNG(S) equivariant; allowing us to writegxfor the classThen

1.P?={g ∈G:gx=x,?x∈?}( [24] 7.4.4).

2. Forα∈Φredandu∈Uα{1}we have ( [24] 7.4.5)

3. For anyg ∈Gthere exists an∈NG(S) such thatgx=nxfor anyx∈A∩g?1A( [24]7.4.8).

We callXthe Bruhat-Tits building ofG. The subsets of X of the formgAwithg ∈Gare called apartments. Two points x and y inAare called equivalent if each affine root is either positive or zero or negative at both points; the corresponding equivalence classes are called

facets. A subsetis called a facet if it is of the formfor someg ∈Gand some facetF ?A. Open facets in apartments are calledchambers. Furthermore the following holds -

(B3) Any two points and even any two facets in X are contained in a common apartment( [24] 7.4.18).

(B4) Ifandare two apartments, there is an elementg ∈Gsuch thatandgfixespointwise; in addition,is a closed union of facets inand in. ( [24]7.4.8).

Remark 5.1.The building X defined above is sometimes called the semi-simple building and there is a notion of expanded building (see [25] 4.2.16) which is used in the theory of Iwahori Hecke algebra.

[32] Chapter 1 contains a quick summary on the building of a reductive group over a local field.

[12], [11], [88] contain some results on the algebraic topology of buildings.

§6. Compactification of buildings

LetGbe a connected semi-simple non-compact Lie group with finite center,Kbe a maximal compact subgroup ofG. The quotient spaceX=G/Khas the Riemannian structure of a non-compact symmetric homogeneous space (see [46]), and in some particular cases this has a complex structure and is a bounded symmetric domain, in this case we writeDforG/K. Let Γ be a discrete subgroup ofGof cofinite volume. Then the orbit space ΓXor ΓDis locally symmetric [9], [46], [73], [86], [110], [111], [112]. People are interested in explicit constructions of the compactificationsandToroidal compactifications was carried out by [1]; see also [63]. Compactification of symmetric spaces has a long history, see [10], [58].

In the following table we give a short summary.

For the rest of this section we consider a similar problem for buildings. We follow [64].See [75], [76], [77] for another method of compactification using Berkovich spaces.

6.1. Compactifying apartments

We take a root system Φ inV ?. (Note that we had takenVto be Euclidean and often identifyVwithV ?.) As in Chap 1§4.2, forx,y ∈V, we putx～yif and only if for allα∈Φ,the following condition is valid:α(x) andα(y) have the same sign or are both equal to zero. In this way Φ defines a Coxeter complex Σ inVsuch that its faces are the equivalence classes with respect to～.

A base Υ of Φ determines a positive set Φ+and then a chamber

This is a canonical bijectionbetween the set of chambers in Σ and the set of bases of Φ. There is also a bijection between the set of facesFcontained in the closureand the set of subsets Υ(F) of Υ(C) with Υ(F)={α∈Υ(C):α|F>0}. The inverse map is denoted asθ ?Υ

as basis to define the the topology on

For any chamberC ∈Σ the corner of a chamberCis defined to be

wheredenotes the subspace spanned byF. Take any subsetU ?V, for a faceFcontained indefine subset

We define a topology onV Cby choosingwhereFruns over the faces ofCandUopen subsets ofV.

Now we try to ‘realize’ a corner - suppose Υ(C)={α1,...,αn}. Define a mapby

Thenfis a homeomorphism ( [64]§2, Lem 2.4).

LetDeclareU ?VΣto be open ifU ∩V Cis open inV Cfor all chambersC ∈Σ. ThenVΣis compact Hausdorff andVis dense inVΣ( [64]§2, Prop 2.8).

Now letAbe an affine space underV, so that there exists a group homomorphismν:NG(S)→Aff(A)extendingν:ZG(S)→V(see§4.3). We putwhere(a,x)～(b,y),if there is a vectorv ∈Vsatisfyinga+v=bandy+v=x. And we take the product-quotient topology onThenis compact Hausdorff,Ais a dense, open subset ofand the actionνofNG(S) onAcan be extended uniquely to a continuous action ofNG(S) on

6.2. Metric

Letdbe the metric we have defined on the apartmentAusing theW-invariant scalar product onV(see§4.4).

On the apartment=gAwithg ∈G(K) the map

defines a metric on

There exists a unique metricd:X(G)×X(G)→R such that the restriction to any apartmentcoincides with the metricgiven above andG(K) acts by isometries ( [64] p.130).

6.3.

Onthere is an equivalence relation defined by: (g,x)～?(h,y), if there is an elementn∈Nwithy=ν(n)(x) andg?lhn∈Ux.

Letdenote the setequipped with the product-quotient topology,whereG=G(F) carries the topology coming fromFandthe topology defined ( [64] p.133).

Theorem 6.1.(1) The topological space

of G and the induce topology on X(P/Ru(P))coincides with the metric topology.

(3) The closure of X(P/Ru(P))in(Q/Ru(Q))where Q are K-parabolicsubgroups contained in P.

(4) The action of G on(G)induced by the map

is continuous and this action extends that of G on X(G).( [64] Cor. 14.16, Prop 14.17, Cor 14.30, Thm 14.31).

§7. Congruence subgroup

In this section we catch up on the materials which should have been discussed earlier when the preference was given to the presentation of the construction of buildings over the logical order.

Letbe a smooth group scheme overwith generic fiberX. Then for eachn≥0, there exists a smooth modelsuch thatMoreover,

(ii) The Lie algebrafor alln≥0.( [113]§2.8).

But if G is a reductive group overFwe would like to have a group schemeover(which depends on some parameters) such that the generic fibre ofis G and then we useto define congruence subgroups.

7.1. Models

LetX/Fbe a smooth separated scheme. A Néron model forXis a smooth separated schemewhich satisfies the Néron mapping property: the natural map

is a bijection, for any smooth schemeoverwhereis the generic fibre of

As a special case of the Néron mapping property, we see that the natural mapis a bijection, i.e., allF-points ofXextend to-points ofThus, from the perspective ofF-points, the Néron model behaves as if it were proper. This is not true forF-points ifF/Fis a ramified extension!

The definition given above of a Néron model for a smooth separated scheme overFis in( [18]§10.1) where this is referred as locallt of finite type Néron model, in contrast with ( [18]§1.2) where the Néron models are supposed to be of finite type. Chapter 10 of [18] deals with unipotent groups and tori.

For a torusdefined overFwe shall letbe the smooth model with connected generic fibre such thatis the maximal bounded subgroup ofT(Fsh); this model is of finite type over O. (See [35]).

For the rest of this subsection we consider an affine schemeXof finite type overF. By a model ofX, we mean a flat-schemeof the form SpecAsuch thatis a sub--algebra of finite type overwithA modelXis smooth ifis smooth. The following proposition helps us to understand ( [25]§1.7).

Proposition 7.1.Assume that X,Y are smooth affine schemes over F.

(1) Letbe smooth models of X such thatIn fact we have

(2) Letbe models of X and Y. Assume thatis smooth, and φ:X →Y is aF-morphism such that φ(X())?Y(Osh), then φ extends to a uniqueO-morphism

7.2. Smooth models of root subgroups

In subsection§5.2 we implicitely used models to construct filtrations on root subgroups and we have introducedUα,r,xwithαa root,r ∈R andxa point in the building.

Proposition 7.2.Let G/F be a connected reductive group. Let S be a maximal F-split torus of G. Assume that S ?F Fsh is a maximal Fsh-split torus of G?F Fsh. Fix a point x in the building of G. We replace F by Fsh to define Uα(Fsh)r,x for α a root and r ∈R.

(1) For any α∈Φ, any r ∈R, there exists an unique smooth-schemewhich is amodel of Uα such thatMoreover, we have

(i)is connected and its closed fibreis unipotent.

(ii) The congruence subgroupis equal to Uα(F)n+r,x for any integer n≥0.

(2)For any α∈Φsuch that2α∈Φ,and r,s∈Rsuch that2r ≥s,there exists an unique smoothmodel schemeof Uα such thatMoreover,

(i)is connected andis unipotent.(ii) The congruence subgroupis equal tofor anyinteger n≥0.(See [25]§4.3).

7.3. Filtrations on tori

For a torusTover henselian local fieldFdefineas follows: First putNext letr>0. IfTis an induced torus i.e.Tisthen put

whereωiis the valuation onFiextendingωonF. In general, choose an induced torusIcontainingT, put

and then check that this is independent of the choice ofI.

By anadmissible filtrationwe mean an assignment from tori over henselian local fields to sequence of groups:satisfying the following conditions:

F0.for alls≥r ≥0.

F2.ifTis an induced torus.

F3. Ifis a morphism between tori overF, thenmapsintofor allr ≥0.

F4. Ifis unramified, then

Say an admissible filtration isschematicif

S1. For eachr ≥0, there is a unique smooth group schemeoversuch that

S3. For anyT/F, there is a strictly increasing sequence{ri}i≥1of non-negative real number such thatri →∞asi→∞, andonly whenr=rifor somei.

A schematic filtration is calledconnectedif the following condition is satisfied

CN. For eachr ≥0, the group schemeis connected.

A schematic filtration is calledcongruentif the following condition is satisfied

Definition 7.1.Let T be a torus over a henselian local field F.

For0

For r ≥1, we write r=n+r0with n∈Zand0≤r0<1. Then we putequal the n-thcongruence subgroup

Proposition 7.3.is an admissible filtration which is schematic, con-nected and congruent ( [113]).

7.4. Smooth models associated to concave functions

LetG/Fbe a connected reductive group. LetSbe a maximalF-split torus ofG,Φ=Φ(G,S)the relative system. LetTbe the centralizer ofSinG. WhenGis quasi-split,Tis a maximal torus ofG. IfFis strictly henselian, thenGis quasi-split.

We fix a schematic admissible filtration on

In order to haveα=0 we set up the following convention - for allr ≥0 and any pointxin the building ofG/F, we put (U0,r)(F)x=T(F)rand

We say thatis aconcave functionif for any non-empty finite familyof elements in Φ∪{0}such thatbelongs to Φ∪{0}we have

For a pointxin the building ofG/Fand a concaveletG(F)f,xbe the subgroup ofG(F) generated by (Uα,f(α))xfor allα∈Φ∪{0}. Ifα∈Φ is such thatwe letIfα∈Φ is such that 2α∈Φ, we let(see§7.2).

Theorem 7.1.Assume that F is strictly henselian. Fix a point x in the building X of G.Given a concave functionThen

(i) There is a unique smooth modelsuch that

(ii) For each α∈Φnd the schematic closure ofUα in

(iii) The multiplication morphism

is an open immersion, and induces an isomorphism on the special fibre if f(0)>0. Herethe two productscan be taken in any order.

(See [113] Thm 8.3; [88] prop I.2.2; [25] 4.6.4, 5.1.3).

Theorem 7.2.Assume that the schematic filtrationis congruent. Then

for all integer n≥0. In particular,

for all r ≥0, integer n≥0.(See [113] Cor. 8.8).

§8. Bounded subgroups

It is well known that all maximal compact subgroups of a connected semisimple Lie groupGare conjugate under inner automorphisms ( [46] VI§2) andGhas Cartan decomposition and Iwasawa decomposition ( [54] VI§3,4, VII§3; [53]). It is important to have the analogues of these properties forp-adic reductive groups.

8.1. Maximal bounded subgroups

Every bounded subgroup ofGis contained in a maximal bounded subgroup and every maximal bounded subgroup is the stabilizer of a point in the building ofG.

IfGis semisimple and simply connected, the maximal bounded subgroups ofGare precisely the stabilizers of the vertices of the building ofG; they formconjugacy classes,where1denote the relative ranks of the quasi-simple factors ofG. This is clearly different from the case of real Lie groups.

8.2. Parabolics

LetGbe a connected reductive group defined over a fieldk. Aparabolic subgroupofGis defined to be a closed subgroupPofGdefined overksuch thatG/Pis a projective variety. The minimal parabolic subgroups ofGare conjugate overk( [14] 5.9 Cor). Choosing a minimal parabolic subgroupP0corresponds to fixing a basis Υ of the root system ofG, then the parabolic subgroups containingP0corresponds to the subsets of Υ ( [14] 5.12 - 5.14). ( [93]§3.6, 5.9) contains a summary on parabolic subgroups; (see [14], [37], [3], [40], [78]) for some proofs; (see [54] VII§7; [107] I§2.2) for proofs in case of Lie groups.

The combinatorics of parabolic subgroups are important for the theory of Eisenstein series,induced representations and trace formula. The theory of parahoric subgroups are local field analogue of that of parabolic subgroups.

8.3. Decompositions

LetFbe a non-archimedean locally compact field, O its ring of integers, p its prime ideal andκits residue field withqelements.

Let G be a connected reductive group defined overF. Fix a minimal parabolic subgroup P of G. Let A be a maximal split torus contained in P, M the centralizer of A. Let Φ be the roots of G with respect to A and Φ+the positive roots determined by P.

Put

Kis a maximal bounded subgroup ofG. The groupGhasIwasawa decomposition

and hasCartan decomposition

where(See [24] 4.4.3; [31] p.392; [32] Lemm 1.4.5; [66]§2.6).

These decompositions are important for theory of spherical functions and induced representations of p-adic Lie groups (see for example [7], [21], [31], [32], [33], [66], [67], [68], [71]).

§9. Hecke algebra

To study the representations of a reductive group over local field (or of its Hecke algebra)we need information on the structure of the group, in particular, of its compact subgroups.Theory of building provides this information; thus we see buildings and Hecke algebras together sometimes - see examples in [61]. In this section we present different versions of Hecke algebras.

9.1. Hecke algebra as a matrix algebra

Hecke algebra was first introduced by Hecke [45] as an algebra of endomorphisms on finite dimensional spaces of modular forms (see for example [38]). Here we give a matrix formulation according to Shimura [90] and Tamagawa [98].

Forwe have a finite disjoint coset decomposition

( see [91], prop 3.1, lem 3.10).

Letdenote the Z-module of all formal sumswithcj ∈Z andαj ∈?. We define multiplication by:α,β ∈?,

Here

andis the number of (i,j) such that Γαiβj=Γξ. With this multiplicationH(Γ,?) is an associative ring with identity which we call the Hecke ring of (Γ,?).

Proposition 9.1.The following identities hold in H(Γ,?).

(1)

(2) For a prime p, T(1,p)2=T(1,p2)+pT(p,p)( [91] Thm 3.24).

9.2. Hecke algebra of p adic groups

We give a reformulation of the above Hecke algebra for p adic groups following Cartier [30];see also [79], [57], [2]. This is important for [51].

LetFbe an non-archimedean local field. LetGbe the group ofF-rational points of a connected reductive group overF. LetKbe a compact open subgroup ofG. Letdenote the complex vector space consisting of complex valued functionsfonGsatisfying the following conditions:

(1)for anyg ∈Gand

(2)fis zero outside of a union of a finite number ofKgK.

Fix a Haar measureμonG.H(G,K) is an associative C-algebra using the following multiplication

Choose a double coset decomposition

Choosexi,yjto give disjoint unions

Letμ(K)uαbe the characteristic function ofKgαK. Then the family{uα}is a basis of the vector spaceH(G,K) and

whereis the number of (i,j) such thatbelong toK.

PutH(G)=∪KH(G,K) whereKruns through a neighbourhood basis of 1 consisting of compact open subgroups ofG. CallH(G) the Hecke algebra ofG.

If we have a group homomorphismπ:G→GL(V)whereVis a C-vector space,we say(π,V)is a representation ofG. For two representationssay a linear mapis aG-homomorphism iffor allg ∈G.

ForH ?GputIfwithKgoing over all the compact open subgroups ofGthen we sayπis a smooth representation. Forthere exists a linear mapπ(f):V →Vsuch that

holds for anyv ∈Vandv?in the dual space ofV. This means thatVis now a-module.From this it follows that the category of smooth representations ofGis the same as the category of non-degenerate-modules. This is at least one reason why we are interested in Hecke algebras.

9.3. Iwahori subgroup and buildings

LetFbe a local field,the ring of integers ofF,κthe residue field withqelements andqis a power of a primep. Let G be a split connected reductive algebraic group overF. WriteG=G(L). Denote by C the connected component of the center of G. Choose a maximalF-split torus T of G.

The Bruhat-Tits buildingXofGwe constructed in section 5.3 will also be called a semi-simple building. The apartment determined by the torusTisA=X?(T/C)?ZR.

LetX?(G) be the group ofF-rational characters of G. In [25]§4.2.16 the extended Bruhat-Tits building is defined to beX1=X×HomR(X?(G)?R,R) andpr:X1→Xis the projection.

For each facetFofXthere exists a smooth-group scheme GFsuch that

(See [101]§3.4.1; [27]; [64], chap II,III,IV; also 7.)

whereRudenotes the unipotent radical ( [8] IV,§11.). ThenIFis the pro-ppSylow subgroup ofPF. By aparahoric subgroupofGwe mean a groupPFfor a facetFofX.

The origin of the apartmentAis taken to be a hyperspecial vertexx0. As the residue field is finite,K:={g ∈G:g·x0=x0}is a maximal compact subgroup ofG. Take a chamberCinAsuch thatx0is a vertex ofC. Writecallan Iwahori subgroup ofG,Ia pro-p Iwahori subgroup ofG. ThenIis a maximal pro-p subgroup ofK.

For an account on parahoric subgroups see [42], [70], [44]. There are two kinds of parahoric subgroups in use : connected and non-connected - (see [104]§1.24).

9.4. Iwahori-Hecke algebra

[1]For any commutative ringRwith identity. LetR[IG/I]denote the set of functionsf:G→Rwith the properties that the support offis compact and for allg ∈Gandthe following holds

With respect to the convolution product:

R[IG/I] is aR-algebra; call this apro-p Iwahori-Hecke algebraofG.

[2] Take the compact induction of the trivial representation ofItoput

By means of Frobenius reciprocity ( [103] Prop I.5.7 (ii)) we get

where charIis the characteristic function ofI. Ashence we an algebra isomorphism

( [105]; [70]§2; [106]).

[3] LetT0denote the maximal compact subgroup ofT,T1be the unique pro-p Sylow subgroup ofT0. Define

[4]is a freeR-module generated byand its ring structure is determined by the following relations:

( [105] theorem 1; [70]§4.8; [5].)

[5] We replaceIbyin the above discussion and we obtain

CalltheIwahori-Hecke algebraofG.is a freeR-module generated by{τw:w ∈W};and its ring structure is determined by the following relations:

braid relation: ifthenτvτw=τvw.

quadratic relation:such thatσsbelongs to a chosen set of generators ofWaff.

( [65] 2.1, 3.2 [70]§4.8.) This algebra was introduced by Iwahori (1926 - 2011) in [50].

9.5. Hecke algebra and Coxeter group

Given a Coxeter graph Π with a vertex setI, Coxeter matrixMand Coxeter groupWgenerated byS={wi:i∈I}.

We introduce variablesqi,i∈Iand impose the conditions:wi, wjare conjugate inW, thenqi=qj. LetAdenote the algebra generated byover the integers ring Z. Write(i) for the length ofwi ∈W.

We introduce an associativeAalgebraH(W,S) with identity generated byti,i∈Isubject to the following conditions:

braid relation: fori,j ∈I, ifthen settitj=tij.

quadratic relation: fori∈I, setCall(W,S) theHecke algebra of the Coxeter system(W,S).

If we set allqiequal toq, then(W,S) is a Z[q±1]-algebra. IfWis an affine Weyl group then(W,S) is called an affine Hecke algebra.

§10. Sheaves on buildings

There are two situations in which the word sheaf are used - one, when we take the building as a simplicial complex, in this case it is also called a coefficient system (see [81]§1; [88]§II.2);two, when the building is taken as a topological space (see [88] IV.1).

10.1. Coefficient systems

Say a locally compact topological groupGis a local profinite group if any open neighbourhood of identity contains an open subgroup ( [29]§1.1.).

LetRbe a commuative Noetherian ring,Ga profinite group. A smoothR-linear representation ofGon aR-moduleVis anG-actionπ:G×V →Vsuch that:

(1) For anyg ∈G,g:V →VisR-linear.

(2) For anyv ∈V, the stabilizer{g ∈G:g·v=v}is an open subgroup ofG.

Letdenote the category of smoothR-linear representations ofG.

LetLbe a local field, G be a connected reductive group splitting overL. PutG=G(L).

LetXbe the Bruhat-Tits building ofG. Acoefficient systemofR-modules onXiswhere we are given for any facetFofXaR-modulefor any facetssuch thatwe are givenR-linear mapssatisfying the following -

By a homomorphismof coefficient systems we meanwhereFruns over all facets ofX, and ifthen we have commutative diagram

Let Coeff(X) denote the category of coefficient systems ofR-modules onX.

Forg ∈G, defineGiven a homomorphismput(g?f)F=fgF,and get

AG-equivariant coefficient system issatisfies

Let CoeffG(X) denote the category ofG-equivariant coefficient systems ofR-modules onX.

Takeis the inclusion map. Hence

Forg ∈Glet

Thus we get a functor

For more information on coefficient systems and representations see [88], [48].

10.2. Sheaves

We give a construction of a sheaf on the buildingX.

We begin with the definition of the groups

For any concave functionf,letUfbe the subgroup ofGgenerated by allUα,f(α)forα∈Φredand allforα,2α∈Φ (cf. [24] 6.4.9).

Take ? to be a facetFofA, we then have the concave functionfF. Next we defineby

The concave functionis defined by

For any integere≥0, we use the concave functionhF+eto define

(see [88] p.21.)

Proposition 10.1.

1.is profinite.

2. Thefor e≥0(and F fixed) form a fundamental system of compact open neigh-bourhoods of1in G.

3.for any two facets in X such that

4.here x runs over all the vertices inand the product can take anyordering of the factors.

5. Assume x to be a special vertex. Then for any point z in the intervalwe have

We shall now build a sheaf onXout of a smooth representationVofG.

For any open subgroupU ?Gwe writeVUfor the maximal quotient ofVon which theU-action is trivial;VUis theU-coinvariants ofV. WritevmodUfor the image ofv ∈VinVU.

Fix an integere≥0. The projection mapinduces an isomorphism

For any two facetsinXsuch thatthe projection maps form commutative diagram

We use the representationVto construct a sheafon the buildingX. For any open subset??Xletbe the complex vector space of all mapssuch that:

(i)for anyz ∈?.

(ii) there is an open coveringfor anyz ∈?iandi∈I.

Then (i) for anyz ∈Xthe stalk

The cohomology with compact support ofis computed in [88].

For more information on coefficient systems and representations see [88], [72], [104].

§11. Conclusions

To summarize we can say the Bruhat-Tits theory consists of 3 parts. Part 1 consists of(a) the theory of reductive groups over local fields, their tori, root subgroups and parabolic subgroups, (b) Néron models for tori and root subgroups. Part 2 consists of (a) the construction of the affine buildingXof a reductive groupGover a local field together with aG-action, (b)to study the algebraic topology of the buildingX. Part 3 is to give applications of this theory.

Though we hope this presentation helps,this is still a very demanding chapter for an average graduate student in China. To understand this chapter you need some knowledge on local fields(see for example [89], [74], [39]) and linear algebraic groups (see for example [94], [93], [92], [18];Grothendieck et. al. SGA3 and[36],[8],[14],[15],[16],[17],[13],[95],[96],[85],[87],[100],[102]).There is no one book that gives complete proofs of what we need and it is unfortunate that no one in China will find it rewarding to write such a book.

Many of the textbooks on buildings(for example[20],[80],[108],[109])are mainly interested in the combinatorial structure of buildings without much reference to the detailed structures of reductive groups. Thus our priorities here are different. In applications references are often made to the original papers [21], [22], [23], [24], [25], [26], [27], [28], [101]. These are not easy papers for students - part of the reason is that the authors often make very general assumptions and the students are at a loss as to how to interpret them in specific situations. However for anyone who is serious about this theory, without the benefit of an expert nearby, there is no alternative to learning french and reading the originals until a textbook covering all the materials appears.