Compactification of the Bruhat-Tits Building

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen

1619

Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore

Tokyo

Erasmus Landvogt

A Compactification of the Bruhat-Tits Building

~ Springer

Author Erasmus Landvogt Mathematisches Institut Westf~ilische Wilhelms-Universitat EinsteinstraBe 62 D-48149 Mtinster, Germany E-mail: [email protected]

Cataloging-in-Publication Data applied for

Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Landvogt, Erasmus: A compactification of the Bruhat Tits building / Erasmus Landvogt. - Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 1995 (Lecture notes in mathematics ; 1619) ISBN 3-540-60427-8 NE: GT

Mathematics Subject Classification (1991): 20G25 ISBN 3-540-60427-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1996 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10479471 46/3142-543210 - Printed on acid-free paper

Preface

The aim of this book is the definition and study of the properties of the polyhedral compactification of the Bruhat-Tits building of a reductive group over a local field. For consistency of presentation, I have decided to present comprehensively the construction of the Bruhat-Tits building. I hope that this approach will make this technical work more accessible. I wish to thank M. Rapoport and U. Stuhler for their interest in this work and for their critical commentary. Thanks also go to D. Gonzalez Afonso for her energetic support in the translation of the German version and T. Chinburg for remarks on the translation. I owe special thanks to P. Schneider for detailed suggestions about how to improve an earlier version of the manuscript, and for many helpful comments. I am convinced that I would have been unable to submit this work without his help. Erasmus Landvogt

Miinster, 1995

Y

Contents

Introduction w 0 Definitions and notations

Chapter I The apartment w 1 The empty apartment w 2 The compactification of the empty apartments

14 14 21

Chapter II The oK-group schemes in the quasi-split case w 3 The group scheme T w 4 The group schemes Lt~,, w 5 A smooth model of the big cell in G w 6 The group schemes ~3n

31 32 34 50 61

Chapter III The building w 7 w 8 w 9

in the quasi-split case The full apartment The groups Un and Pn The definition of the building

67 67 76 87

Chapter IV The building w 10 w 11 w 12 w 13 w 14 w 15

and its compactification The 6tale descent The full apartment in the general case The groups Un and Pn in the general case The building in the general case The compactification of the building Example: SL,~

References Index

98 98 118 122 126 133 148

150 152

Vii

Introduction

The investigation of locally symmetric spaces can draw on different forms of compactifications. For example, one has the compactification of W.L. Bally and A. Borel (see [BaBo] and IS 2]) and the polyhedral compactification of A. Borel and J.P. Serre (see [BoSe 1] and [S 1]). One can view the Sruhat-Tits building of a reductive group over a local field as the p-adic analog of a symmetric space (see [Ti 2] 5). Thus a natural question is to find p-adic compactifications of such buildings which are analogs of various compactifications of symmetric spaces. A. Borel and J.P. Serre constructed in [BoSe 2] a compactification which differs fundamentally from any other in the classical context. To my knowledge, only P. G6rardin has published results concerning p-adic versions of polyhedral compactifications (see [Ge]). However, G6rardin considers only the case of a split reductive group G over a p-adic field K, and he focuses only on the set of special points. The object of this work is to define compactifications for an arbitrary reductive group G over a local field K. This will be done in two steps: 1. A compactification A of an apartment A of the Bruhat-Tits building X(G) will be constructed in complete analogy to the classical case (e.g. see [AMRT]). This part of the construction follows the ideas os G~rardin, though I will focus more closely on the connection of the topology of a corner and combinatorial properties of the Coxeter complex. 2. In the second step, the compactification A is used as a "local model" for the compactification of X(G). Starting from the equality X(G) = G(K) xA/,-,, where ,~ is a suitable equivalence relation (see [BT 1] w I will define X(G) as G(K) • A/ ~*, where ~* is a natural extension of ,,~. If we equip G(K) with the p-adic topology X(G) will carry the corresponding product-quotient topology. It turns out that the topological space X-(G) is Hausdorff, compact and contractible, that X(G) --- [,J X(P/R~(P)) (9 Per denotes the set of K-parabolic subgroups of G and R~,(P) the unipotent radical of P) and that the topology on X(P/R,,(P)) induced by X(G) coincides with the canonical building-topology. In this global part this work differs to a high degree from [Ge] as the latter used Chevalley lattices to define X(G), a method which can not be generalized in a suitable way. Furthermore I consider the whole Bruhat-Tits building and not only the subset of special points. The second purpose of this work is didactic. Many of the properties of X(G) and some of the decomposition theorems for G(K) can only be found in the original works by F. Bruhat and J. Tits, at least in the generality which is needed

here. Because of the axiomatic character of these results, I did not think that merely citing their proofs would make the constructions in this paper easier to understand. Therefore I decided to present a nearly complete account of the construction of X(G) and of the necessary theorems when G is a connected reductive K-group and K is complete with respect to a discrete valuation. I have refrained from presenting the axiomatic framework. To reduce the technical difficulties even further, some theorems will only be proved for the case. in which the residue class field k of K is finite. This makes it possible to consider only reductive groups, so that one does not need to consider quasi-reductive groups. (In constructing the compactification, the finiteness of k will be assumed anyway.) Apart from a few exceptions, the reader should be able to follow this text without the aid of the original works [BT 1,2]. Those properties that have not been proved (with the exception of the proof of the commutator-rule for quasi-split groups) are not essential for the understanding of this work. Even though at some points of the text ad-hoc proofs would have shorted the arguments, I decided to follow on the whole the strategy mapped out in [BT 1,2]. I hope that this approach has made the presentation more clear, and that it will facilitate comparisons with the original texts. To carry out the approach described above, it is necessary to copy some theorems and proofs word-for-word from [BT 1,2]. At some other points, the ideas of the proofs are the same, but because I have avoided such notions as 'valuated root data' and 'quasi-concave functions' the particular phrases and/or proofs themselves are different. To avoid making reading the quotations more difficult than reading the proofs, quotations are given only at the beginning of each paragraph. At the beginning of each chapter I give a detailed summary of the contents of the chapter.

w Definitions and notations In this p a r a g r a p h we will give the fundamental definitions and notations used throughout this work. Furthermore, we will present some properties of group schemes and reductive groups. As usual IN, 2Z, Q, ~ denote the sets of natural, integral, rational and real numbers. By IN>0 we will denote the set of positive natural numbers and by l~+ the set of positive reals. In this work all rings are assumed to be commutative and to have an identity element 1. Ring h o m o m o r p h i s m s should always preserve the identity elements and all modules are assumed to be unitary. All fields are commutative. If R is a ring, then a ring R ~ together with a ring homomorphism R --+ R ~ is called an R-algebra. If R is an integral domain, then we will denote the quotient field by Quot(R). For a ring R and an R-module M, we denote by M* the dual module. If S C_ M is an arbitrary subset, then we will write (S) for the submodule of M generated by S. We abbreviate the group of R-module isomorphisms M --+ M as G L ( M ) . Let G be an abstract group, let M C__ G and let X be a G-set. Then we let X c = {x G X : gx = x for all g E G} and let (M) be the subgroup of G generated by M. For A , B C G, we denote the c o m m u t a t o r subgroup < { a b a - I b - ' : a 9 A,b 9 B}> by (A,B). If H is a further group and ~ : G --+ A u t ( H ) ( = group of automorphisms of H ) is a group homomorphism, then we will write H>~ ~G for the semi-direct product of G and H with respect to T. If ~ is clear in the context, then we will abbreviate this as H >~ G. For a field K and a field extension L / K , let [L : K] be the degree of the field extension. If moreover L / K is Galois, then the Galois group will be denoted by G a l ( L / g ) . For 7 e G a l ( L / K ) and x 9 L, we will also write x ~ instead of 7(z). As usual a local field is a field which is complete with respect to a discrete valuation and locally compact. If K is a field which is complete with respect to the discrete valuation w : K • --+ JR, then we let OK = {x 9 K : ~ ( x ) >_ 0} and m g = {x 9 K : w(x) > 0}. Here we let K x = K \ { 0 } and w(0) = oo > 0. We will denote the residue class field by the same but small Latin letter as the local field. Finally, we will write K sh for the strict Henselization. By [CaFr] II 7 we know t h a t K is locally compact if and only if k is finite. In order to make the notations more clear we will denote schemes over a field (in particular varieties) by capital Latin letters and schemes over arbitrary rings by capital Gothic letters. A separated, reduced scheme of finite type over a field is called a variety. 3

Let R be a ring, let :E be a scheme over Spec(R) (also: 3~ an R-scheme or 3C//z) and let R ~ be an R-algebra. Then let 3~(R~) be the set of morphisms Spec(R') --4 :E over Spec(R). We will suppress the base ring in the following as well. For 3E• ~ = 3EXspec(n ) Spec(R~), we also write ~n'. Following the usual notations from the theory of varieties we will denote P(:~R', Ox~, ) by R'[:E], if :E is an affine R-scheme and OxR, denotes the structure sheaf of the scheme 3~n,. If ~ is a further R-scheme and f : :E --+ ~ a morphism of R-schemes, then we will write f* for the R-algebra homomorphism R[~] -~ R[iE] induced by f. If R is a local ring and :E an arbitrary R-scheme, then the special fibre of :E will be denoted by :E. 0.1. Let R be a complete, discrete valuation ring with residue class field k and let :E be a smooth R-scheme. Since, in particular, R is Henselian, we get that the canonicaJ map

X(R)

X(k)(-- X(k))

is surjective (see [BLR] 2.3 Prop. 5). 0.2, Let R be an integral domain, K = Quot(R) and let :~ be a flat, affine R-scheme. Then the map R[:~] --4 K | RfX] is injective. If ~ is a further affine R-scheme and if f, g : :E --4 ~ are two morphisms of R-schemes, which coincide on the generic fibre of :E, then f = g.

Proposition 0.3. (Extension principle, see [BT 2] 1.7) Let R be a complete, discrete valuation ring with separably closed residue class field k and let 3C,~ be two a]fine R-schemes of finite type, where X is smooth over R. If g : 3~• --4 ~ • is a moTThism of K-schemes with g(~(R)) C ~3(R) where K = Quot(R), then g can be extended uniquely to a morphism [7 : X -+ ~3. P r o o f . Since/E is smooth over R, in particular fiat, it follows/E(R) C_ X(K). Now the uniqueness follows immediately from (0.2). In order to prove the existence, it obviously suffices to show that R[:s = {f E K[X]: f(iE(R)) C_ R}. Here the inclusion R[X] C_ {f e K[X]: f ( X ( R ) ) C_ R} is clear. Thus let f E K[iE] with f(iE(R)) C R and suppose that f r R[X]. Let m be the maximal ideal of R and let 7r be a uniformizer. Further let k = R / m be the residue class field and Choose the minimal n E IN such that r n f E R[/E]. Obviously, we have n > 1 and therefore Irnf(X(R)) C m but 7r'~f ~ m. Hence the image of ~rnf in k[:E] does not vanish. Since 3~ is smooth, it follows from (0.1) that X(R) -~ X(k) is surjective. Hence the image of r n f vanishes on X(k), which has a dense image in ~ by [Bo] AG 13.3 (k is separably closed). Contradiction. [] 4

Supplement: By (0.3) we obtain the following uniqueness statement: Let R be a complete discrete valuation ring with separably closed residue class field, let g --- Quot(R), let X be an affine K-scheme and let M C_ X ( K ) be an arbitrary subset. If there exists a smooth, affine R-scheme Y- of finite type with generic fibre X and Y-(R) = M, then this is up to a unique isomorphism uniquely determined by this data. T h e d i s t r i b u t i o n m o d u l e o f a n afilne s c h e m e (see [Ja] 1.7): Let R be a ring, let Y- be an affine R-scheme and let x E Y-(R). If I , denotes the ideal in R[Y-] defining the closed immersion x : Spec(R) --4 Y-, then Dist(y-,x) := {p e R[Y-]* : there exists a number n e ~q with #(I~ +1) = 0} is called the R-module of distributions on Y- with support in x. Obviously, this construction is functorial, i.e. a morphism f : Y- --4 ~ of affine R-schemes Y-, ~ induces an R-module homomorphism Dist(y-,x)

-4 ~

Dist(~,f(x)) #of*

for all x E Y-(R) (see [Ja] 1.7.2).

Proposition 0.4. Let R be a discrete valuation ring, K = Quot(R) and let Y- be an irreducible, smooth afffine R-scheme with irreducible generic fibre. Furthermore, let x E Y.(R). Then R[Y-] = {f e K[Y-] : # ( f ) e R for all # e Dist(y-, x)}.

P r o o f . The proof can be copied word for word from [Ja] I 10.12, since the property of being a group scheme is not needed there. []

The W e i l r e s t r i c t i o n (see [BLR] 7.6): Let R be a ring and let R ~ be an R-algebra which is projective and of finite type as an R-module. If Y- is an affine R'-scheme, then the functor

{ R-schemes a ne }

~ E n s (= category of sets)

can be represented by an affine R-scheme R g ' ( ~ ) (see [BLR] 7.6 Theorem 4). So we obtain the Weil restriction functor affine RI-schemes}

~

{ R-schemes affine }

R' A simple exercise in universal algebra and [BLR] 7.6 Prop. 5 show that the Weil restriction has the following properties: 0 . 5 . If :E is an affine Rr-scheme, then:

(i)

If :E is of finite type over R', then TrR' (:E) is of finite type over R.

(ii) If :E is smooth over R', then Ten n' (:E) is smooth over R. (iii) If S is an arbitrary R-algebra and S' := S | s'

R', then

R'

7r s (9r x n, S ' ) = 7e n (X) • n S

Let K be a topological field. We can define on K-varieties next to the Zariski topology a finer (in general strictly finer) topology. We will call this the Kanalytic topology, following the terminology in the cases K = ~( and K = @. This topology is defined exactly in [We] App. III. Only the most important properties will be recalled here. 0.6. For each K-variety X, there exists a unique topology on X(K) such that the following conditions are valid: (i)

If X ~-4 A N (= N-dimensional affine space) is a closed immersion, then the topology on X(K) is induced by the product topology on K g via X(K) ~-+ A N ( K ) = g N.

(ii)

If (U~)ie~ is a covering of X by open, affine subvarieties, then the inclusion ui(g) -4 X(K) induces the K-analytic topology on ui(g).

(iii) If f : X --4 Y is a morphism of K-varieties, then the induced map f :X(K )--+ Y(K) is continuous with respect to the K-analytic topologies. (iv) The K-analytic topology on X(K) is finer (in general strictly finer) than the Zariski topology. (v)

If Y is a further K-variety, then the canonical bijection X(K) x Y(K) -+ (X XK Y)(K) is a homeomorphism, if X(K) x Y(K) is equipped with the product topology.

In addition, if K is a local field, then we have: (vi) X(K) is Iocally compact and (vii) X(K) is compact, if X is complete.

Let R be an integral domain. Then statements like "@ is an R-group scheme with generic fibre G ..." should always mean that the group law on G is induced by the group law on @ for the case that G is a Quot(R)-group scheme. If @ and f) are two R-group schemes, then a homomorphism of R-group schemes f : @ -4 5) will be called an R-group homomorphism. As usual we let ~ = Spec(2~[T]) and ~ m = Spec(~[T,-~]). If R is a ring, then we also write ~ a / R and ~ m / n for ~ a xm R and G,n • R, respectively. Furthermore, we let

GL,~/R

=

Spec(R[Tll,..

.

1

and

S L n / n = S p e c ( R [ T m , . . . , Tnn]/(1 - det(T~j))) In case t h a t the group schemes are defined over an integral domain the following three lemmas intend to make clear, in how far the group law, homomorphisms etc. can be extended from the generic fibre to the whole scheme. Let R be an integral domain and let K = Quot(R). Lemma

0.7.

Let @ and f) be two fiat, aft/fine R-group schemes. If f : @ -4 f) is a morphism of R-schemes which induces a K-group homomorphism {~g -4 f)g, then f is an R-group homomorphism. P r o o f . Let m : @ • @ -4 @ and m ' : 5) • Y) be the group laws and let e : Spec(R) -4 @ and e' : Spec(R) -4 Y) be the 1-sections. Then the assertion follows by applying (0.2) to the following diagrams: x ~

fx{

ml

f) x fj

I ~' f

Spec(R)

and

~-

Spec(n)

le'

el f

[] Lemma

0.8.

Let @ be a fiat, aJfine R-scheme and let m : @ x @ -4 @, s : @ -4 and e : Spec(R) -4 @ be morphisms of R-schemes which make @K a K-group scheme. Then @ is an R-group scheme. P r o o f . This follows immediately by applying (0.2) to the diagrams which will be used in the definition of a group scheme (see [DG] II w 1.1). []

L e m m a 0.9. Let 6 be an affine R-group scheme. If 5) is a fiat, closed subscheme of 6 such that 5)g iS a K-subgroup scheme of 6 K , then 5) is an R-subgroup scheme.

P r o o f . Let m : ~ • 6 --~ 6 be the group law and let s : 6 -+ 6 be the "inverse-morphism". By the assumption these induce morphisms 5)g X 5)g 5)K and 5)g --+ 5)g. Since 5) is flat, the canonical map R[5)] --+ K[5) x n K] is injective, hence 5)K has a dense image in 5). Therefore m and s induce morphisms 5) x 5) --+ 5) and 5) --~ 5). []

P r o p o s i t i o n 0.10. Let R be a complete discrete valuation ring with separably closed residue class field k and let K = Quot(R). Let 6 and 5) be two affine R-group schemes, suppose 6 has connected fibres and is smooth over R, and let f g : ~)g "~ 5 ) K be a K-group homomorphism. If there exists an open neighbourhood H of the 1-section in 6 and an extension of f K to an R-morphism f : ~ -+ 5), then we can extend f g uniquely to an R-group homomorphism 6 --~ 5).

P r o o f . Since 6 is flat, the uniqueness follows from (0.2). Let x E 6 C 6. Then it suffices to show that we have f ~ ( g ) E 0~,~ for all g e R[5)], since R[6] = A O~,~zE~

Since ~5 is connected, we obtain 6 = H(k). H ([So] AG 13.3 and 1.3). 6 is smooth and R is Henselian, hence the canonical map H(R) --+ H(k) is surjective (see (0.1)), i.e. there exists an element y in H(R) with x e y . H. Obviously, the morphism yH -~ 5), y . u ~-~ f ( y ) 9 f ( u ) extends the K-group homomorphism f g . Hence f~c(g) e 0~,~ for all g e R[5)]. Finally, it follows from (0.7) that f is even an R-group homomorphism. []

For a ring R and an affine R-group scheme 6 with 1-section e : Spec(R) -4 6, we will abbreviate Dist(~, e) as Dist(~). By [Ja]I 7.7 we know that Dist(6) is an associative (in general not commutative) R-algebra. If R ~ is an R-algebra which is projective and of finite type as an R-module, then an affine R'-group scheme 6 becomes an affine R-group scheme T ~ ' ( 6 ) by applying the Weil restriction. If R is a ring and ~b is a smooth R-scheme, then the identity component of 6 will be denoted by 6 ~ (see [SGA 3] Exp. VIB 3.1 if). By [SGA 3] Exp. VIB 3.10 we know that 6 ~ can be represented by a (smooth) open R-subgroup scheme of 6. This will be denoted by ~b~ too.

0.11. Let R be a discrete valuation ring and let O be a flat, separated R-group scheme of finite type with affine generic fibre. Then according to [SGA 3] Exp. XVIII App. III Prop. 2.1 (iii) we know that 0 is affine. 0.12. Let R be a discrete valuation ring with residue class field k and let K = Quot(R). Let 0 , ~ be two affine R-group schemes of finite type and let f : 0 --+ ~ be an R-group homomorphism. If 0, ~ are of the multiplicative type (in the sense of [SGA 3] Exp. IX 1), then by [SGA 3] Exp. IX 2.9 the following assertions are equivalent: (i)

f is a monomorphism;

(ii) the K-group homomorphism OK ~ J~K induced by f is a monomorphism; (iii) the k-group homomorphism Ok --+ ~k induced by f is a monomorphism. P r o p o s i t i o n 0.13. Let R be a complete discrete valuation ring with residue class field k and let 0, ~3 be two affine R-group schemes where 0 is of finite type and ~3 is smooth. If 0 is of the multiplicative type, then every k-group homomorphism f : 0 --+ ~ can be extended to an R-group homomorphism f : 0 ~ ~.

P r o o f . By [SGA 3] Exp. XI 4.2 the functor S ~-~ Homs-sroups(OS,~S)

can be represented by a smooth R-group scheme $)oma(O, 2)). Hence by (0.1) the canonical map 2)omR(O, $))(R) ~ ~omR(O, ~))(k) is surjective and therefore there exists an R-group homomorphism f : 0 ~ ~ extending f. [] 0.14. Let R be a ring and let M be a free R-module of finite type. Then the functor

{ R-schemes a oe } SpecR I

~, Ab (= category of abelian groups)

~ '~

R~|

can be represented by a smooth, anne R-group scheme 3,t of finite type (see [DG] ILl 2.1). The underlying scheme will be called the canonical R-scheme associated with M.

R o o t s y s t e m s a n d C o x e t e r c o m p l e x e s (see [Bou 1]): 0.15. Let X be a metric space with metric d. For x E X and e > 0, we let

Bs(x) = {Y e X : d(x,y) < d" If A is a real affine space, then we denote by Aft(A) the group of affine bijections A --+ A. For x, y E A, we denote by ]x, y[ and [x, y] the open and closed segment in A with end-points x and y, respectively. For an arbitrary subset U C_ A, we

-

denote by U the topological closure of U in A and by

+

the interior of U.

Let ~ be a root system in a finite-dimensional K-vector space V. A root a E is called divisible, if 89 E 9. 1 r ~}. I f a , b E 9, For an arbitrary subset q C_ <~, we let q r , a {a E if2 : ga then we let (a, b) = {pa + qb : p, q E IN>0} n 9. A subset q C_ ~ is called closed, if (a, b) C_ q for all a, b C ~P. If in addition lies in an open half-space of V, then ~P is called positively closed. Let k~ c_ ~ be a positively closed subset. A root a E q is called extremal, if the intersection of ~ + a with any system of generators of the convex cone generated by q is non-empty. An arbitrary total ordering of ~]~red will be called simply an ordering of ~. An ordering of q2 is called good, if every a E ~I/red is an extremal root for the set of all roots which are greater than a. (See [BT 1] 1.3.15 for the existence of good orderings.) In order to distinguish these (total) orderings from orderings of r with respect to a basis, we will call the latter ones orders on 9. Now let ~ be a root system in V*. Then 9 defines a Coxeter complex ~ in V such that its faces are the equivalence classes with respect to the following equivalence relation ,.~: For x, y E V, we have x ~ y if and only if for all a C ~, the following condition is valid: a(x) and a(y) have the same sign or are both equal to zero. According to [Bou 1] VI 1.5 there exists a canonical bijection between the set of chambers in ~ and the set of bases of 9. Let C E ~ be a chamber and let A(C) be the basis of 9 defined by it. Obviously, there exists a bijection A:

{ F E P,: F C_ C} F

---+ ,

gI(A(C)) = set of all subsets of A(C)).

~ A ( F ) = {a e A ( C ) : alF > 0}

If 0 C_ A, then A - I ( o ) will also be denoted by Fo. Since we will consider finite and infinite (affine) Coxeter complexes simultaneously, the word "vector" (in words like "vector face", "vector chamber" etc.) will indicate in paragraphs in which both types arise that we consider the finite Coxeter complex. 10

A l g e b r a i c a n d r e d u c t i v e g r o u p s (see [Bo], [BoTi] and [Hu]): Let K be a field and let G be a K-group. As usual we denote by

X*(G) the group of characters; X,(G) the group of 1-parameter subgroups; X*K(G) the group of K-rational characters; :D(G) the derived group;

C(G) the connected centre; R(G) the radical; Ru(G) the unipotent radical. If H c_ G is a closed subgroup, then we will denote by Afa(H) and Za(H) the normalizer and the centralizer of H in G, respectively. For closed subgroups H I , . . . , H,~ C_ G, we denote by (HI,. 9 Hn) the closed subgroup of G generated by H I , . . . , Hn in G. If A, B C_ G are closed subgroups, then we will abbreviate the commutator of A and B as (A, B). 0.16. A reductive K-group is called split over K (or K-split), if there is a maximal torus which is defined over K and splits over K (see [Bo] 18.7). A reductive K-group is called quasi-split over K (or K-quasi-split), if there is a Borel subgroup defined over K. In this case the centralizer of a maximal K-split torus is K-Levi subgroup of a Borel subgroup and therefore equals a maximal torus (see [Bo] 20.5 and 20.6 (iii)). Now let G be a reductive K-group, let let r = r S, K) be the root system there denoted by Kr If g is the adjoint representation and a C r then

S be a maximal K-split torus in G and of G with respect to S (see [Bo] V 21.1, Lie algebra of G, Ad : G --+ GL(g) the we let

ga -- {X E 9: (Ad s)(X) = a(s)X for all s e S}. 0.17. By use of [Bo] 21.9 and 14.5 (*) we obtain the following characterization of the root groups: (i)

If a E q~, then there exists a unique closed, connected, unipotent Ksubgroup Ua of G which is normalized by Za(S) and has Lie algebra ga+g2~ (if 2a ~ ~, then we let g2~ = 0).

(ii) If ~ C ~ is positively closed, then there exists a unique closed, connected, unipotent K-subgroup U~ of G which is normalized by Z c ( S ) and has Lie algebra ~ ~a. aE~

11

(iii) If k9 C (~ is positively closed, then the product morphism

1-I u~-~ u~

aE~2red

is an isomorphism of K-varieties for each ordering of q~. (iv) Let a, b E (I) and suppose that a and b are linear independent. Then (a, b) is positively closed and we have

(u~, ub) c_ u(a,b) Here Ur -- {1} (see (ii)). If 9 C_ ~ is positively closed, then we let G~ = (Up, U-v, Zc(S)). For an order on ~, the groups U~+ and Ur will also be denoted by U + and U - , respectively. If E denotes the Coxeter complex in X . ( S ) | ]R defined by 9 and if C E E is a chamber, then C defines an order on ~. The groups U~+ and U~- will also be denoted by U+ and Uc, respectively. 0.18. According to [BoTi] 5.15 we have the following decompositon of G(K): Let C, C' E ~ be two chambers. Then: (i)

G(K) = U+(K)AfG(S)(K)U+,(K).

(ii) For n, n' E A/'G(S), we have n = n' if and only if the double cosets U+nU +, and U+n'U +, are equal. (iii) U+U~ NAfc(S) = {1}.

L e m m a 0.19.

Let a E 9 and u E U~(K). Then AfG(S)(K) n U_~(K)uU_~(K) =: {m(u)} consits of one element. For u ~ 1, the element m(u) induces the reflection r~ in X . ( S ) and in X*(S) (see [BoTi] w P r o o f . There exists at most one n E AfG(S)(K) with u E U_a(K)nU_~(K) by (0.18). On the other hand, by applying (0.18) to the subgroup of G generated by U-a, Ua and Zc(S) the existence is clear, and the supplement also holds. [] 0.20. Let A C ~ be a basis of 9 and let 8 C_ A. Then (8) denotes the set of all roots in 9 which are linear combinations of roots in 8. If we let Se = ( N ker a) ~ aE8

then ZG(Se) normalizes the group U~+\(e), and Pe := ZG(Se)Ur is a parabolic K-subgroup of G (see [Bo] 21.11). Pe is called the standard parabolic

K-subgroup of type 8. 12

Proposition 0.21. We fix an order on q~. Then the product morphism U-

x

Zc(S)

x

U + -+ G

is an open immersion. P r o o f . Let P be the minimal standard parabolic K-subgroup associated with the given order on q~. If we let P - be the uniquely determined parabolic K subgroup in opposition to P with Z a ( S ) C_ P - , then the product morphism R~(P-)

x

P -+ G

is an open immersion which is obviously defined over K (see [Bo] 14.21). By [Bo] 21.11 and 14.18 we obtain R ~ ( P - ) = U - , P = Z G ( S ) U + and Z a ( S ) N U + = {1}, from which the assertion follows. [] This survey of properties of reductive groups and root systems can by no means be and of course should not be a complete description of the theories which will be needed. Still, we have fixed the notations by recalling some well known facts. If nothing else is said, we will assume in this work t h a t K is a field which is complete with respect to the discrete valuation w : K x --+ ]R and that G is a connected reductive K-group.

13

Chapter The

I

apartment

The Bruhat-Tits building (in the sequel abbreviated as "the building") of a reductive group G over a field K is a (poly-) simplicial complex together with a G(K)-action, which is build up by more simple (poly-) simplicial complexes, the apartments. These apartments are affine Euclidean spaces equipped with a (poly-) simplicial complex. Often we will make statements like "locally the building looks like an affine Euclidean space...". In this case "locally" does not have the strong topological meaning but intends to make clear that the building is a composition of more simple (poly-) simplicial complexes which "cover" it (see the axioms (B0)-(B2) in [Br 1] IV). In addition to this inner structure, we also have a group action on the apartments. For example, if A is such an apartment, then N ( K ) acts on A and preserves the (poly-) simplicial complex; here N is the normalizer of a suitable maximal K-split torus in G. For this chapter, we let G be a connected, reductive K-group, S a maximal K-split torus in G, N = N'G(S) its normalizer and Z = ZG(S) its centralizer in G. The root system of G with respect to S will be denoted by r = ~(G, S, K). An affine space A (associated with S) under a quotient V of V1 = X . ( S ) | N together with a N ( K ) - a c t i o n will be defined and examined in w We will call this pair, which is given canonically, the empty apartment (empty, because the (poly-) simplicial structur is missing). Starting with an abstract N-vector space V and a root system r C_ V*, we will construct a compactification in w which is well known in the classical context (e.g., see [AMRT]). This compactification V r~ will yield a compactification of A in a natural way, namely A = A x v V x. Note that we will construct the apartment actually for the derived group :D(G), i.e. we restrict ourselves to the semi-simple case (hence, in general, A is not an affine space under V1). To avoid unnecessary restrictions of the applicability, we let the group of K-rational points of the full reductive group act.

w The empty apartment For the definition of the empty apartment, we follow [Ti 3] 1.2 (see also [BT 2] 4.2.7, 4.2.16 and 5.1.22). By [Bo] III 8.11 we have the following perfect pairing of abelian groups:

<, > : x . ( s ) • x*(s) where is the integer such that X o A(t) = t <~'~>. Now let V1 = X . ( S ) | N. In the sequel we will identify Vt* and X * ( S ) @~ N and we will also write < , > : V1 x VI* --+ N for the canonical pairing which extends < , > (see above). 14

By [Bo] 21.1 the group N ( K ) / Z ( K ) is the Weyl group of the root system ~ and therefore acts on 171 and VI* in a natural way. Since K is in addition a discrete valuation field, we can define an affine action of N ( K ) on V1 which lifts the action of N ( K ) / Z ( K ) and where Z(K) acts by translations. Lemma

1.1.

There is a unique group homomorphism vl : Z(K) --4 V1 such that <.,(z),

x> =

for all z e Z(K) and X e X*K(Z ). In the following ker(ul) will be denoted by Zb(K). P r o o f . By [BoTi] 2.3 the homomorphism C(Z) x :D(Z) T-~t Z is an isogeny. T := C(Z) is a torus, since Z is reductive ([BoWi] 2.15 (d)). Let T~ C_ T be the maximal K-anisotropic subtorus. Then S x T~ , ~ t T is an isogeny ([Bo] 8.15) and we obtain an isogeny S x T~ x D(Z) T-~t Z. Thus we can identify X~-(Z) with a subgroup of finite index in X*(S). Hence the dual space of V1 is generated by X~:(Z), and the uniqueness is clear. Then the existence is also clear. [5 In the case t h a t K is locally compact, the following proposition gives a purely topological description of the group Zb(K).

Proposition 1.2. We consider the K-analytic topology on Z(K) and the induced one on Zb(K). Then: (i) Zb(K) is an open subset of Z(K). (ii) If K is locally compact, then Zb(K) is the maximal compact subgroup of

Z(K). P r o o f . For this proof, we fix a metric on 171 we induces the canonical A-vector space topology. Then ul is continuous and since w is discrete, there exists an r > 0 such that u~I(B~(O)) = zb(g). Hence Zb(K) is an open subset. Ad (ii): By using the same notations as in (1.1) we have the isogeny

15

Let Z~ = TaT)(Z). Since Z is a Levi subgroup of a minimal parabolic Ksubgroup ([]30] 20,5 and 20.6 (iii)), Z~ is a K-anisotropic, reductive group. Hence Z,~(K)is compact ([Pr]). Now S is isomorphic to (G,n/K) '~ and by (1.I) the group S ( K ) N Zb(K) is isomorphic to (o~7)~, hence also compact, since K is locally compact. Since Z~(K) _C Zb(K), we have (S(K) N Zb(K)). Za(K) = (S(K)Z~(K)) (q &(K). In order to show that Zb(K) is compact, it therefore suffices to show that S(K)Z=(K) has a finite index in Z(K). Let G = ker(S • Za ,,~,,z>tZ). Then

0 --+ G(K) --+ S(K) x Z~(K) --+ Z(K) --+ HI(K, G) is exact ([CaFr] X 2.1). Hence it is sufficient to show that H I ( K , G) is finite. First of all, G is a finite K-group of the multiplicative type. If the characteristic of K equals p > 0, then the order of G is prime to p by [Hu] 16.2. Without loss of generality we may pass to a finite Galois extension L/K. Thus we can assume that G is the product of some #n (= K-group of the n-th roots of unity). Without loss of generality we may finally assume that G = #n. By K u m m e r theory (e.g., see [CaFr]) we know that

HI(K, G) = K•215

'~

Let rr be a uniformizer of OK and assume w(K x) -=- 2~. Let N = 2 w ( n ) + l . I f a 9 (I + rcNoK) and f(T) = T'~-a 9 oK[T], then f'(T) = aT n-1 # 0 because of the condition on the characteristic. Moreover, co(f(1)) _> X > 2w(n) = co(f'(1)2), so by [La] II w Prop. 2 we get that a is a n-th power. Hence every higher 1-unit is a n-th-power, and it follows that K•215 n is finite. Finally, let H be a further compact subgroup of Z(K). Since Ul is continuous the image'of H in V1 = X.(S) | ~ is a compact subgroup and therefore trivial. Hence H is a subgroup of Zb(K) = ker(ui). [] Let n 9 N(K) and z 9 Zb(K). Since Z(K) is a normal subgroup of N(K), we obtain nzn -1 9 Z(K). For X 9 X~:(Z), we have x>

since X' :=

(z'

~

=

=

X(nZtTg--1)) E X~(Z). Hence Zb(K)

N(K). 16

X ' > -- o ,

is a normal subgroup

of

Therefore we obtain the group extension

0

~ Z(K)/Zb(K)

>N(K)/Zb(K)

>N(K)/Z(K) -----+1

(*)

While the group law on N(K)/Zb(K) and N(K)/Z(K) is given by "multiplication" the group law on Z(K)/Zb(K) is given by "addition". The following lemma will justify this notation. L a m i n a 1.3.

We have: (i) N(K)/Z(K) is the Weyl group of 9 and 5i) Z(K)/Zb(K) is a free abelian group of rank dim V1. P r o o f . (i) is clear by [Bo] 21.1. Vl(Z(K)) is a discrete, torsion-free subgroup of V1, since w is discrete. Hence Vl(Z(K)) is finitely generated, which follows by induction on dim V1 in the following way: For dim V1 = 1 the assertion is clear. So let dim V1 > 1. We can find a vector v in Vl(Z(K)) such that v is of minimal length in (v) A Vl(Z(K)). By induction hypothesis the image of Vl(Z(K)) under 7r : Y --+ V/(v) is finitely generated, hence Vl(Z(K)) is finitely generated, too. Therefore vI(Z(K)) is a finitely generated, free abelian group, and in order to prove (ii), it suffices to show that V1 is generated by gl(Z(K)). Let X 1 , . . . , X,~ be a basis of X*(S) and let )~1,..-, AN C X.(S) be the dual basis with respect to < , >. Each A~ induces a group homomorphism ( K • = ) GIn(K)

, S(K) - L Z(K) ,

which will be denoted by A{, too. For 1 _
-w(xj(zi)) = -w(xj(Ai(t))) = -w(t ~'j) = { -w(t) 0

if we let zi = ;~i(t). Thus we have proved (ii).

for i = j else

' []

Now let us consider the subspace V0 :-- {v E V1 :a(v) = 0

for all a e ~}

of V1. It is the group of the co-characters of the centre of G tensored with ~=t. If G is semi-simple, then V0 = 0. If G is reductive, then V0 is not necessarily trivial, but N(K)/Z(K) acts trivial on V0. In the following we will not consider V0 (even for reductive groups), i.e. every construction will be made actually for semi-simple groups, but we let the K-rational points of the full reductive group act. 17

Notations

1.4.

We let (i)

V = V(G,S,K)

= V1/Vo,

(ii)

z/: Z ( K ) -+ V be the composition Z ( K )

(iii) v W = N ( K ) / Z ( K ) ,

vl> VI - ~ V and

A = Z ( K ) / Z b ( K ) and W = N ( K ) / Z b ( K ) .

T h e image of the canonical group homomorphism ~'W ~ GL(V1) (1.3) acts trivial on V0. Hence the group homomorphism induces a group homomorphism j : v W -+ GL(V). Now let A be an arbitrary affine space under V. Since Aft(A) ~- V>~ G L ( V ) and because of the existence of the group homomorphisms L, : A --+ V and j : v W --+ G L ( V ) , we know in which way the first and the third group in (*) are acting on A. By putting them together we will construct a group homomorphism v : N ( K ) ~ Aft(A). The following l e m m a makes the connection clear between the action of vW on A and the canonical action of G L ( V ) on V. Lemma

1.5.

For ~ e A and w e W , we have L,(w. ~. w - i ) --- j ( ~ ) ( v ( ~ ) ) , if we let ~ = pr(w). P r o o f . Let n E N ( K ) and z E Z ( K ) be representatives of w and ~. Then it suffices to show t h a t ~ l ( n z n -1) = ~ ( v t (z)). By [BoTi] w we have


X > = - - 0 2 ( x ( n z n - 1 ) ) ~--- --w(~ -1 (X)(Z))

=

= <W(/21(Z)) , ~ >

for all X E X~,(Z). Proposition

[]

1,6.

There is a group homomorphism f : W --+ V>~ G L ( V ) which makes the diagram 0

--+

A

4 0

> V

>

W

t">

I

"W

~, 1

tJ

$f ---+ VxGL(V)

commutative. 18

P~> G L ( V )

-----+ 1

P r o o f . The isomorphism classes of group extensions of "W by A are in one-toone correspondence with the elements of Ext2(~W, A) ([Br 2] IV 3.12). By (1.5) and by the functoriM properties of Ext ([Br 2] IV Ex. 1) we obtain E x t 2 ( ' W , A) --+ Ext2('W, V) , since V = (A | ~:l~)/Vo and " W acts trivial on V0. Since V is a real vector space, we obtain Ext2(vW, V) = {1} ([Br 2] Ill 4.2), i.e. up to isomorphy there is only the trivial extension. By using the canonical homomorphism V ~ ~ W ~ V ~ G L ( V ) we obtain the required map. []

Lemma

1.7.

Let g : V ~ aL(V) -+ V~ GL(V) be a group h o m o m o r p h i s m which makes the diagram 0 ----4 V ~ V>~GL(V)

0 commutative.

-----4 V

4--4

V>4 G L ( V )

Then there is a vector v C V such that g((O, ~)) = (~(v) - v, ~)

for all (0, ~) e V >~G L ( V ) . Proof.

First of all, we can expand the diagram given above in the following

way: 0

> V

~

V>~GL(V)

0

> V

> VxGL(V)

> GL(V)

---+

1

> GL(V)

Let ~ E G L ( V ) and let v C V. Then (~(v), id) = g((~(v), id)) = g((0, ~). (v,id). (0, ~-1)) = (w,~(~)). (v, id)- (-~(~)-l(w),~(~o) -1) = (~(T)(v),id)

,

if g(0, ~o) = (w, ~(~)). Hence ~ = id. 19

Next we define a : G L ( V ) --* V by g(0, ~) = (a(~), ~o). A simple calculation shows t h a t a(~ o r = a(~) + ~(a(r for all ~, r E GL(V). By letting v = a ( 2 . id) we obtain for all ~ E GL(V): a(qoo(2.id))=a(~)+~(v)

and a ( ( 2 . i d ) oqo) = v + 2 - a ( ~ )

Hence a(qp) = ~(v) - v.

[]

P r o p o s i t i o n 1.8.

Up to a unique isomorphism there is a unique affine space A under V together with a group homomorphism v: N ( K ) ~ Aft(A) extending v: Z ( K ) -+ V. P r o o f . Let A be an arbitrary affine V-space. Since Aft(A) ~- V>~ GL(V), the existence is clear by (1.6). In order to prove the uniqueness, we will prove the following Claim: For two group homomorphisms f, f ' : W --+ V>~ GL(V) satisfying f[A = f'lh = u, there is a unique isomorphism g : V:4 GL(V) --+ V>~ GL(V) such that f ' = g o f . For all a E A we choose an element m~ E W such that p r ( m a ) = r~. By (1.5) we know that p r ( f ( m ~ ) ) = p r ( f ' ( m ~ ) ) = j(r~). If a" E V\{0} satisfies ra(a') = -a" for all a E A, then there are real numbers Aa E ~ such that f'(m~) = ( g1f ( m ~2) + A ~ . a ' , j ( r ~ ) ) , since f'(m~) 2 = / ' ( m ~ ) = f(m~). It follows from the definition that V is generated by { a - : a E A}. By (1.7) and by definition of the reflections r~ we have to find an element v E V such t h a t A~ = - a ( v ) for all a E A. Obviously, there is exactly one v E V satisfying these properties, hence the existence and uniqueness are clear. If we had considered V1 instead of V, the pair (A, u) would have been unique up to isomorphy, only.

D e f i n i t i o n 1.9. The affine space A(G, S, K ) := A s := A together with the group homomorphism v : N ( K ) ~ Aft(A) (1.8) will be called the empty apartment of G (with respect

to S). Let P be a parabolic K-subgroup of G. In the following we will examine the connection between the a p a r t m e n t of G and the a p a r t m e n t of P / R u ( P ) . Let A be a basis of ~, let 0 C A and let P = Po be the standard parabolic K - s u b g r o u p with respect to 0 (0.20). Then S is a maximal K-split torus of P, too. Let ~r: P --+ P / R u ( P ) be the canonical projection. 20

L e m m a 1.10.

The vector spaces V ( P / R~( P ), ~r(S), K) and Y / ( Fo) are canonically isomorphic. P r o o f . By (0.18) (iii) we know that S n R~,(P) = {1}. Hence 7r induces an isomorphism S ~ ~r(S), and we can identify V1 and X,(~r(S)) | JR. Then we obtain V(P/R~,(P),~r(S),K)

= V 1 / { v E V1 : a ( v ) = 0 for all a 6 0} = V(G,S,K)/{v =

e V : a ( v ) = 0 for all a e O}

v/ []

by definition of Fo (0.15).

Now Zp(S) = Z a ( S ) N P = Z N P and Z C_ P by [Bo] 20.5, hence Zp(S) = Z. By [Bo] 11.14 Corollary 2 we know that the projection ~r induces an isomorphism

z ~ z~/R~(v)(~(s)). P r o p o s i t i o n 1.11.

The apartment A ( P / R~, (P ), ~r(S ), K ) can be identified with A / (Fo } . If vWo is the subgroup o f ' W , generated by {ra : a e O} and if Wo denotes the pre-image of "Wo under the canonical projection W --4 v w , then y(A[p/a~(p)(Tr(S))) C Aff( A / ( Fo} ) is canonically isomorphic to the image of Wo in Aft(A). P r o o f . Because of the previous remark we cannot only identify X ~ ( Z ) and also Zb(K) and Zp/R,,(p)(Tr(S))b(K ). By (1.8) and (1.10) we get that A(P/R,~(P), 7r(S), K) = A(G, S, K)/(Fo) and that the action of Afp/R,(p)(Tr(S))(K) on A(P/Ru(P), 7r(S), K ) i s "induced" by the action of N ( K ) on A. Since P = Po, the assertion follows. []

X~(Zp/R.(p)(Tr(S)) ) but

w The compactification of the empty apartment In this paragraph we will present a central tool for the compactification of the Bruhat-Tits-building: the compactification of the apartments. In a slightly more specific situation than in [AMRT] and [Ge] we will compactify a finitedimensional R-vector space V together with a root system ~ C_ V*. Many topological properties are still clear in the simple topological space which will be define in the following. Let oo be a symbol not contained in ~ and let a C JR. Then ]a, oo] denotes the set {x E ~ : a < x} U {co}. 21

By ~ we denote the set ~ U {co} together with the topology defined by the following basis: {]a,b[, ]c, cc]: a,b,c E 13~} The following properties can be proved easily: * ~ is a second countable topological space, * ~ is Hausdorff and * ~:[>o := {x E ~ : x ~ O} U {co} is compact. 2.1. By using standard methods from analysis it is easy to show that the function r : ~_>0 • [0, 1] --+ ~>0 defined by

r(x,t)-

( 1 - t ) x f o r x E P ~ and l+tx co for t = O r(co, t ) = !_~ else ' is continuous for the topology on ~_>0 induced by ~:1,. Hence the topological space lR>0 is contractible.

Definition 2.2. Let n E ~>0. The continuous map r,~ : ~[>0 x [0,1] ~ IR>0 defined by r,~(zl,..., z~; t) = (r(xl, t ) , . . . , r(xn, t)) is called the n-fold product of the retraction r. Now let V be an arbitrary finite-dimensional R-vector space and let ~ C_ V* be a root system. The Coxeter complex in V defined by 9 (0.15) will be denoted

byE. Definition 2.3. (comp. [Ge]) (i) For any chamber C E E, we will call V C = 0

V/(F)

the corner defined

FEw.

FEE

by C. (ii) VW. := 0

V/(F} will be called the compactification of V defined by q~.

FEW.

Let C E E be a chamber and let A(C) be the basis of 9 with respect to C. The following lemma makes the connection clear between ~ and V c. Then the structure and topology of VW. can be described in terms of the topology on V c and the combinatorial properties of the Coxeter complex E. 22

L e m m a 2,4.

Let C E E be a chamber and suppo~e A ( C ) = { a l , . . . , a ~ } . Then:

(i)

The map f : V c --+

defined by for ai E A ( F ) "~

f ( x + ) =

ai(x)

else

J l
is bijeetive. (ii) Let F E E such that F C_ C and let U C_ V. We let

C~= [_J (U+F)+ F' EE

F'CP

Then {C F : F a face of C and U C Y an open subset} is a basis of a topology on V C for which f becomes a homeomorphiam.

Examples of open subsets (0 of type A1 x A1) This topology will also be studied in [AMRT], but it is characterized by sequences of points converging to the boundary (using the roots) there. P r o o f . Ad (i): Let F E E be a face of C. For all x E F und ai ~ A ( F ) , we have ai(x) 0 by construction. Hence f is well-defined. Now we define g : ~ " --+ V c b y g ( ~ k l , . . . , ~ n ) = z + ( F ) w h e r e F = A - ~ ( { a , E =

23

A(C) : Ai = e~}) and x E V such that ai(x) = Ai for all ai with Ai r co. Since the space ( V / ( F ) ) * C_ V* is generated by {al E A(C) : Ai r ec}, g is well-defined. Obviously, f and g are bijections, inverse to each other. Ad (ii): First of all, the Ft-linear map ~p : V ~ IR"~ defined by x ~ ( a l ( x ) , . . . , a,~(x)) is a topological Ft-vector space isomorphism. Let I 1 , . . . , I,~ C_ ~ be non-empty, open subsets satisfying the following condition: For all i, we have Ii =]xi, y~[ or Ii =]zi, oc] for suitable xi, Yl, zi E Ft. Next let F = A - l ( { a i : oo E /i}). Since the canonical Ft-vector space topology on Ft '~ coincides with the topology induced by ~ n , the subset U = ~-1(I1 • ... x In M Ft n) C_ V is open. Now m

m

+ F) = v ( u ) + v ( F ) = (I1 • =I1 •

•

• In n Ft n) +

'~=~(U),

since

-

~(F)=W1

x... xWn

{ {0} cc[

whereWi=,[0,

Therefore U + F =- U and by (i) it follows that f - l ( I t the continuity of f .

for o0 r else x...

(1)

x In) = C F, hence

Conversely, let F E E such that F C_ C and let U _C V be a non-empty, open subset. Then, for (A1,..., An) E f ( C f f ) , there exists a point x E U + F C_ V and a face F ' E E such that f ( x + ( F ' ) ) = (AI,... , An). Since U + F C_ V is open, there exist xi, yi E F t (1 < i < n) such that ~(Z)

e

]xl,yl[

x i , yi[ Now we let I ~ = { ]]x~,oo]

X ...

Aii=# oo co ffor orA

X

]Xn,Yn[ ~ ~(U+F) for 1 < i <

n.

We conclude by

(I):

I1 x . . . x I,~ C_ lcl.'~ is an open subset and (AI,... , An) E 11 x . . . x I,~ C f ( C f f ) . Hence f is an open map. [] For a chamber C E E, we always assume V C to be equipped with this topology.

Definition 2.5. The standard topology on V ~ is defined by the following property: U C V ~ is open, if U M V C C V c is open for all chambers C E E . Some examples of compactifications for the root systems A1, A1 x A1, A2, B2 and A1 • A1 • AI: 24

A~

A~ ~ A 4

A

.-

l

I

I

n

,

I

i . . . .

~-!---

J

_2'__1__

>/I

J I 25

Comparison between the polyhedral and the Borel-Serre compactification: P

f Our compactification

Borel-Serre compactification [BoSe 2]

In both sketches, the topology is characterized by lines converging to the boundary. In our topology, parallel (non identical) lines converge to different points in the boundary, while these lines converge to the same point in Borel-Serre topology. On the other hand, different lines through the origin lying in the same chamber converge to different points in Borel-Serre topology but to the same point in our topology. First of all, we will prove some fundamental properties of the corners. L e m m a 2.6. Let C E ~ be a chamber. Then: (i)

The topology on V C defined in (2.4) and the topology induced by V ~ coincide,

(ii)

V C is an open subset of V ~,

(iii) the closures of C in V C and V ~ coincide and (iv) the closure of C in V ~ is compact.

P r o o f . In order to prove (i) and (ii), it suffices to show that the canonical inclusion V c --+ V ~ is an open embedding. Let C1,C2 E ~ be two chambers, let F C }2 be a face of CI and let U C_ V be 26

an open subset. T h e n

c~ n v c~

=

c F - n v c~ U+F

=

(U

c vF' +~)n

vC~ =

F' EE

lJ C vF'_ +F F' E~

F'C~

F'C~ F' C C 2

E I

where Cu+-f is an open subset of V C1 and V C2 for F ' C F M C1 N C2- Hence

Cff N V C2 is an open subset of V C2. Ad (iii): Let 91(C) be the topological closure of C in V C. By (i) it suffices to show t h a t 91(C) is a closed subset of V z. According to (2.4) we know that

~(c)

- " = [3 c + ( ~ ) = f - ~(~_>0) FE~

Fc_U If C ~ is a further chamber, then we have to show that 91(C) N V C' is a closed subset of V c ' . In order to do this, we consider f(gg(C)M V C') C ~'~ where f is the m a p defined in (2.4) with respect to C ~. Let A = { a ~ , . . . , a~} be the basis of r with respect to C . For a i b > 0, we have f(9.1(C)M Vc')i = [0, oo], which is a closed subset. For a~lc < 0, we have f(gg(c) M vC')i =] - oc,0], which is a closed subset, too. Therefore 9.1(C) is dosed in an open covering of V z. By use of (iii), (2.4) (ii) and the fact that R>0 is compact the assertion (iv) follows. [] Now let " W be the Weyl group of the root system ~. Then " W acts in a natural way on V and Z. So we obtain an action of ~W on V Z "W x V z --+ V ~

( w , x + i F ) ) , - > w ~ + <~F> extending the action of " W on V. If we equip " W with the discrete topology, then this action becomes continuous.

L e m m a 2.7.

Let C E ~ be a chamber. Then 92(C) (=topological closure of C in V ~) is a fundamental domain for the ~W-action on V z. P r o o f . Let x + (F) E V z. By [Bou 1] V 3.3 the closure U is a fundamental domain for the " W - a c t i o n on V. Hence there is exactly one F E E such that C_ C and F E " W . F. Without loss of generality we may assume F C_ C. By [Bou 1] IV 1.8 we know t h a t " W E ( = stabilizer of F in ~W) is a Coxeter group, since vW is a Coxeter group. Now C +
Now we can formulate and prove the central proposition of this paragraph.

Proposition

2.8.

We have: (i)

V ~ is a second countable topological space and

(ii) V ~" is Hausdorff, compact und contractible. Let F E E. Then: (iii) The topology on V / ( F ) induced by V ~ and the canonical R-vector space topology coincide, V is dense in V ~ and (iv) the closure of V / ( F ) in Y ~ equals

0

V/(F'}.

F~ EE

-~' ~_g P r o o f . Since l=~ has a countable topological basis and G is finite, (i) is clear by (2.4) and by definition of V s. Ad (ii): Let (xi) be a sequence in V n converging at least to one x E V ~'. Since E is finite, we may assume without loss of generality by passing to a subsequence t h a t there is a c h a m b e r C E E with x~ E 91(C) for all i. Since 92(C) is closed ((2.6) (iii)) and Sausdorff, we conclude t h a t Y ~' is Sausdorff. By (2.6) (iv) we know t h a t the closure 92(C) is compact for all chambers C E E. Moreover, 92(C) is a f u n d a m e n t a l domain for the ~W-action (2.7) and E is finite, hence the compactness of V ~" follows. Let C E E be a c h a m b e r and let r e : 92(C) x [0, 1] -+ 91(C) be the map defined by r c ( x + ( F } , t ) = f c l ( r , ~ ( f c ( x + (F)),t)) where f c : Y c --+ ~'~ is the map f defined in (2.4) and rn : 1R.>0 x [0,1] -+ ~ > 0 is the n-fold product of the retraction r (2.2). T h e n we define R:V ~'x[0,1]

~V s (rc(w

-1 9

where w E v W is an element satisfying w -1 - x E 92(C). In order to prove t h a t this m a p is well-defined, we have to show: If x E 92(C) and w E " W such t h a t w . x = x, then we have for all t E [0, 1] r c ( x , t) = r c ( x , t) . m

For instance, let x E F1 + ( F 2 ) such t h a t F1,F2 E E and F1,F2 C_ C. According to the description in (2.4) we can also assume that F2 C_ F1. First of all, w E v W F 2 (= stabilizer of F2 in vW), since C is a fundamental domain (2.7). 28

Since vWF 2 is a Coxeter group, too ([Bou 1] IV 1.S), we obtain w E vWF 1 (--stabilizer of F1 in vW). So by definition we obtain re(x, t) E F1 for all t E]0, 1] and therefore w- re(x, t) = re(x, t), showing the contractibility of V ~. Let F be a face of C. Then the topology on V/(F) induced by V c is the canonical one. Moreover, V is dense in V C. So by definition of the topology on V ~ the assertion (iii) follows. Ad (iv): Let C E ~ be a chamber. Then

v Cn

U

V/(F')) =

U

F' E~

V/(F')

F' E F' C_-C

-f' ~_F

-~'~F which is the interior of V C \ ( V / ( F ) ) in V c by (2.4). Hence the assertion follows.

[]

By definition of the topology on V ~ we get t h a t the V-translation on V can be extended to a continuous action V x V~

~V~

(v, x + (F)) ~-+ x + v + (F) Now let V be the vector space defined in (1.4) and let 9 = ~(G, S,K) C V* be the root system of the reductive group G.

D e f i n i t i o n 2.9. T h e set A -- A x V V ~ (:= A x V ~ / . ~ , where (a,x) ..~ (b,y), if there is a vector v E V satisfying a + v = b and y + v = x) equipped with the product-quotient topology will be called the compactification of A with respect to r The following proposition will give an overview of the important properties of A.

P r o p o s i t i o n 2.10.

We have: (i)

-A is a second countable topological space,

(ii) -A is Hausdorff, compact and contractible, (iii) A = 0 A / ( F ) , the induced topology on A/(F) coincides with the canoniFEE

cal topology and (iv) A is a dense, open subset of-A. P r o o f i All statements follow from (2.8) by using the following lemma. 29

[]

Lemma 2.11. L e t o E A be an arbitrary point. T h e n the map a : A x v V r,

) Vz

(a,x+
(a-o)+x+(E)

is a h o m e o m o r p h i s m .

P r o o f . Let us consider the m a p fl : V r' --+ A x v V r, (x ~ (o, x)). By definition of A the m a p s a and fl are bijections, inverse to each other. The continuity of /3 is clear. Because of the continuity of the V-translation on A and on V ~ the continuity of a is clear as well. [] By (1.8) the group N ( K ) acts on A through ~. Finally, we want to extend this action to an action on A. In order to do this, let o E A and let a : A --+ V z be the homeomorphism considered in (2.11). The exact sequence 0

)V--+V>~'W

>vW--+l

and the V- and vW-action on V ~ can be put together to a continuous group homomorphism (V ~ ~W) x V ~ ) V~ ((v,w),x+),)v+wx+(wF>

Proposition 2.12. The action u of N ( K ) of g(g) on A .

on A can be extended uniquely to a continuous action

P r o o f . Let us consider the canonical homomorphism W --+ V>4 vW. The existence is clear by the previous remark. Since A is Hausdorff and A is dense in A (2.10), the uniqueness is also clear because of the continuity of the action.[] Note t h a t the construction was made without any knowledge of the (poly-) simplicial complex in A.

30

Chapter The

II

oK-group

schemes

in the

quasi-split

case

In this chapter we will construct a smooth, affine oK-group scheme of finite type with generic fibre G for all non-empty, bounded subsets fl C_ V (the ~tvector space defined in (1.4)) for the case that G is quasi-split. Only in chapter III we will generalize this construction to subsets fl C_ A (the affine space in (1.9)) and canonify it in this way. Up to this point, ~fi will depend on the choice of a Chevalley-Steinberg system. In chapter III and IV it turns out that ~a(og) C G(K) is essentially the stabilizer of f~ in the Bruhat-Tits building and that the special fibre ~fi = ~ • oK k gives us information about the local structure near ~ (in the topological sense) of the building.

Let K be a field which is complete with respect to a discrete valuation and which is strictly Henselian, let G be a connected, reductive quasi-split K-group, S a maximal K-split torus, T = ZG(S) and N = AYe(S). Hence T is a maximal torus (0.16). Let K/K be the (Galois) splitting field of T and let E = G a l ( / ~ / g ) be the Galois group. Next let r = r S, K) and ~ = O(G, T , / ~ ) be the root systems of G with respect to S and T, respectively, let /~ be a basis of ~ and let /k be a basis of 4~ consisting of restrictions of roots in /~ to S. Since G is quasi-split over K, there exists a Borel subgroup defined over K (0.16). So we may assume without loss of generality that ~ is ~-invariant. According to [BoTi] 6.4 (2) and 6.8 each fibre of the restriction map a ~-+ c~s is a single Galois orbit in/~. We will denote the roots in ~ by small Greek letters and the roots in r by small Latin letters. For the root groups with respect to a 9 r (resp. a 9 ~) we will write U~ (resp. ~ r ) (see (0.17)).

In w we will start with the construction of a smooth, affine oK-group scheme of finite type with generic fibre T which has a universal property closely related to the universal property of the N@ron model. It turns out that T is the identity component of the N@ron model. Then we will examine the structure of the groups U,~(K) and will give filtrations (Ua,~)ee~ in w For all ~ 9 ~ , a smooth, affine oK-group scheme ~la,e of finite type with generic fibre Ua and ll~,e(oK) = Ua,~ will be constructed. In w we will construct a smooth oK-model of "the" big cell in G together with an og-birational group law on it (see [BLR] 5). Finally, we put together these data in w and obtain for every non-empty, bounded subset C V, a smooth, affine oK-group scheme ~ of finite type with generic fibre G. Furthermore, the root system of ~I/R,~(~) will be determined in case that k is algebraically closed. 31

w

The group scheme T

In opposite to the explicit description in [BT 2] 4.4 we will use the N6ron-model to construct a smooth, affine oK-group scheme with generic fibre T.

L e m m a 3.1. (i)

Let T be a fiat, affine oK-group scheme with generic fibre T. Then ~ ( o g ) C_ T b ( g ) (see (1.2)).

(ii) Let T ~,T" be two tori defined over K and let f : T ~ -~ T " be a K-group homomorphism. Then f ( T ~ ( K ) ) C_ T~'(K). (iii) Let L / K be a finite Galois extension and let X E X ~ ( T ) . Then oJ(x(t)) = 0 for all t E Tb(K), i.e. Tb(K) C_ Tb(L). P r o o f . Ad (i): Obviously, it suffices to show that we have w o Xlz(oK) = 0 for all X E X ~ ( T ) . Since ~(OK) is a group, we only have to show that w o X[z(or) has a l o w e r bound. Let X E X ~ ( T ) and c E K x such that c X E og[~]. Then x(t) E c - l o g for all t E ~(og). Hence w o X[Z(oK) >- -w(c). Ad (ii): Let X E X~:(T") and t E T[(K). Then X o f E X~c(T') and therefore

0 = w((x o f)(t)) = w(x(f(t))). E X~ E X k ( T ) . Since K is complete, w o X = ~eaal(L/K) W o X~ for all a E G a l ( L / K ) (e.g., see [CaFr] II w Hence for t E Tb(K), we obtain w(x(t)) = [L: K] - 1 . w(~i(t)) = O. [] Ad (iii): First of all, ;~ :=

P r o p o s i t i o n 3.2. Up to a unique isomorphism there exists a unique smooth, a]fine oK-group scheme T of finite type with generic fibre T, ~ = T ~ and the following universal property: For every smooth, aJ:fine oK-group scheme ~ of finite type with generic fibre T and ~t = Tro, there is a unique oK-morphism f : ~ ~ 9 which extends the identity on T. P r o o f . Let fit(T) be the (lft-)N~ron model of T ([BLR] 10.1 Prop.6) and let ~7 = fit(T) ~ Hence ~7 is a (smooth) open subgroup scheme of 9l(T) (see w which is of finite type, since 91(T) is locally of finite type (see construction in [BLR]). By (0.11) the group scheme ~7 is affine, since 9l(T) ~ is separated. Now let T r be an arbitrary oK-group scheme with these properties. Because of the universal property of the N6ron model there is a unique o g - m o r p h i s m f : ~ --+ 92(T) which extends the identity on T. Since ~Y = ~yo this morphism induces a unique o g - m o r p h i s m ~7' -+ T (see (0.2)) which extends the identity on T. The uniqueness can be proved by using the usual universal arguments. [] 32

Definition 3.3. The oK-group scheme ~Yin (3.2) is called the canonical oK-group scheme asso-

ciated with T. R e m a r k 3.4. (i)

By (0.7) the morphism f in (3.2) is an oK-group homomorphism.

(ii) Obviously, the construction in (3.2) is functorial, i.e.: If T' is a further K-torus and ~r the canonical oK-group scheme associated with T ~, then every K-group homomorphism T' --+ T extends uniquely to an oK-group homomorphism S t ~ ~. (iii) In the case of wild ramification of f~/K it is possible to give examples where the inclusion T.(og) C_Tb(K) (see (3.1)0)) holds strictly ([BT 2] 4.4.13). Now let G be the canonical oK-group scheme associated with S. By (0.12) and by [SGA 3] Exp. IX 2.5 the extension ~ --+ ~ of the canonical morphism S ~ T is a closed immersion. Thus we will identify ~ with its image in ~ in the following. If ff is the canonical oK-group scheme associated with T, then we know by (3.4) (iii) that ~(oK) C_Tb(K). Later there will arise cases in which we have to decide whether an element lies in T.(OK) or not. Therefore we will examine the case where T is a torus of a simply connected group.

Proposition 3.5.

If G is simply connected, then ~(oK) = Tb(K). P r o o f . By [SGA 3] Exp. XXIV 3.13 the torus T is a product of tori of the form RL(~m/L) where L / K is a finite field extension. So we can restrict ourselves to the case T = TiL(~,,~/L). First of all, we will show that 7r176(~m/oc) has the universal property of the canonical oK-group scheme ~ associated with T. Let ~r be a smooth, affine oK-group scheme of finite type with generic fibre T and ~r = ~ o . Hence we have to show that there is exactly one extension ~' -'+ T~~ (Cm/On) of the identity on T. By the universal property of the Well restriction it suffices to show that there exists a unique extension

33

of

~(id) : T~ XoK K XK L --+ t~m/L where

~1: MorK(?~' •

K, 7~L(~m/L))-+ MorL(T' •

K •

L,~m/L)

is the canonical bijection. Here we denote by MorK( , ) and MorL( , ) the sets of K- and L-morphisms, respectively. Besides one should notice that the generic fibre of the Weil restriction is the Weil restriction of the generic fibre (0.5). • Now (~Im/oL)(OL) = oL = (~,VL)b(L) (1.2) and by (3.1) we know that the image of (~' X o~ OL)(OL) under 7l(id ) lies in (~3m/oL)(OL). By using (0.3) we OL obtain the existence and the uniqueness of the lifting, hence T = ~oK (G-~/OL)' Obviously, the norm map X L is an element of X*K(T ). Hence if t C Tb(K), then x it follows that NL(t) E Olc [] x and therefore that t E oL = T(OK).

w

T h e g r o u p s c h e m e s ~l~,e

In this paragraph we will follow [BT 2] 4.1 and 4.3. We will differ from this in so far as we will not deal with the dependence of the filtrations of the groups Ua(K) on the choice of the Chevalley-Steinberg system. A canonification will be given in w

The Galois group E acts on the set of all U~ (a C ~). If a E ~ and a E E, then a(U-~) = ~;~(~). Let E~ be the stabilizer of a in E and let L~ = / ; / ~ . Then ~)~ is defined over L~. First of all, we want to examine the types of the Dynkin diagrams of G which can arise. Let us consider a connected component of the Dynkin diagram of G. Then the classifcation theory (see [Ti 1] Table II) yields the following possible types of root systems: 1) (split case): X,~ where Xn is the type of a reduced, irreducible root system (hence An, Bn, C,~, Dry, E6, E7, Es,F4, G2). 34

2)

(quasi-split case): The following types remain:

-

2A2,~(n >_ 1) :

(relative root system: BCn);

- 2A2~+l(n > 1) :

(relative root system: Cn+l);

2Dn(n >_4)

:@-@-.

(relative root system: Bn-1);

2E 6

-

aD4'6D4

(relative root system: F4);

:

N

@~-I~1

(relative root system: G2);

Here every I denotes a simple root in ~ and two I's are connected if and only if the root groups, associated with the corresponding roots in/~, have a non-trivial commutator. The Galois group E acts on the full Dynkin diagram by diagram automorphisms. If we consider the stabilizer of the connected component chosen above, then the I's in the same Galois orbit have a border. (For the details of the notation see [BT 2] 4.2.23 and [Ti 1].) Note that multiple roots only arise in the relative root system of type 2A2n in the quasi-split case. If L is a field and V is a finite-dimensional L-vector space, then we will denote 35

by W L ( V ) the canonical L-scheme associated with V (0.14) and we equip it with the natural L-group scheme structure. The Lie algebra of WL(V) is canonically isomorphic to V. Any linear map t : V ~ V induces ~ L-group homomorphism t : W L ( V ) ~ W L ( V ) which induces itself the old map t on the level of Lie algebras. Let 0 be the Lie algebra of G, let ~ = 0 ~K /~ and let Ad : T --4 GL(o ) be the adjoint representation of T. For a 6 ~, we let 0~:{ve0| and ~ = 0~ |

for allt ET(La)}

/( (see (0.16)).

P r o p o s i t i o n 4.1. We have: (i)

There is a unique TL~-equivariant L~-group homomorphism exp~ : WL~ (0r --9 GL~ which induces the inclusion O~ --4 0 |

L~ on the associated Lie algebras.

(/i) If X e 0~\{0}, then u ~ e x p A u X ) is a n~-group homomo~phism ~a/L~ ~ GL~, which is TL~-equivariant for the TL~-action on ~a/L~ through a. By this assignment we obtain a bijection o] oa\{O } onto the set of all such La-group homomorphisms ~a/n~ -4 Gnu. (iii) There is a unique Lc,-linear map < , > : Oa |

~--a -+

L~

and a uniquely determined L~-group homomorphism r* : @m/Lr ''') TL~ such that expa (X) exp_~ (Y) Y X = e x p - ~ ( l + < X , Y > ) r * ( l + < X , Y >)expa(1 + < X , Y > ) for all X 6 gc, and Y 6 0-~. P r o o f . If GLr " is replaced by the L~-split, reductive group generated by Ua and U_~ in GL~, then the assertions follow from ISGA 3t Exp. XXII 1.1. By definition of U~ and U_~ the morphisms exp~ in (i) and 6~/L~ --4 GL~ in (ii) have to factor through this group so that we have proved the assertions. [] 36

Obviously, we can replace L~ by an arbitrary field extension and the last theorem still holds.

D e f i n i t i o n 4.2.

(i)

A system ( ~ ) ~ e s

of K - g r o u p isomorphisms ~

: ~//(

~ Ur will be called

a [(-dpinglage of G (with respect to T). (ii) A system ( ~ ) ~ s called a Steinberg

of L~-group isomorphisms ~ : ~a/L~ ~ Va will be dpinglage of G (with respect to T), if ~ ( ~ ) = a o ~ o a -1

for all a E /~ and all a E Z.

Obviously, the R-~pinglages ( ~ ) ~ E s

with ~ ( ~ ) = a o ~

o a -1 for all a E /k

and a E Z are exactly the R-~pinglages which arise by a base change from a Steinberg ~pinglage. Therefore we will identify these in the sequel. According to (4.1) the R-~pinglages in our sense exactly correspond to the R-~pinglages in the sense of [SGA 3] Exp. X X I I I 1.1. Now if ( ~ ) ~ e s is a K ~pinglage in our sense and (X~)~e h is the associated one in the sense of [SGA 3], then (~a) is a Steinberg ~pinglage if and only if Xa(~) = and a E ~..

a(X~) for all a E

~'alR --+ Un and ~_~ : ~alR --+ ~f-~ will be called associated, if there is a R - g r o u p monomorphism e~ : SL2/R --+ G In the following two R-isomorphisms ~

such t h a t for all u E ~ ( / ~ )

~(u)

= e~

:

= R , the following conditions hold:

and 5:_~(u) = e~

--U

.

Obviously, e= is determined uniquely by this condition.

Note t h a t if ~ is a L~- isomorphism, then x - a is a La-isomorphism, too, hence m s is a L~-rational element (4.I) (iii).

Now if a E ~ and a = a[s E r is divisible, then by the classification given above there exist j3,/31 E ~ such that j3+~31 = a and /3Is = ~']s = ~. Obviously, L~ = L~, is a quadratic (separable) extension of La.

37

D e f i n i t o n 4.3. (i) A system ( i s ) ~ e ~ of K-group isomorphisms i s : ~ / R -+ Us is called a fi[-ChevaUey system of G (with respect to T), if the following conditions hold: a)

i s and i _ ~ are associated for all a E (~ and

b)

for all a,/3 E ~, there exists an e E {• /(, the following condition holds: ir~(/3)(U ) :

such that for all u E ~I~(/() =

m s . i/3(~/t)'

m21

.

(ii) A/~-Chevalley system (ia)~E ~ of G is called a Chevalley-Steinberg system of G, if the following conditions hold: a)

If a E ~ and a[s E 9 is a non-divisible root, then i~(~) = ~ o i s o a -1 for all a E E (in particular, i~ is defined over La) and

b)

if a E ~ and his E '~ is a divisible root and i f / 3 , ~ E ~ and L~, = L~ are as above, then, for all a E E, there exists an e E {4-1} such that the following condition holds for all u E K: ~ ( i ~ ( ~ a - l ( u ) ) ) = i~(s)(u) (in particular, i s is defined over LZ). If a E G a l ( / ( / L s ) , then c = - 1 if and only if ~r induces the non-trivial automorphism on L B.

By (4.1) t h e / ( - C h e v a l l e y systems in our sense correspond to t h e / ( - C h e v a l l e y systems in the sense of [SGA 3] Exp. XXIII 6.1. If ( i ~ ) s @ is a / ( - C h e v a l l e y system in our sense and (X~)~Ei denotes the associated K-Chevalley system in the sense of [SGA 3], then ( i s ) is a Chevalley-Steinberg system if and only if for all a as in (4.3) (ii) a) and all a E E, we have X~(~) = a(Xs) and for all a as in (4.3) (ii) b) ,we have: For all a E E, there exists an ~ E {• such that Z~(~) = r and for a E Gal(/~/L~) we have r = - 1 if and only if a induces the non-trivial automorphism of L B. P r o p o s i t i o n 4.4.

There exists a Chevalley-Steinberg system of G. The element in gs associated with i s will be denoted by Xs in the following. Now we choose representatives a E /~ for all Galois orbits in zX and we choose L~-isomorphisms i ~ : I[la/L~ --40-c~ extending this system in sense of (4.2) (ii), hence we obtain a Steinberg ~pinglage ( i ~ ) ~ ? , of G. By [SGA 3] Exp. XXIII 6.2 we can extend this to a/~/-Chevalley system (i~)sE~. 38

L e m m a 4.5.

If a E E acts trivial on 7x, then we have a o 5c~, o a -1 = i s for all a E ~. P r o o f . There exists a root/3 E /~ and an element w E " W (= Weyl group of ~)) such that w./3 = a. Since " W is a Coxeter group, there exist 5 1 , . . . , 5s E /~ with w = r~: o ... o r ~ . By (4.2) (ii) and by the definition of m~, we obtain (r(m~,) = m~(~,) = rn~ for all i. Hence for a suitable ~ E {• we obtain o

o o-l(u)

=m~...m~

=

.a(S:~(cr-l(~.u))).

9 ? r t -~ -1

by definition of the Steinberg 6pinglage.

m ,

..m~ 1-~,(u)

)

for a l l u E / ~ []

L e m m a 4.6.

Assume ~ to have one of the possible types in the quasi-split case (seethe classification given above) and let a,/3 E ~) with a ~ fl and a + ~ E ~. Then [X~, X~] = • X~+~. P r o o f . By [SGA 3] Exp. XXIII 6.5 we have [X~, X~] = + p . X~+~ where p is the smallest positive integer such t h a t / 3 - p . a 9~ ~. Let us consider the a-string/3 + 2~ 9 a through/3. By [Bou 1] VI 1.3 Corollary of Proposition 9 the length of/3 + 2~. a is given by -n(/3, a) where ( n ( - , - ) ) denotes the Cartan matrix of ~. Hence the length is _ 1. On the other hand, in the quasi-split case the arising irreducible components are of type An, D,~ or E6 (see the classification). By having a glance on the table of the Cartan matrices (e.g., see [Bou 1] VI Appendix) one obtains the validity of the assertion. [] L e m m a 4.7.

Let a E ~ and let a E E with cr(a) = a. If there is a root 5 E fi~ such that 5 ~ a(5), 5 + a ( 5 ) ~ a a n d a - h E ~, t h e n a - 5 - a ( 5 ) E ~. P r o o f . First of all, a, 5, a - 5, a - a(5) E ~. Since 5 # a(5), only the quasi-split cases can arise. By using an argument on the length (as in (4.6)) one can show that a + 5, a + a(5) r <~. Thus by [Bou 1] VI 1.3 Prop. 9 (iii) we get n(a, 5), n(a, a(5)) > 0 ( ( n ( - , - ) ) denotes the Caftan matrix of ~ again). If 5 + a ( 5 ) E ~, then n(5+c~(5), a) --- n(8, a ) + n ( a ( 5 ) , a) > 0. Hence a - 5 - a ( 5 ) E + ( [ B o u 1] VI 1.3 Theorem 1). For 5 + ~r(5) ~ ~, we have n(5, a(5)) = 0, since obviously 5 - a(5) r ~. So n ( a - 5, a(5)) ---_n(a, a(5)) - n(5, a(5)) = n(a~ ~(5)) > 0 and therefore a - 5 - a(5) E 9 ([Sou 1] VI 1.3 Theorem 1 again). [] 39

P r o o f of proposition 4.4. In complete analogy to [St] 3.2 we use sign changes like X~ F-~ - X a (a E ~) to construct a Chevalley-Steinberg system out of the ChevaJley system. The Galois group E acts on the Dynkin diagram by diagram automorphisms. Hence we can choose representatives for every E-orbit. If we can prove the properties in (4.3) for roots which are linear combinations of these roots and for the subgroup of E which stabilizes a component, then according to (4.3) we are done. Finally, we can argue for every connected component of the system of representatives separately. So let us assume that the Dynkin diagram of G R has a type which occurs in the classification given above. Of course, it suffices to consider the positive roots, since the negative ones are completely determined by these (see (4.3) (i) a) and (4.1) (iii)). Let n be the order of the image of E in Aut(/~) (= group of automorphisms of /~ corresponding to automorphisms of the Dynkin diagram. Note that E acts on the Dynkin diagram by diagram automorphisms.) By the classification given above the following cases can arise: 1. Case: n - 1; there is nothing to prove. 2. Case: n = 2; let r E E be an element which generates the image of ~ in Ant(/~). If { a l , a 2 } is an orbit in ~+ consisting of two elements, then one can obtain a ( Z ~ 1) -= X~ 2 and a(X~2) :- X ~ by changing signs (see (4.5)). So let {a} be an orbit in ~+ consisting of one element. a) There is a root • e ~ with a =/3+a(13). Hence [X~, X~(~)] = +X~ by (4.6). If we apply a to both sides of this equality, then we get a(X~) = - X ~ . b) There is no fl 9 ~) with a = /3 + a(/3). Therefore we have to show that a(Xa) -- Xa. Let /~ = {(~1,..., 5n}. For an arbitrary root r = Enihi we let ht(r = Eni (the height of ~). We will prove the assertion by induction on

ht(a). For ht(a) -- 1, there is nothing to prove according to (4.2). So let us assume that ht(a) > 1. Then there exists a root 5 E/~ with a - 5 E ~. If 5 = a(5), then we apply a to both sides of the equality [X~_~, X~] = =t=X~ (see (4.6)) and obtain a(X~) = Xa by induction hypothesis. Note that the classification forces - - the case of multiple roots is impossible because of ~ = a(d) - - that a - 5 is a root itself, and so we are able to use the induction hypothesis. If ~ ~ cr(~), then a - a(~) e ~ and therefore a - a(~) - (f e ~ according to (4.7). If there is a root Z 9 ~ with Z + a(Z) = a - (~- a(~), then fl + ~ 9 or a(fl) 4- ~ 9 ~. Indeed, if fl + 5, cr(fl) + (f ~ ~, then n(fl, ~), n(a(fl), (~) > 0 ([Bou 1] VI 1.3 Prop. 9 (iii)) and therefore n(fl + a(/3), 5) > 0, in contrast to Z+a(t3)+(~ E ~. By using fl' := Z + ~ or Z' := a(/3)+~ we get a = fl'+a(fl'), in contrast to the assumption. Hence there is no such ft. Now we consider [[X~_~_~(~), X~], Xa(~)] = -t-X~ (again (4.6)) and apply a to both sides of the equality. Then the induction hypotheses yields a(X~) = X~. 40

3. Case: n = 3; let cr ~ ~ be an element which generates the image of E in Aut(/~). If {a} is an orbit consisting of one element, then by using a(X,~) = -4-X~ and a3(X~) = X~ (4.5) we obtain o(X~,) = X a . If { a l , a 2 , a 3 } is an orbit consisting of three elements, then we can obtain a(Z~,~) = Xr a(X~2) = X~ 3 and cr(X~) = X ~ by changing signs again (4.5). 4. Case: n -- 6; let al, a2 E ~ be two elements of order 2 which generate the image of ~ in Aut(/~). If {al, a2, a3 } is an orbit consisting of three elements, then a suitable change of signs yields a l ( X ~ ) = X ~ , a l ( X ~ ) -- X ~ , a 2 ( X ~ ) -- X~ 3 and a 2 ( X ~ ) --X If {a} is an orbit consisting of one element, then it follows a~(X~) = X,~ and a2(X~) -- X~ or a l ( X ~ ) --- - X ~ and a2(X~) -- - X ~ as in the second case. [] ~:~ 2 "

In the following we will examine the structure of the root groups. Let a E ~r'ed. Without loss of generality we may assume that a E A. Finally, let / ~ be the associated Z-orbit in /x. By [Bo] 22.6 it suffices to consider the universal semi-simple covering 7r : G ~ -~ (U~, U_~> and its unipotent subgroups to study the group U~. Two cases can arise: 1. C a s e : 2a r ~; the above classification shows immediately that G ~ x K / ~ is isomorphic to a product of S L 2 / R indexed by Aa and that the Galois group a c t s o n G a x g ]~ by permuting the components.

We fix a root a E /~a. The simple factor G ~ C_ G ~ x g /~ with respect to a is obviously defined over L~. Now the inclusion G ~ C_ G ~ x g/~ induces a canonical morphism T ~ ~ (G ~) --+ G a which is an isomorphism by descent theory (e.g., see [BLR] 6.2). In the sequel we will identify ~L~ (G o) and G ~. In the same way one can show that the inclusion ~ r C_ U~ x K /~ induces a canonical isomorphism 7~L~(Lr~) --+ U~. Note that ~r maps the root groups of G ~ isomorphically onto the root groups of (U~, U-a) ([Bo] 22.6). Thus we will also identify 7~L~ (0n) and U~. Therefore we get isomorphisms 9

(V,o/Lo) --+ V =o

=

which induce group isomorphisms z-4-~ : L,~ --+ U+a(K) (recall L~ = L - s ) . 41

N o t a t i o n s 4.8.

(i)

Let ~ a : Ua(K) --+ ~U {oc} be defined by ~p,~(u) = W(Xal(U)) and let

(ii)

F~ := F'~ := ~ ( U ~ ( K ) \ { 1 } )

C_ ~ .

For t E ~ , we let (iii) Ua,l = ~-1([~, co]) and (iv) U,,o~ =

Ua,~ = {1}, U~,-oo = U u~,e = U~(K) and U~,e+ = U u~,i.

~

Obviously, we have F~ = F~ = w(L,~). Lemma

4.9.

The filtration (U~,~)~E~ of U~( K) is independent of the choice of a ProoL

If cr E E, then the isomorphism

0 "-1

:

L~(~) ~

E

;X~.

L~ induces a K -

RKL~

i s o m o r p h i s m i : R L~(")(~a/L,(")) --+ (l~a/L.). Now if we replace ~ by ;}~(~) = a o ~ o (~-1 and L~ by Lo(~), then we have to replace x~ by x , o i. By [CaFr] II w we have w = uJ o T for all ~- E E, since K is complete. Hence

w o x-j l = w o i - ' o x~ 1.

[]

4.10. In order to give a description of the action of the torus on U,~(K), we first consider the following L ~ - g r o u p h o m o m o r p h i s m s :

y+

:

tI?~/L~

~

SL2/Lo

with

u~

Y-

:

~/c~

-+

SL2/L~

with

u~

z

:

GraIL,,

-")" SL21L,:,

with

t r-~

(1 ;) 0

( 1

01) '

(; o) t- 1

F u r t h e r let ~ : SL2/L,~ ~ G '~ be the L~- isomorphism defined by ~•

= Tr o ~

o y+.

Finally, we let (~

=

71"

0 ~ ' ~ (~ct 0 Z) : T~LKa ((l~m/Lc' ) -4 (Ua, U_a}

and for u E L~\{O}, we let

Tn,a(U) = X_a(U--1)Xa(U)X_a(U -1) C_ (Ua, U-a} 42

'

.

Hence the element ma(u) is given by the matrix

(o o) _u_ 1

.

4.11. If ma :--- m~(1), then a simple matrix calculation shows that we obtain for all u E L~:

(i) m~(u) E N(K), (ii) x_~(u) = m~x~(u)m-d 1, (iii) ma(u) = ~z(u)m~ = m ~ ( u -1) and (iv) m~4 = i d .

Proposition

4.12.

Let ~ E JR. Up to a unique ~somorphism there exists a unique smooth, affine oK-group scheme L[~,~ of finite type with generic fibre U~ and ~l~,t(og) = U~,t. P r o o f . The uniqueness follows from the supplement of (0.3). Let L:~,t be the canonical OK-Scheme associated with L~,, = {u E La : w(u) > ~} and we equip it with the natural group scheme structure (see (0.14)). Since Ua,~ = x~(L~,e), we obtain the existence of s [] 2. C a s e : 2a E ~; the above classifications shows that G a • is isomorphic to a direct product of SL3/R indexed by the family of sets {t3,/3 r} with/3, fl' E and fl + fir E ~, and that the Galois group E acts on G ~ x g the components.

/~

by permuting

Now we fix a, a r E /X~ with a + a r E ~. Then L~ = L~, is a quadratic (separable) extension of L~+~,. To simplify the notations, we let L = La and L2 = L~+a,. For the non-trivial a E Gal(L/L2), there exists the following Hermitean form on L 3:

h: (z-l,z0,zl) ~ x~_lxl +z~z0 + z~z_l. By SU3 we denote the special unitary L2-group associated with h. The simple factor G a,~' C_ G a • with respect to {a, a r} is defined over L2, since Gal(/~/L2) stabilizes the set {a, a'}. Besides it is L2-isomorphic to SU3, since on the one hand G ~,a' is a non-trivial Gal(L/L2)-form of SL3/L and on the other hand there exists a unique group homomorphism of Gai(L/L2) into the group of diagram automorphisms of the Dynkin diagram of G ~ from which the assertion follows (see [CaFr] X 2.2). The inclusion G ~,~' C G ~ • K / ( induces a canonical morphism 7"4L~(G ~,~') ~ G ~ which is an isomorphism (descent theory, e.g., see [BLR] 6.2). In the sequel we will identify T~L2(G~,~') and G ~. Obviously, the unipotent subgroup D-~D-~+~,D~, is defined over L2, and one 43

can show in t h e s a m e w a y - - t a k i n g into a c c o u n t t h a t 7r m a p s the_ _r~176groups i s o m o r p h i c onto each o t h e r ([Bo] 22.6) - - t h a t t h e inclusion U~U~+~,U~, C U~ • i n d u c e s a c a n o n i c a l K - i s o m o r p h i s m 7~L2 (D~U~+~,U-~,) --+ U~. We will i d e n t i f y t h e s e K - g r o u p s , too. O u r n e x t a i m is to u n d e r s t a n d t h e s t r u c t u r e of t h e L2-group U~U~+~,Uo,. L e t u, v, w 6 L. T h e n b y definition of t h e C h e v M l e y - S t e i n b e r g s y s t e m we o b t a i n

~(~(~)~+~,(~)~,(w)) = ~,(~)~+~,(-~)~(w~). In o r d e r t o e x t r a c t a r e l a t i o n of u, v, w from this, we will c a l c u l a t e b o t h sides of t h e e q u a l i t y in S L 3 / L ( L ) ( n o t e [So] 22.6 again):

(iui)01 (1 01 0 v )(1)000001 0 = v+u )00 and

(i00) (10v)(1 1 0

u~ 1

.

0 0

1 0

0 1

9

0 0

1 0

0)(i w

0 1

=

1 0

u: 1

H e n c e we o b t a i n u = w a a n d v * = - v + u w = - v + u u % T h e r e f o r e it suffices to c o n s i d e r t h e r e l a t i o n v + v * = uu% We can deal with - a and - a ' in a similar

way. L e m m a 4.13. Let Ho(L, L2) = { ( u , v ) e L • L : v + v ~ = uu~}. If we equip Ho(L, L2) with the following action

Ho(L, L2) x Ho(L, L2) ---+ Ho(L, L2) ((u, v), (u', v')) ~-~ (~ + u', v + v' + u%')

,

then it becomes a L2-subvariety of L • L which defines a L~-group scheme also denoted by Ho(L, L2). P r o o f . Obviously, we can identify Ho(L, L2) t o g e t h e r w i t h t h e group law w i t h t h e following closed s u b g r o u p of SU3:

u v) 1 0

u 1

:u, v E L

and v + v a = u ~ u

}

C SU3(L2)

.

[]

All a s s e r t i o n s follow from this. 44

Now let _

k•177

~

-

: H0(L, L2) --+ U•177177

be the L2-isomorphisms defined by

~•177

~) = ~•

~•177

9~•

We let H ( L , L2) = T~L2 g (Ho(L, L2)) and obtain canonical K-isomorphisms X•

:

~r~gL~ ( X-• 1 7 7

:

H ( L , L2) --+ 7~L2(u+~u+~+~,U•

which induce group isomorphisms x•

{(u,v) 9 L • L : v + v ~ = u"u} -+ U+,~(K) and

x•

{(0, v) 9 L • L : v + v ~ = 0} --+ U+2,~(K) ,

since one can show t h a t T~L~ (U• t h e o r y [BLR 6.2].

Notations

is isomorphic to U+2a by using the descent

4.14.

(i)

Let ~ : Ua(K) --+ Ft U {oc} be defined by ~o~(u) = 89 for x~(fi, 9) = u, let ~2~: U2~(K) --+ ~ U {co} be defined by p2~(u) = w(~)) for xa(0,~) = u,

(ii)

let F~ = ~ a ( U ~ ( K ) \ { 1 } ) C_ lR, F ~ = F2~ = ~02~(U2~(K)\{1}) C ~ and r" = { ~ o ( u ) : u E U o ( K ) \ { 1 } and ~ o ( u ) = sup ~o(uU2o(K))}.

For g E ~ , we let - t ([ g ,oc]), Uao~ , (iii) Ua,g = ~ - l ( [ g , cc]), U2a,g = ~2a

['1 U~,g = {1} and gent

u~,_~ = U Ua,g = U~(K) and gent

(iv) u~o,~ :

N u~,~ : {~}, u~o,_~ :

U u~o,~ : U~(K), Uos+ :

and

u~o,~+ = U u~o,~. ~>g

Obviously, U2~,2~ = U2,~(K) N U,~,t for all g e ~ . 45

U u~,~

L e m m a 4.15.

The filtrations (U~,t)ee~ and (U2~,e)~E~ of U~(K) and U2a(g) are independent of the choice of (a, a').

P r o o f . This can be shown in the same way as in (4.10).

[]

In order to give a description of the structure of the sets of values F~ and I ' ~ , we have to go a little deeper into the valuation theory of L. But since L/L2 is ramified ( K is strictly Henselian), we do not need an approach as general as in [BT 2] 4.2 and 4.3. We let L~ ={v 9

~=0}

,

L1 = {v 9

~'=1}

and

nlm~x = {A 9 L 1 : w(A) = s u p { w ( x ) : x 9 L1}} . L e m m a 4.16.

(i)

There exist elements t 9 L and r, s 9 L2 such that L = L2[t], t 2 - rt + s = 0 and w(at -1) < w(2) (if a 7~ 0).

(ii) Let )t =

1

/ora=O

1

2 Then A 9 L.~a~. ta -1 for a r O " (iii) F o r ,~ 9 L m1 a x we h a v e c.J( )Q ~ 0 and t.o( )~ - )~o.) ]> O.

P r o o f . For (i) and (ii) see [BT 2] 4.3.3. The assertion (iii) is a direct consequence of [BT 2] 4.3.3. (iii) and (iv). [] L e m m a 4.17.

We have: (i) L = L 2 A + L (ii) If A 9

1 Lma~,

~ x 9 L2 and y 9 n ~ then w(xA + y) = inf(w(zA), w(y)).

P r o o f . (i) is clear, since )~ ~ L ~ Without loss of generality we may assume that x r 0. Then A + x - l y E L 1 and therefore w(A) _> co(A + x - l y ) , hence co(xA) > co(xA + y). Next we have co(y) > inf{w(xA + y),w(-xA)} = w(xA + y) and therefore w ( x A + y ) > inf{w(xA), w(y)} > w(xA+y), from which the assertion (ii) follows.[] The following lemma yields another description of the unwieldy condition defining the set r ' a " 46

Lemma

4.18.

1 ~. Then L e t u , v E Ho(L, L2) and let )~ E Lma

o(xo(u, v)) <

+

and equality holds if and only if ~ ( x ~ ( u , v)) = sup ~a(Xa(U, v)U~a(K)). P r o o f . We have (u, v) = (u, Auu*')(O, v - Auu~). By (4.17) we obtain 1 sup ~ ( x ~ ( u , v)U2~( g ) ) = -~ supw(Xuu ~ + L ~ = ~w(X) + w(u) []

Hence the assertions are clear.

Thus we are able to give a complete description of the structure of the sets of values F~,F~ etc. P r o p o s i t i o n 4.19. 1 Let )~ E Lma ~. Then:

(i)

F2~ = F~2~ = w(n~

(ii) r~ = 89

•

a n d r " = 89

+w(L•

If w is normalized in such a way that w(L • = 2Z, then: (iii) If a = O, then F'2~ = 1 + 2 ~ and F: = 2~. (iv) If c~ r O, then F~a = 2 ~ and F~a = 89+ TZ. So F~a and 2F~ are always disjoint. P r o o f . (i) follows immediately from the definition. The inclusion F~ C_ l w ( L • clear. By (4.17) we have w(L x ) = (w(A) + w(L•

U w(L~

= (w(X) + 2w(LX)) U w(L~

C_ 2to ,

since we have (u,)~uu ~) E Ho(L, L2) for u E L • Finally, the second equation in (ii) is clear by (4.18). Ad (iii): Obviously, L ~ = L2 9t and w(1-~ 2/ E 22Z. By using (ii) we obtain both equalities. Ad (iv): Let x + t y E L ~ with x , y E L2. Hence 2 x + a y = 0 and therefore x + ty = x(1 - 2ta-1). Then according to (4.16) (i) it follows F~a = 22~. On the other hand, by (4.17) (i) and (ii) we obtain

w(x + )~y) = inf(w(x) + w(),)w(y)) • 22Z for suitable x E L2 and y E L ~ since w(x) E 2 ~ and w(y) E F~a = 2~. So A 9~ 22Z, from which the remaining equality follows. [] 47

Now we want to examine the action of the torus on the unipotent subgroups. 4.20. We consider the following L2-group homomorphisms: --V y+

"

H o ( L , L2)

--+SU3

with

(u,v)

~-~

y-

:

Ho(L, L2)

--~SU 3

with

(u,v)

~

z

:

T'~L(Gm/L)

-"~SU3

with

t

~

m

:

"~L2(~m/L)

---+SU3

with

t

0 0

1 0

;

u --V

1 --U a

;

(10 ) o)

~+ t

t-it ~

0

o

(t~) -~

-t-lff " 0

0 0

;

9

Next let ~ : SU3 --+ G a'a' be the L2-group homomorphism defined by ~,,•

= 7r o ~ o

y+

Finally, we let a = ~ o n ~ , ( ~ o ~): ~ : : ( ~ m / ~ ) - ~

(Ua, U-o)

and for ( u , v ) E Ho(L, L2)\{(O,O)}, we let ma(U , V) : X_a(UU -1, (Va)--I)Xa(U, V)X_a(U(VCr) -1, (Va) - I )

.

4.21. If we let ma = m(1), then one can show by some matrix calculations that, for all (u,v) e H0(L, L2)\{(0,0)}, the following assertions hold: (i)

m~(u,v) E N(K),

(ii) x_o(u, v) = moxo(u, v)m-: 1, (iii) m ~ ( u , v) = 5 ( v ) m ~ = m~5((v~) -1) and (iv) mo = id. In order to construct the group schemes again, we have to give another description of H o ( L , L2). 48

Lemma 4.22. Let ik E L 1 . Then: (i) L • L ~ equipped with the action (x, y) . (x', y') = (x + x', y + y' - ~xx '~ + ~ x ~ x ') is an algebraic L2-group and

5i) the map : onto L • L ~

(u,

is a L -group isomorphism o/ Ho(n,

P r o o f . Both assertions can be proved by simple straight forward calculations.[~ Let H ~ be the K-group 7r the above action.

(L • L ~ equipped with the group law induced by

L e m m a 4.23. 1 x and let 7 = --lw()~). If Lt+7 := {u E L : w(u) > ~ + 7} Let ~ E I:~, )~ E Lma and LOt :-- {u E L ~ w(u) > 2~}, then up to a unique isomozThism there exists a unique smooth, a]fine oK-group scheme 7-I;~ of finite type with generic fibre H A, 7"/~(oK) = Lt+7 • LOt and a group law, which induces the group law in (~.22) on the generic fibre.

P r o o f . The uniqueness follows from the supplement of (0.3). Obviously, Lt+7 and LOt are free OLd-modules of finite type. Let L;t+7 and L;~ be the canonical OLd-schemes associated with Lt+7 and LOt and we equip them with the canonical group scheme structures (see (0.14)). We let 7~0~ = L;t+7 x L;~ The map L x L --+ L ~ (u, u p) ~-+ Auu r~ - )~'ue'u ' can be extended uniquely to a morphism L~t+7 x L;t+-r --4/2~ Hence the group law can be extended to 7/0~. The 1-section of L x L0 extends uniquely to a 1-section of 7/0~. Finally, if s : L x Lo --4 L x Lo (u, v) ~ ( - u , - v -4- uu~'()~ a - A)) is the "inversemorphism", then according to (4.16) (iii) and (0.3) we obtain an extension to By (0.8) the scheme ~0~ is a smooth, affine OLd-group scheme of finite type. According to (0.5) the scheme 7/~ := 7r1762 (7/0~) has the desired properties. [] P r o p o s i t i o n 4.24. (see [BT 2] 4.3.5) Let g E ~:~. Up to a unique isomophism there exists a unique smooth, aJfine oK-group scheme Lia,t of finite type with generic fibre Ua and s = U~,e.

49

Proof. The Let A and 7 the "image" it suffices to

uniqueness follows immediately from the supplement of (0.3). be as in (4.23) and let s be the smooth, affine oK-group which is of 7-/~ under the isomorphism x~ o 7~L~ (j~-l) : L x L0 ~ Ua. Thus show that s = U~,e.

In order to prove this, we will show that

--- {x~(u, v ) : (u, v) 9 Ho(L, L2) with a~(u) > e + 3' and w(v - %u"u) >_ 2e} ---- U a , ~

.

Let (u,v) C Ho(L, L2) and let x - Xa(U,V). T h e n x = x ' - x t' where

X' = Xa(U,,)IU~

=

x a ( j ; l ( u , O ) ) E Ua(K) and

x" = xo(O,v - ),u%) = z

(j;l(O,v - A u % ) ) e U2 (K)

The first equality follows from this. Since p~(x') = 89 = w(u) - 7 and ~2~(x") = w(v - A u % ) , it is clear that the second set is contained in the third one. So let us prove the converse inclusion: Let x E U~,~ and consider the decomposition x = x I 9x t~. As in (4.18) one can show t h a t f < ~o~(x) _< s u p ( ~ ( x . U2a(K)) = w ( u ) - 7 = ~a(Xt). Hence x' e Ua,e and therefore x" E U~,eAU2~(K) = U2a,2e, from which w ( v - A u % ) >_ I. follows.J3

Proposition 4.25. Let ~ E F~. Up to a unique isomorphism there exists a unique smooth, affine OKgroup scheme Li2a,e of finite type with generic fibre U2~ and }.12a,*(OK) = U2~,e. P r o o f . The uniqueness follows immediately from the supplement of (0.3). By using T~L~(L;~ instead of 7/~ (see (4.23)) one proceeds as in (4.24).

[]

Obviously, we can identify :g2a,2e with a closed OK-Subgroup scheme of s since ~12~,2~ can be identified as a scheme (via 1-section) with the second factor of il~,~.

w

A s m o o t h m o d e l o f t h e b i g cell in G

In order to construct an oK-group scheme t3 with generic fibre G, we will first put together the group schemes LI~,t and T to construct a smooth model of the big cell in G and to define an oK-birational group law on it. In the first part we will follow [BT 2] 3, but then we will avoid the representation theory. 50

First of all, we will put together all ~l~,e (a E ~+ for a given order on ~) to obtain a unipotent group scheme with generic fibre U~. Of course, this need not work for arbitrary choices of g E N. By using a non-empty, bounded subset ft of V ( = R - v e c t o r space defined in (1.4)) we will obtain conditions which enable the construction of the unipotent group scheme. The more general method of quasi-concave functions as in [BT 2] will not be considered here. Let X be the canonical oK-group scheme associated with T and for g E N, let ga,e (a E q~) be as in (4.12) resp. (4.24) resp. (4.25). In order to simplify the notation, we will define an addition on N U {4-oc} extending the addition on N. We let - c o + z = - t o for all z E N tO { - o o } and +oo + z = +oo for all z E 1RU {:t:oo}. Moreover, let the infimum of an unbound-below set be - o o and let the infimum of the empty set be +oo. For an arbitrary (not necessarily bounded), non-empty subset ~2 C_ V, let ffl : ~ --+ ~ U {:koo} be defined by fa(a)=inf{eEN:a(x)+g>>_0 = -sup{a(z):x

for a l l x E ~ 2 }

E fl} .

L e m m a 5.1.

Let a E ~. Then: (/)

/ a ( 2 a ) = 2fa(a), if2a E ~.

#i)

If b E 9 with a + b E 02, then ffl(a) § f•(b) >_ f a ( a § b).

(iii) Ia(a) + la(-a) >__0. (iv) For a finite set of roots {ai} in q~, we have f f l ( ~ ai) <_ 2 ffl(a~). i

4

P r o o f . The assertions (i), (ii) and (iii) follow immediately from the definition. As in [BT 1] 6.4.5. one can show that (iv) follows from (i) - (iii). [] In the following U~,a and ~l~,a will denote U~,fa(~ ) and 5.1a,fa(a), respectively. Let a, b E 9 with b ~ - N + a . Then (a, b) is positively closed and the commutator map Ua • Ub --+ U(a,b) induces a K-morphism % , b : U ~ • Ub .....~

1-I Uc cE(a,b)~ed

for an arbitrary ordering of (a,b) (see w especially (0.17)). depends on the choice of the ordering of (a, b).

Of course, %,b

The following proposition gives us one fundamental property which will be needed for the examination of the geometry of the Bruhat-Tits building. 51

Proposition 5.2. Let a, b E q~ with b f~ - ~ + a "/a,b : Ua,r

X

Ub,s

--}

and let r, s E ~ U {• (Upa+qb,pr+qs

:

Then %,b induces a map

P, q E lN>0 and pa + qb E r

.

P r o o f . If r, s E ~ , then the assertion follows from the explicit calculation in Appendix A in [BT 2]. Note that no valuation root data are used there. The general case follows easily from this case and the definition of U~,+~. []

Corollary 5.3. Let a, b E O~a with b ~ -]R+a, Then "Ya,b induces a map

~'a,b: Uo,a x gb,a ----+

I-I

g~,a

cE(a,b) ~d

[]

P r o o f . The assertion follows from (5.1) and (5.2). L e m m a 5.4.

Let Y be a K-group scheme and let X 1 , X 2 be closed subgroup schemes. Let be a fiat oK-scheme with generic fibre Y and let X l , X 2 be fiat oK-group schemes with generic fibres X1 and X2, respectively. Finally, we assume that the product morphisms X1 x )(2 ~ Y and X2 x X1 ~ Y extend to oK-isomorphisms a : :~l x ~2 -+ ~ and/3 : X2 x Xl --+ ~ . Then ~ is an OK-group scheme with generic fibre Y . 9

[]

P r o o f . See [BT 2] 3.3.1. In addition let ft C V be a bounded subset. Then obviously fa((I)) C_ ~ .

Proposition 5.5. (see [BT 2] 3.3.2) Let qs C_ q~ be a positively closed subset and let g : 9 --+ ~ be a map, which satisfies 5 ) - ( i i i ) and therefore also (iv) in (5.1). Then there is a unique smooth, a]fine oK-group scheme ~l~,g of finite type with generic fibre U~ and the property that for every good ordering ofq~ ~ed (0.15), the K-isomorphism I~ Ua -'+ U~ aEq2 r~d

1-I ~la,g(a) -+ 12~,g.

can be extended to an oK-isomorphism

aEq2 ~ d

52

P r o o f . Since all H~,g(~) are smooth, the uniqueness follows from (0.3). We will show the existence by induction on card(62) (= number of elements of 62). For card(62) -- 1 there is nothing to prove. So let us assume that card(62) > 1. Fix a good ordering of 62 and let us consider two extremal elements a, b E 62. We get a K-isomorphism U a x U~_{a } mul~ U~ m - ~ 1 U',~-{b } X U b ,

and if we let U~,g = Ua,g x Hr oK-isomorphism

(*)

then we have to show that (*) induces an

U~,g x g,~_{~},g ~

U,~_{b},g x gb,g

We distinguish two cases: 1. C a s e : a r b; by induction hypothesis we obtain

g,~,g x g~_{,~},g ~~ g(,,g x ge-{a,b},g • Ub,~ ~> ge-{b},g X gb,p . 2. C a s e : a = b; by the extension principle (0.3) it is sufficient to find a morphism U~,g x U~,_{~},g --4 He_{~},g x ~la,g which extends (*), since one can proceed with the inverse m a p of (*) in the same way and obtains as composition the identity. So let b' E @ be an extremal element with b' # a and let 621 = 62 N (a, b'). Since Ha,g and Ub',g are smooth, we obtain by (0.3) and by induction hypothesis a unique extension

a : g~,g x Ub,,g ~ gv~,g x Ub,,g x H~,g of

vo x vb,

u , , • vb, • uo ((x, y)

(xyx-ly -1, y, x)).

Now let us consider the composition x id Ua,g X ~'~--{a},g id • mu)It -I Ua,g X U~_{a,b,},g X Ub',gmult "--)" U'cE-{b'},g X Ub,,g

mult-Z>x id HCt--{a,b'},g X "~-[a,g X Ub',g id • m u ~ id

UT~_{a},g X Ua,g ,

which extends (.). By (5.4) and by induction hypothesis the existence of the oK-group law on s follows. [] 53

C o r o l l a r y 5.6.

Let q~ C q~ be positively closed. Then there exists a unique smooth, affine OKgroup scheme ~l~,,a of finite type with generic fibre U~ and the property that for every good order of qj~d the K-isomorphism 11 U~ ~ Us can be extended aE~I/~ed

to an oK-isomorphism

[I

[l~,a --+ t2,~,~.

P r o o f . T h e assertion follows directly from (5.1) and (5.5).

[]

C o r o l l a r y 5.7.

Let q~' C_ q~ C ~ be two positively closed subsets. Then the inclusion U~, --~ Uv extends to an oK-group isomorphism of ~.l~,,n onto a closed oK-subgroup scheme

of tl,~,a. P r o o f . One can show immediately by induction on card(qd) that every good ordering of ~ ' is induced by a good ordering of 9. Hence the assertion follows from (0.3) and (5.6). [] Now we assume that we have given an order on ~ and we will write ~lfl resp. 1.1+ resp. LI~ for ~l~,fl resp. L[e+,a resp. ~ - , f l . Lemma

5.8. (see [BT 2] 4.4.19)

Let q~ C_ ~ be positively closed. Then the action o f T on Us extends to an action of T on ~l~,f~ compatible with the group structure on t.lq,,n. P r o o f . According to (5.6) and (5.7) it is obviously sufficient to show the existence of the action for 9 = {a} with a E r So we consider the morphism r x u~ --+ u~ (t,u) ~-~ tut -1. Let t E T(OK) and x E ~l~,a(OK). By (3.1) we know that T(OK) C Tb(K) and hence ~(t) E o Lx~ for all c~ E / ~ (see (3.1) and -w As in w we distinguish two cases: 1. C a s e : 2a ~ ; we obtain

with x~ as in w (before (4.8)) and u E L~ w i t h x ~ ( u ) = x

tx~(u)t -1 = x~(c~(t)u) . 2. C a s e : 2a E ~5; with x~ as in w (before (4.14)) and u E L, v E L2 with x~(u, v) = x and id ~ a E Gal(L/L2) we obtain

tx~(u, v)t -1 = x~(c~(t)u, c~(t)c~(t)~ v) According to (0.3) we obtain an action ~ x 12~,n ~ LI~,n. The compatibility with the group structure follows from the fact that T x U~ ~ Ua is compatible with the group structure on ETa (see (0.2)). [] 54

Hence we can equip ~ x ~l~,,f~ and g[~,~ • ~: with the group structure of a semidirect product (see [DG] II w 3.10). This oK-group scheme (with generic fibre TU~ resp. U~T) will be denoted by ~:gt~,,~ resp. s We will identify these group schemes. For a 9 ~.~d let Ws be the pre-image of U _ a T U s under the product morphism Us x U_~ --+ G~. Here Gs denotes the group G{s,2s}ne (see w By (0.21) we know that Ws is an open neighbourhood of the "l-section" of Us x U-a. Hence the inverse of the product morphism induces a morphism/~ : Ws --+ U-s x T x Us of K-varieties. Now we want to lift /~s to the level of oK-schemes. distinguish two cases as in w

In order to do this, we

1. Case: 2a ~ ~; let the notations be as in the first case of w Then (10

1)"

1

(-u'

01)= (

uu ~ -u' 1)

1 -

and

(_1721~)" (t0 tO1)"(~ 1)= I--IV' t--1~--~tVVt)~ from which follows that Ws is the standard open subset of Us • U - s with respect to the function ds: (xo(u), x - s ( u ' ) ) ~* N ~ ~ (1 - uu') and that

~s(Xs(U), X_s(U'))

= (X_s((1 -

uu')-lu'),

5(1 - u u ' ) , X s ( ( 1

-

uul)-lu))

.

In order to simplify the notations, we have given an element of K[Us x U-s] as a function ("variety-point-of-view"). L e m m a 5.9.

We have ds 9 og[gts,f~ x gt-~,f2].

x ~_a,f~(OK))

P r o o f . By (0.3) it suffices to show that d~(gts,~(OK) C OK. For u, u' 9 L~ with w(u) >_ fa(a) and w(u') >_ f n ( - a ) , we obtain w(uu') > 0 (see (5.1)), hence 1 - u u ' 9 OLd. Therefore ds(xs(u), x_s(u')) = N L" ( 1 - u u ' ) 9 OK.t2 Now let ~2~Ta,a be the standard open subset of s Obviously, the generic fibre of ~ s , a equals Ws. 55

x gt_s,a with respect to ds.

5.10. (see [BT 2] 4.5.8)

Proposition The morphism

fl~ : W ~

--+ U_~ • T x Ua extends u n i q u e l y to a m o r p h i s m

P r o o f . By (0.3) it suffices to show that

#r

C u-,,,n(o,r

• ~:(o~:) • u o , . ( o K )

. •

Let ( x ~ ( u ) , x _ ~ ( u ' ) ) E ~,a(og), i.e. d ~ ( x ~ ( u ) , x _ ~ ( u ' ) ) E o K, from which w(l - u u ' ) = 0 follows. By (3.6) we know t h a t 7~~ ~ ( ~ m / o L , ) is the canonical oK-group scheme associX OLc~ ated with ~L~ ( ~ m / L ~ ) , hence 1 - u u ' e OL~ = T~oK (Gm/oL~)(oK). It follows from the functorial properties of T ~-~ ~ (3.4) that 6(1 - u u ' ) E T ( O K ) . Hence w((1 - u u ' ) u ' ) = w(u') >_ f n ( - a ) and w((1 - u u ' ) u ) > f n ( a ) , from which the assertion follows. []

2. C a s e : 2a C if; let the notations be as in the second case of w

(i U l)(

01)( (1 oo)(,o 0)(1 -v

1

9

0

1 u'

0 1

0

~'U t

- - U t~

0 0

t - l ff" 0

1-

=

u ~ u ' + vv' u' - uv'

=

1 -w"

0 1

.

0 (ta) -1

.

- - U 'c"

0 0

t

- t w c"

-tx

-tw'w ~ + t-tt ~

-twx + t-lt~w t x x I - t - l t a w w ' ~ + (t~) -1

x'tw a - w'at-lt ~

1)

-v

x)

1 0

w't \-x't

- u ~" + v u '~ 1 - u u 'a

--'Ut

and

w' -x'

Then

)

from which follows that W~ is the standard open subset of U~ • U_~ with respect to the function da : ( x ~ ( u , v ) , x _ ~ ( u ' , v') ) ~-+

NKL~(I-ur

' + vv')

.

Let t = 1 - u'~u ' + v v ' . T h e n

#o(x~(u, v), z_o(u', r = (x_~(t-t(u

' - uv'),t-lv'),&(t),x~((t~)-l(u

56

- v~u'),t-lv)).

L e m m a 5.11.

We have da E OK[Lta,fl x s P r o o f . By (0.3) it suffices to show that d~(t.l,~,a(OK) x H-a,r~(OK)) C_ OK. Let ( u , v ) , ( u ' , v ' ) E H ( L , L 2 ) ( K ) with x~(u,v) E tl~,,n(OK) and x_a(u',v') E ll_~,a(OK). By (4.23) and (4.24) we obtain (using the notations given there) w(u) >_ f~(a) + V, w(v) > fn(2a), w(u') >_ f a ( - a ) + V and w(v') > f n ( - 2 a ) . Hence according to (5.1) we obtain t = 1 - uau ~ + vv ~ E OL,~ and therefore d,~(x~(u,v),x_~(u',v')) = YL~(t) E OK. [] Now let g/J~,n be the standard open subsubset of Lt~,a x Lt_~,a with respect to d~. Obviously, the generic fibre of ~lJ~,a equals We.

P r o p o s i t i o n 5.12. (see [BT 2] 4.5.8)

The morphism t3~ : W~ -+ U_~ • T x U~ extends uniquely to a morphism #~ : fg1J,,a ~ t.1_~,a x ~ x ~.la,a. P r o o f . By (0.3) it suffices to show that c a_o,a(oK) •

(oK) •

to,a(oK) .

Let (x,~(u, v), x_,~(u', v')) E fllJa,a(OK). Then co(t) = 0 for t = 1 - u~'u ' + vv'. As in (5.10) one concludes that a(t) E T.(OK). On the other hand,

co(u' - uv') > i n f { f a ( - a ) + 7, fa(a) + 3' + f a ( - 2 a ) } = f a ( - a ) + 7 , since we know by (5.1) that

fa(a) + 7 + / a ( - 2 a )

- f a ( - a ) - V : fa(a) + f a ( - a ) >_ O .

In a similar way one obtains w(u - v~u ') > fa(a) + 3', from which the assertion follows by making use of the explicit description of ~a. []

Since ~, tAa,fl and t/~,,n (9 C ~ positively closed) are smooth, we can identify Dist ~ (resp. Dist ~l~,a, Dist He,n) with o/<-subalgebras of Dist T (resp. Dist U~, Dist U~) and these with K-subalgebras of Dist G. Let Dist(D) be the subalgebra generated by Dist a2 and Dist t/~,a (a E ~ d ) . 57

Proposition 5.13. (see [BT 2] 3.4) If 9 C_ ~ is positively closed, then the following product maps are bijections: (i)

|

a~red

Dist H~,n -4 Dist il~,,n (for every good ordering of ~ d ) ;

(ii) Dist ~ | Dist s

-4 Dist ~H~,,n and Dist L[v,n | Dist ~ -4 Dist ~g~,,~;

(iii) Dist H~ | Dist ~ | Disti[ + -4 Dist(/)). (iv) The canonical map K | Dist(lP) -4 Dist G is a bijection, too. P r o o f . By (5.6) and (5.8) the assertions (i) and (ii) are clear. Ad (iii): The injectivity is clear because of the injectivity of U - • T • U + ~ G and the previous remark. So we have to show the surjectivity. Therefore it is sufficient to show that the image D of Dist H~ | Dist 2~ | Dist H+ in Dist(2P) is idependent of the order on ~, since then D is closed under multiplication from the left by Dist H~, Dist ~1+ and Dist 2[. So let ~+ be the set of positive roots of with respect to a further order. Then we will prove the assertion by induction on card(~2+~ ~ ~ - ) (= number of elements of ~+ ;3 ~ - ) . For card(~+l ~ ~ - ) -~ 0 there is nothing to prove. So let card(~ + ;3 ~ - ) > 0. Then there is an extremal element a of ~+ lying in ~ - . Let kg+ = ~ + \ { a } and kg- = ~ { - \ { - a } . Then r + = ~ + U { - a } is a system of positive roots (see [Bou 1] VI Proof of Prop. 17). By (5.10) and (5.12) we obtain Dist 114~-,a Dist ~ D i s t L[4+,a = Dist H~,- ,n Dist ~ - a , a Dist ~ Dist H~,n Dist [1~+ ,~ = Dist ~ r

,~ Dist [la,a Dist ~; Dist H_~,a Dist ~2v+,a

= Dist [14~,a Dist ~ Dist Hr

,

from which the assertion follows by using the induction hypothesis. Finally, (iv) is clear by (iii), since the product map U4- • T • U4+ -4 G is an open immersion (see (0.21)). [] In order to simplify the notation, H • will denote LI~ and H - ~ H + resp. L~+~Awill denote H - • (E • [1+ resp. i[ + x ~ x H - . By (5.13) we know that Dist(l/-Z~l +) = Dist(~t+~s According to (0.21) we know that U - T U + and U+TU - are open affine neighbourhoods of the 1-section in G. Hence there exist functions f 6 K [ U - T U +] and f' 6 K[U+TU -] such that

K [ U - T U + ] f = K[U+TU-]f,

.

Without loss of generality we can choose f e OK[U.--~H.+]\~rOK[~2-7~+] and

f'e 58

Proposition 5.14. In K[U-TU+]f

= K[U+TU-]I,

we have og[52-T~+]f = OK[Lt+TH-]f,.

[]

P r o o f . This follows immediately from (0.4). So we will identify ( H - T H + ) / a n d

(H+TL[-)/, in the following.

In order to construct a smooth oK-group scheme with generic fibre G from ~A-TH+, we will define an og-birational group law on H-~:H + first. In order to do this, we need the notions of oK-rational maps and oK-dense, open subschemes (see [BLR] 2.5). We will recall briefly the definitions: Let R be a ring and let X be a smooth R-scheme. An open subscheme ~3 of :~ is called R - d e n s e , if all fibres of ~3 (over R) are Zariski-dense in the corresponding fibres of :~. If 2] is a further, not necessarily smooth R-scheme, then an R r a t i o n a l m a p 3~ - -+ ~ is an equivalence class of R-morphisms ~3 --+ ~ where ~3 is an open, R-dense subscheme of ~. Here two morphisms ~3 --+ 3C and ~ --+ are called equivalent, if they coincide on an open R-dense subscheme of ~ N ~Y. If finally ~ is smooth, too, then an R-rational m a p f : : ~ - --+ ~ is called Rbirational, if there is an R-rational m a p g : 2] - ~ X such that g o f and f o g are equivalent to the identity. D e f i n i t i o n 5.15. (see [BLR] 5.1) Let R be a Ring and let :E be a smooth, separated R-scheme of finite type with n o n - e m p t y fibres over R. An R - b i r a t i o n a l group law on :E is an R-rational m a p m : :E x R Y. - --+ :E

( x, y ) ~--~x y

such that the following conditions hold: a) T h e R-rational maps

r

(x,y)~(x, xy)

~1/: ~ x : E - - - + X x :E

(x,y)~-+(xy, y)

are R-birational and b) m is associative, i.e. (xy)z = x ( y z ) , in any case when both sides are defined. Let :E = Li-TL[ +. Since :EK = U - T U + is an open neighbourhood of the 1section of G, there exists a unique K-birational group law on the generic fibre of :E. We want to extend this to X. 59

According to (5.8) we can identify ~11+ and g + T , hence also ~3A+~I- and ~ / + ~ / - . In • x :~ = 11- x ( ~ x s + x11-) x~11 + we consider the open subscheme .tl.- x (~+~11-).,,, x ~ x .tl.+ = ~ - x (g.-"s

x ~ x g.+

C_K- x g l . - x ~ x g .

"-~g.-

+ x ~.s +

x 9 x U.+

So we obtain a morphism t l - x (~l+ x ~ l - ) f , x I[~ + --+ :~. Since ~ has irreducible fibres over OK and f ' q~ zroK[11+l:t/-], we obtain an oK-rational map m:i~x:~--~:~

,

which coincides with the rational m a p induced by G in the generic fibre. Proposition

5.16.

T h e o K - r a t i o n a l m a p m is an o g - b i r a t i o n a t group law on Y~.

P r o o f . Assertion b) in (5.15) holds for the generic fibre (G is a K-group) and therefore by the extension principle (0.3) also for :~. In order to prove a), it is obviously sufficient to construct oK-rational inverse morphisms of @ and ~. Therefore it suffices to check that the inverse map of the generic fibre can be extended to an oK-rational m a p 3C---+ :~. First the inverse morphism induces a morphism U - T U + --+ U + T U -

which induces an oK-isomorphism 11-~:11+ -+ ~1+f711according to (0.3). Since ( l l + T l l - ) f , = (11-~ll+)f is an open, oK-dense subscheme o f ~ , we can find a function g E OK[il-?fli+]\~OK[t~-?fll+t such that the inverse morphism induces a morphism -+

=

c_

,

i.e. we have found a representative for the inverse morphism as an oK-rational m a p 3 ~ - - ~ :t. [] 60

P r o p o s i t i o n 5.17.

Up to a unique isomorphism there exists a unique smooth, separated oK-group scheme ~) of finite type (with group law ~ ) together with an open immersion :E -4 ~5 such that the restriction of ~ to 2~ is m. P r o o f . We glue together the schemes G and :E along ~K and obtain a smooth, separated (see [EGAI] 5.5.6) OK-Scheme :E' of finite type with generic fibre G (a K-group). Obviously, the og-birational group law m defined in (5.16) can be extended to :Er. By applying [BLR] 5.1 Theorem 5 to :E' the assertion follows. []

w

T h e group schemes ~ a

In this paragraph we formulate and prove some properties of the oK-group schemes defined in (5.17). We will follow [BT 2] 4.6. Let f~ C V be a non-empty, bounded subset and fix an order on ~. T h e o r e m 6.1.

Up to a unique isomorphism there exists a unique smooth, affine oK-group scheme ~f~ of finite type with ~ = ~ such that the following conditions hold: (i)

~

has generic fibre G,

(ii) the inclusions T C_ G and U~ C G (a E ~ e d ) can be extended to isomorphisms of T and ~ , ~ (a E ~ e d ) onto closed OK-SUbgroup schemes of (iii) if ~ C ~ is positively closed, then the product morphism induces an isomorphism of the scheme I-I tla,a onto a closed OK-subgroup scheme of aCff2~ed

~5~ which is idependent of the (not necessarily good) ordering of ~P, (iv) the product morphism induces a group isomorphism of ?~[~,~ onto a closed OK-subgroup scheme of ~ and (v)

the product morphism induces an isomorphism of s open subscheme of ~ .

• ~ • H+ onto an

P r o o f . Let ~ be the oK-group scheme @ defined in (5.17). Since H - ~ . / + has connected fibres, we obtain @~ = @~. By (0.11) the scheme @n is affine. The assertions (i) and (v) follow directly from the construction and (5.17). According 61

to [BLR] 5.1 Lemma 4 the topological closures of T and ~Aa,n in On are equal to T resp. lga,n. Hence (ii) is clear. We have proved the assertion (iii) for any good ordering. By [BT 2] 1.4.5 there is a faithfull representation On ~ GLn/oK. Here we can choose the representation in such a way that the image of ~ lies in the subgroup scheme associated with the diagonal matrices of GL,~ and that if T is the root system of GLn/K with respect to the diagonal matrices, then C_ ~+. Of course, in Gnn/oK the assertion analogous to (iii) holds (see [BT 2] 2.1.4 (iv)), hence the image of I-I ~ , ~ is a closed subscheme of the subgroup aE~a

scheme associated with the upper triangular matrices of GLn/oK and therefore a closed subscheme of On. Since (iii) holds for the generic fibre (0.17)(iii), the general assertion (iii) follows from (0.9). Finally, the assertion (iv) is clear. The uniqueness follows directly from the uniqueness in (5.17).

[]

In the following we will explain the connection between On and Off, for ~' C_

Proposition 6.2. Let ~, ~' C_ V be non-empty, bounded subsets with fl C_ fl'. Then the identity G ~ G extends uniquely to an oK-group homomorphism R e s t ' : On, ----+ r P r o o f . Let a E (~red. Because of the smoothness of Ha,n and s the identity Ua -+ Ua extends uniquely to an oK-group homomorphism ~-[a,n' -+ H~,n (see (0.3)). Hence the identity G -~ G can be "extended" uniquely to an og-morphism .t.l.~, x ~ x a +, ~ On

(see (6.1) (iii)). Then the assertion follows from (6.1) (v) and (0.10).

[]

Now let us consider the special fibres of the group schemes On. As in w we will identify the canonical oK-group scheme ~ associated with S with a closed OK-Subgroup scheme of ~.

Proposition 6.3. We have: (i) G is a maximal k-split torus in r (ii) ~ is the centralizer of ~ in On and -~ is the centralizer of-~ in -0~. 62

] ) r o o f . Suppose 6 is not a maximal k-split torus in On. Then there is a number n > dim S and a k-group homomorphism ~ : (~,~/k) '~ --+ ~Sn which is a closed immersion. By (0.13) we know that ~ extends to an oK-group homomorphism u : (ff~m/oK) '~ --+ ~5n. According to (0.12) this is a closed immersion. Hence we obtain a contradiction to the maximality of S, and (i) follows. By [SGA 3] Exp. XI 5.3 the functor

~; ~ z~(~)(~(x)) from the category of OK-Schemes to the category of groups can be represented by a smooth, closed subgroup scheme Z ~ ( | of ~Sn. By [Bo] 11.12 the fibres of Z ~ ( O ) are connected, since ( Z r = Z~n,.(Os ) for all s 9 Spec(og). Next T C_ Z ~ n ( G ). Suppose this inclusion holds strictly. Then d i m T < dim Zvn ( 6 ) and since

z~.(O)

n ~E~

= ':'.'Z ,

we obtain a contradiction (by using (6.1) (v)), since | characters on the Lie algebras of L[~ and L[+.

acts through non-trivial []

C o r o l l a r y 6.4.

We have: (i)

Lia,n is a maximal closed, connected, unipotent subgroup of r acts on its Lie algebra through the character a.

such that

(ii)

For a positively closed subset q~ C_ ~ , we get that s162 is a maximal closed, connected, unipotent subgroup of r such that | acts on its Lie algebra through the characters in q~.

(iii) The unipotent radical of T is a maximal closed, connected, unipotent subgroup of t3n such that G acts trivial on its Lie algebra.

P r o o f . Up to the maximalities all assertions follow from the last proposition and the way ~ is acting on L[a,~. According to (6.1) (v) the Lie algebra of qSn --+

is the direct sum of the Lie algebras of ~, LI~ and L[n. By dimension reasons the maximalities follow. []

L e m m a 6.5.

Let ~, [ e F~ with ~ + [ > O. Then s

,i(k) is a group. 63

P r o o f . By construction (in w and w the group schemes ~,s smooth. According to (0.1) it is sufficient to show that

and ila, ~ are

M : : U-a,w~(OK)~.(OK)~.[a,~(OK) is a group. By (5.8) we only have to show that M is closed under multiplication from the left by ~a,~(OK). Since g + ~ > 0, this follows from the explicit calculations in (5.9)-(5.12). [] In the sequel k is assumed to be algebraically closed. P r o p o s i t i o n 6.6. (see [BT 2] 4.6.10)

Let R be the unipotent radical of ~a und let a E ~. [ = f ~ ( - a ) . I l i a denotes the image of Ua,t+ U~,e

We let g = fa(a) and

for g + g = 0 and g E Fa else

under the map ~la,n(oK) --+ g~,a(k), then R n ~l~,a(k) = I~. P r o o L Let p : ~fl ~ ~Sa/R be the canonical projection. Obviously, we can restrict ourselves to the case that G is generated by U~, U_~ and T. Let ~ E lr~ and suppose p(() r 1. Since p : ~ a ( k ) --+ (r surjective (k ist algebraically closed), there exist ~', ~" E ~-~,a(k) such that p(~'~(") normalizes p(~) and induces the reflection ra in the Weyl group of ~ (see (0.19)). By using (6.5) one obtains

p(r

e p(~_o,e)(k)p(~)(k), p(~o,~)(k)

Note that for g r Fa, there is an e > 0 with U~,~ = U~,e+~. By (6.3) and (0.18) we know that p ( ~ ' ~ " ) centralizes p(~), in contrast to the assumption. Hence

e n

n

lla,a(k).

Conversely, let us show that for ~ E ll~,t(k) with ~ r I~, we have ~ r R. First ~ + ~ = 0 and g E r~, and because of the surjectivity of Lt~,t(og) --+ ll~,t(k) (see (0.1)) there is an element x E lla,t(Og) with z ~ ~l~,t+~(o~r) for all r > 0. Now we distinguish four cases (notations as in w 1. Case: 89 2a ~ r then there is an element u E L~ with x = Xa(U) and w(u) = g. Let x' = x" = x_~(u-1). Then x'xx" E ~hn(OK), it normalizes G and 64

induces the reflection r , in the Weyl group of 9 (see (4.10), (4.11) and (0.19)). Hence the image of x'xx" in ~fl/R does not centralize the torus p ( ~ ) , from which p(~) # 1 follows. 2. Case: 2a E 9 and x r U,,~+~ 9 U2,,2t for all ~ > 0; then there is an element (u,v) E Ho(L, L2) with z = x,(u,v), w(u) = t + 7 and w(v) = 2l (see (4.23) and (4.24)). Let x ' = x_,(uv -1, (v~ -1) and x " = x_~(u(v~') -1, (v~)-l). Then x':cx" E ~a(OK), it normalizes | and induces the reflection r , in the Weyl group of ~ (see (4.20), (4.21) and (0.19)). Hence the image of x'xx" in r does not centralize the torus p(| from which p(~) # 1 follows. 3. Case: 2a E 9 and x E U,,~+~ 9 Ue~,2t for an r > 0; without loss of generality we may assume that x E U2~,2t. Hence there is an element (0, v) E Ho(L, L2) with x = xa(O,v) and w(v) = 2~. With x' := x" := x_~(O,(v~) -1) one proceeds as in the second case. 4. Case: a ~ ~ d ;

the proof can be done in the same way as in the third case.D

C o r o l l a r y 6.7.

If the notations and assumptions are as in (6.6), then R(k) is generated by all Ia (a E ~) and Ru(~)(k). P r o o f . By [Hu] 7.5 the subgroup of R=(-~fl)(k) generated by Ia (a E q~) and R~(~)(k) is closed and connected. Obviously, its dimension is greater or equal than the sum of the dimensions of R=(T) and all I~ (a E r By (6.4) (iii), (6.6) und (6.1) (v) this sum of dimensions is greater or equal than the dimension of the unipotent radical of ~ . [] 6.8. For a non-empty, bounded subset ~ C_ V, we let ~

= {a E ~ : /fl(a) +

fa(-a) = 0 and ffl(a) E F'a}

= {a E ~ : ah is constant with value in F'~} P r o p o s i t i o n 6.9. __

i

a2~ is the root system of ~ a / R u ( ~ f l ) . P r o o f i By (4.19) we know that Fa = F" U ~F2a and that this union is disjoint. Therefore Off is reduced. In the proof of (6.6) we have shown more exactly that in case a, 2a E 9 and l E F~\F'a we have 1

!

i.e. ~ "only" acts through 2a on the Lie algebra.

On the other hand, the

~I~,~(k)/(R~ M~I~,~(k)) (a E ~fl) are connected, unipotent subgroups which have to be all root subgroups by dimension reasons (see (6.1)). 65

[]

Lernma 6.10.

Let p : - ~ n -+ ~ f l / R , ~ ( ~ n ) be the canonical projection. If Q is the norrnalizer of p ( ~ ) in ~ a / R , , ( ~ n ) , then the canonical map N(K)nOSa(OK) --+ Q(k)/(p(~)(k)) is surjective. Proof.

It is sufficient to show that for all a E ~ a C_ (I), there is an element n E N ( K ) N Oa(OK) which is mapped onto the reflection ra. But this follows immediately from the proof of (6.6). []

66

Chapter The

III

building

in the

quasi-split

case

In this chapter we will construct the Bruhat-Tits building for the case in which K is strictly Henselian and G is quasi-split over K. In addition we will define a (poly-) simplicial structure in the e m p t y apartment. Some constructions will be done in a more general context in order to enable us to proceed in chapter IV in a completely analogous way as here. In chapter III we make the same assumptions as in chapter II. A (poly-) simplicial structure in an a p a r t m e n t will be defined in w Then we are able to generalize the constructions made in chapter II for f~ C V (the P~vector space in (1.4)) to the case fl C_ A (the affine space in (1.8)) and to make t h e m independent of the choice of a Chevalley-Steinberg system. Furthermore, the remaining "V-axioms" ([BT 1] 6.2) will be proved; these are (next to some general properties) the only statements which will be needed later on. Then the groups Pfl for f~ C_ A (see (2.9)) will be defined and studied in w Besides some simple decomposition lemmas, we will prove the existence of the mixed Bruhat- and Iwasawa decomposition. In w the Bruhat-Tits building X(G) will be defined as G(K) x A~ ~. It turns out that Pa (see w is the stabilizer of f~ C_ A. T h e notions of face, chamber and a p a r t m e n t will be generalized to X(G) and some fundamental topological properties of X(G) (hausdorff, metrizible and contractible) will be established. Finally, we will examine the local structure of X(G). If f is a face of X(G), then we will show that X(G) "looks" like the (combinatorial) Tits building of ~)F/R~(~)~) in the neighbourhood of F.

w

T h e full a p a r t m e n t

After having defined the e m p t y a p a r t m e n t in w we fill it now with faces, chambers, walls etc, We fix a point o E A and obtain a canonical isomorphism of affine spaces ~:V

v~

>A

~oq-v .

Let A be a basis of 9 and choose a homomorphism v : N ( K ) -~ Aft(A) such that v(rria)(o) = o for all a E A (for the definition of ma see (4.10) and (4.21)). The following l e m m a implies t h a t we have v(m~)(o) = o for all a E @r~d since for each a E ~y~redthere exist b l , . . . , bn, c E A with r~ = rbl o... orb~ orc orb~lo...or~ 1 . 67

iemma.

Let a,b,a' E ~,~d with a = rb(a'). Then v(m~) = u(mbm~,mbt).

P r o o f . Obviously, rn : = m b m ~ , m [ l m - 2 1 E T ( K ) . Hence it is sufficient to show t h a t m E Tb(K). If c E o ~ d is an a r b i t r a r y root, then we can distinguish two cases: 1) 2c ~ ~: If 3' E ~ with ")'Is = c, then m~ = m~ by (4.14). 2) 2c E ~: If 3` E ~ with "/13 = c, then we get by a simple matrix calculation t h a t m~ = r - m ~ where r E Tb([(). In the special situation of (4.21) we have

0 0) r=

0 0

-1 0

0 -1

Now let a , a ' , 1 3 E ~ with a[s = a, a'ls = a', /3is = b and r~(a') = a. T h e n m o d Tb([(). Therefore it suffices to prove the assertion in the split case.

m =_ rn~rn~,rn~lm21

B y (4.3) and by definition of m~, we obtain

m~m~,rn~ 1 = m,~_~,(1)~,(1)~?_~,(1)m~ 1 = 2_~(s)~(e')~_~(e) for suitable e , r that

E {=t=1}. Since m~rnc,,m~ 1 E

He(T)(/(),

it follows e = r so

m z m ~ , , m ~ 1 = - m s m o d Tb(fi[) . Hence u ( m ) = l.

[]

For a n o n - e m p t y subset ft C_ A let f a : ~5 --+ R U { 4 - o o } be the m a p S~-'fl defined in w T h u s for any non-empty, b o u n d e d subset fl C_ A we can define the groups and group schemes Ua,n, Lta,n, t a , , a etc. according to chapter II.

D e f i n i t i o n 7.1. Let a E ~. An affine function 0 : A -~ ~ is called an affine root (with direction a), if there is a real n u m b e r e E F~ such t h a t 6 = a ( . . . - o) + ~. 68

We will often identify an affine root tO with the half-space tO-l([0, cr If tO : A ~ R is an affine function, then the linear form a : V --+ ~ with tO = a ( . . . - o) + * (for ~ E ~ ) is independent of the choice of o. We will call a the

linear part of tO. If a E r and 1 # u E U~(K), then the elements m(u) in (0.19) and m ( x a l ( u ) ) in (4.10) resp. (4.21) coincide. By (4.11) (iv) and (4.21) (iv) we know that v(m(u)) is a reflection. Notations (i)

7.2.

For a E q~ and u E U~(K)\{1}, let tO(a,u ) : A --+ ~ be the affine function with linear part r~ and the property tO(a,u)-l({0}) = {x E A : v(m(u))(x) = x} .

(ii) For an affine function tO : A -+ ~Z with linear part ra, we let

Uo = {u E U~(K)\{1} : tO(a,u) > tO} U {1}

The canonical h o m o m o r p h i s m N ( K ) --+ "W(= by "v.

Lemma

N(K)/Z(K))

will be denoted

7.3.

Let 0 : A --+ IR be an af-fine function with linear part a E q~ and let n E N ( K ) . Then nUon -1 = Uoo.(n-~).

P r o o f . Obviously, it suffices to show that nUon -1 C Uoo~(n-1). For u E U0\{1} and m(u) = u'uu" with u',u" E U_a(K), we obtain to(a,u) >_ 0 by the assumption. If we can s h o w that n u n - 1 ) -- e(a, u ) o then we are done. We have n m ( u ) n -1 = nu'uu"n -1 = ( n u ' n - 1 ) ( n u n - 1 ) ( n u " n -1) which implies m ( n u n -1) = n m ( u ) n -1 by (0.19). Hence v ( m ( n u n - 1 ) ) = v(n) o v(m(u)) o v(n -1) and therefore O(Vv(n)(a), nun -1) = O(a, u) o v ( n - 1 ) . []

Our next aim is to compare the two filtrations (Us) and (Ua,~). If a E ~, then U - ~ ( K ) ( U a ( K ) \ { 1 } ) U _ a ( K ) N N ( K ) will be denoted by Ma. First we will show t h a t the axioms (V2) and (V5) from [BT 1] 6.2. are valid. 69

Lemma

7.4.

Let a E ~. Then: (i)

For m E M~, the /unction u ~-~ qa_~(u) - ~ a ( m u m -1) is constant on U_~(K)\{1}.

(ii)

For u E U , ( K ) \ { 1 } and u ' , u " E U _ ~ ( K ) with u ' u u " E N ( K ) , ~_a(U I) ~- --~a(U) -~ ~9_a(%t'l).

(iii) For u E V a ( K ) \ { 1 } , we have ~ ( u )

we have

= ~(u-1).

P r o o f . We will prove (i) for the case t h a t a E q5 ~ d and 2a r e2. T h e other cases in (i) and the assertion (ii) follow in a similar way by use of (4.10) and (4.11) rasp. (4.20) and (4.21). Let a E ~ with als = a and let v E L~ with x_~(v) = u. Next let u' E U ~ ( K ) \ { 1 } with m = m ( u ' ) and w E L~ with g ( w ) m a = m (notations as in (4.10) if). Hence by (4.10) and (4.11) we obtain ~_o(~) - ~(m-~m)

= ~_~(x_o(~)) - ~a(~(w)-~m2~_o(~)moa(w)) = ~(~)

- ~o(~(w)-~x~(~)~(w))

= ~(~)

- ~(~(a(~(w-~))~)

= ~(v)

- ~(w-~)

= 2~(w)

which is i n d e p e n d e n t of u rasp. v. Assertion (iii) follows from the fact t h a t all U~,e are groups. Lamina

[]

7.5.

Let a E 9 and m E M~ with m = u' uu" for u', u" E U_~( K ) and let u E Ua( K ) . We let e = ~o(~) If a" de~otes the element in Y dual to the ~oot a (see [Bo~ 1] VI 1.1), then we obtain for all b E (~ and t E Fr mUb,tm -1 = U,.~(b),t-b(a')4

P r o o f . First we know t h a t ~ a ( u ) = - ~ _ a ( u ' ) Now we distinguish three cases:

.

= -~_~(u")

= ~ by (7.4) (ii).

1. C a s e : a , b are linear independent; let 9 = { p a + q b : p E 7Z and let q E IN>0} M ft. If c = pa + qb, then we a b b r e v i a t e ps + qt as h(c). According to (5.6) and (6.1) the set I-[ Uc,h(c) is m a p p e d bijectively onto cE~d

a s u b g r o u p U' of G ( K ) which is i n d e p e n d e n t of the ordering of kg, under the 70

p r o d u c t m a p (notice (4.12), (4.24) and (4.25)). (Since a and b are linear indep e n d e n t , t h e r e exists a point x E A with a(x - o) = - t and b(x - o) = - t . T h e n we can a p p l y (5.6) to f~ = {x}.) We have Ub(K) N g ' = Ub,t and gr~(b)(K) VI U' = Ura(b),h(r~(b)) Since ra(b) = b - b ( a ' ) . a ([Bou 1] VI 1.1), we obtain

gr~(b),h(r~(b))

:

gr~(b),t_b(a').g

9

Finally, g ' is normalized by u, u' and u" ((5.3) and (5.6)), hence by m, too. So we obtain: mUb,tm -1 = mUb( K ) m -1 VI U' = Ur~(b)( K ) fl U' = Ur~(b),t_b(a-)~ 2. C a s e : b = - a ; then u' = ( m u " - i m - t ) m u - 1 ~a(mu'-lm

-1) = - ~ - a ( u ' )

and we obtain

= e = -~_~(u")

= ~ _ ~ ( u " ) + 2e

(7.4).

Since a(a') = 2 (see [Bou 1] VI 1.1), we obtain m U - ~ , t m -1 = Ua,t+2e = U~,t-(-~)(~)4 by (7.4) (i) and (iii). 3. C a s e : b = a; this can be done in complete analogy to the second case.

[]

L e m m a 7.6.

Let a e ~, u e U ~ ( K ) \ { 1 } and let t e T ( K ) .

Then

~ ( t u t -1) = ~a(u) + w ( a ( t ) ) for all ct E ~ with a{s = a.

P r o o f . T h e assertion follows i m m e d i a t e l y from the definition of ~ formulars in (5.8). 71

and the two []

Proposition Lethe

7.T.

~, g E Ft and O = a(... - o ) + g. Then Uo = U~,~.

P r o o f . Let 1 r u e Ua,~. We let g' = ~ ( u ) and m = m(u). T h e n we know t h a t m ~ m e T ( K ) (ma as in (4.11) and (4.21)), and by (7.6) we obtain -1) -

o(u) =

for all a E ~ with Ms -- a. On the other hand, one obtains

~Oa((mam)U(mam) -1) - ~a(U) ------a(aV) " g' by (7.5), if one notes t h a t m~ = m(v) for any v with ~o(v) = 0. Hence w ( a ( m ~ m ) ) = - a ( a ' ) . g' = - 2 g ' (see [Bou 1] VI 1.1). Let g e ~ with 0(a, u) = a ( . . . - o) + g. T h e n we obtain a(u(mam)(o) - o) = 2g. If one regards u as a m a p T ( K ) --+ V (see (1.4)), then a(u(m~m)) = 2g. Since X ~ ( T ) has a finite index in X * ( S ) (proof of (1.1)), there exists a n u m b e r g 9 IN with a N 9 X*I((T ). T h u s (see (1.2))

2 N [ ~ aN (l](mam) ) : --og(aN(mam) ) : --w(o~N (mam) ) = -N.

w ( a ( m a m ) ) = 2Ng'

(by definition of u),

and therefore g = g~.

[]

T h u s we have an invariant description of the filtrations of the groups U~(K). Corollary

7.8.

Let a 9 ~ and g 9 ~ . Then: (i)

For 1 # u 9 U~(K), we have u(m(u)) = r~(... - o) + ~(u)a ~.

5i)

For all t 9 Tb(K), we have tUa,et -1 -= Ua,g.

(iii) For n 9 N ( K ) , we have nU~,tn -1 = U-~(n)(~),~+~(v), /f we let v

=

-

o.

P r o o f . T h e assertion (ii) is clear by (7.6). (i) follows directly from (7.3), (7.5) and (7.7). Ad (iii): Since the assertion holds for n 9 T ( K ) (see (7.6)), we can restrict ourselves to the case t h a t n is of the form re(u1).., re(us) with ui 9 U a , ( g ) \ { 1 } . Let us prove this assertion by induction on s. In the case t h a t s = 0 there is n o t h i n g to prove. So let us assume that s > 0. Let 72

n' := m ( u 2 ) . . , m ( u s ) and let ul 9 Ub(K)\{1}. hypothesis

Then we obtain by induction

nUa,en -1 = m ( u l ) n ' U a , e n ' - l m ( u l ) -1 = m ( u l )U~v(n,)(~),e+a(v,)m(ul ) -1 where v' = u ( n ' - l ) ( o ) - o . Let/9 be the affine function " u ( n ' ) ( a ) ( . . . - o ) + l + a ( v ' ) . According to (7.3) and (7.7) we have

nUa,en -1 = m ( u l )Uem(ul ) - I = Ueov(m(u,)-l) and u ( m ( u l ) -1) = r b l ( . . . -- o) + qOb(ul )b ~. Hence

0 o u ( m ( u l ) -1) = V u ( n ' ) ( a ) ( r b l ( . . . -- o) + qab(ul)b" -- o) + ~ + a(v') = rb o Vu(n')(a)(... -- o) + ~b(Ul)~u(n')(a)(b ") + t + a(v') = ~(n)(a)(...

- o) + e + ~ b ( ~ l ) 9 ~ ( n ' ) ( a ) ( b ' )

+ ~(~')

.

Thus it suffices to prove that

a(v) = qob(Ul). Vu(n')(a)(b') q- a(v') for v = u ( n - 1 ) ( o ) - o. Now we know

~(~) = ~ p ( ~ - l ( o )

- o)

---- a ( v ( n t-1 o / ] ( m ( z t l ) - l ) ( o )

-- o)

= a(u(n'-1)(rb(o - o) + ~b(ul)b') -- o) =

aV~(n'-l)(Vb(~)b" + ~(n'-l)(o) - o)

= Vu(n')(a)(~b(ul)b') + a ( u ( n ' - l ) ( o ) - o) = vb(~)-~(~')(~)(b')

+ ~(~')

[]

Now we are able to define the affine root system in A. Let ( , ) be a ~Winvariant scalar product on V* (~W is the Weyl group of ~, see (1.3)). Such a scalar product exists by [Bou 1] VI w Prop. 3 and is according to [Bou 1] VI w Prop. 7 uniquely determined on every irreducible component of V (resp. ~W) up to a scalar factor. By use of the canonical pairing ( , ) : V x V* ~ R we also obtain a "W-invariant scalar product on V. N o t a t i o n s 7.9. (i) Let 7-/ be the set of all hypersurfaces H in A such that there is an affine root 0 of a with H = 0-1({0}). (ii) Let W~ff C_ Aft(A) be the subgroup generated by all orthogonal reflections s n with H 9 7g (see [Bou 1] V w 73

Proposition 7.10. We have: (i) Wayl is a normal subgroup of finite indez in the image of W in Aft(A) and

(ii) w h e n . P r o o f . Let 0: A --+ N be an affine root and let H = 0-1({0}). If a = a ( . . . - o) + ~ with ~ e F~, then there is an element u e Ua(K) with ~a(u) = L Now the proof of (7.8) shows that u(m(u)) = SH (the reflection at H ) and hence Waif C W. Let a e 9 and let u e U~(K) with e := ~ ( u ) E F~. For n e N ( K ) , let Vv(n) be the image of v(n) under the canonical projection Aft(A) ~ GL(V). By (7.8) and by definition of F'~ we obtain

~,~(n)(a)(nun -1) E Fr.~(n)(a) from which assertion (ii) follows by use of (7.3) and (7.8). Hence Wall is normal in W. In order to prove (i), it is sufficient to show that there is a free abelian subgroup of rank dim V in Waf f. Finally, this is clear by construction. [] Now we are able to make use of the machines of affine reflection groups and of affine root systems (see [Bou 1] V w and VI w

Definition 7.11. (i)

The hypersurfaces H E 7-/are called walls in A. U H are called chambers in A.

(ii) The connected componentes of A \

HEN

(iii) We will call two points x, y E A equivalent, if the following condition is satisfied for all affine roots a: The real numbers a(x) and a(y) have the same sign or are both equal to 0. Obviously, this defines an equivalence relation on A. The equivalence classes are called faces in A. Hence we obtain a (poly-) simplicial complex in A which is a simplicial complex in all irreducible components (with respect to vW). Here "poly" should indicate that we have a direct product of simplicial complexes. Further properties can be found in [Bou 1] V w and VI w 74

L e m m a 7.12.

Let x 9 A. Then there is an ~ 9 I:~+ with the following property: We have x 9 F for all faces F in A with F M Be(x) ~ 0. P r o o f . Since A is locally compact, this follows directly from [Bou 1] V 3.1 L e m m a 1. [] D e f i n i t i o n 7.13. (i) Let f~ _ A. We will call l(a) =

N

{x 9 A: a(x) +~ >_0}

N

(x 9 A : - a ( x ) <_l}

the simplicial closure of ~ in A. (ii) The empty apartment A together with the affine root system and the faces, chambers etc. (hence the (poly-) simplicial complex) is called the full apartment. In the notation we will usually suppress the dependence on the choice of the scalar product on V (see above). Let A C ~ be a basis and let 0 _C A. I f P = Po is the standard parabolic K-subgroup (see (0.20)), then according to (1.11) the empty apartment of P / R ~ ( P ) is given by A/(Fo). Now we choose an o 9 A/(Fol such that the o 9 A is mapped onto it under the canonical projection A -~ A/(Fo). Moreover, we will assume the scalar product on Y/(Fo) to be induced by the scalar product on V. P r o p o s i t i o n 7.14.

We can identify the full apartment of Po with A/(Fo) (notations as above). Let rr : P --+ P / R u ( P ) be the canonical projection. If U~ C_ P with Ua ~ R~(P), then U_a C_ P. So we can define the element m(u) associated with u 9 U~(K) in P/R~,(P), too. Because of the uniqueness in (0.19) this must be our well known element m(u), and we obtain the equality of the filtrations. [] Proof.

Whenever we will in the following consider an apartment of G and simultaniousty an apartment of a parabolic K-subgroup, the scalar products are assumed to be compatible in the sense of what we said above. 75

w The groups U~ and P~ In this p a r a g r a p h we will formulate and prove the most important results from [BT 1] 6.3, 6.4, 7.1 and 7.3. A main part of the structure of the groups U~ and P~ can be understood by considering the case rank(~) = 1. The general case follows then by use of the c o m m u t a t o r rule. Let the assumptions be the same as in w If a 9 9 and u,u' 9 U~(K), then ~a(UU') >_ m i n { ~ ( u ) , ~ ( u ' ) } . ~a(u) ~ ~ a ( u ' ) equality holds.

In case that

Lemma 8.1. Let a 9 r u 9 U~(K) and u' 9 U_~(K) such that ~ ( u ) § ~_~(u') > O. Then there is a unique element (Ul, t, u~) 9 U~(K) • T(K) • U_a(K) with u'u = ultu~. Moreover, we have t 9 Tb(K), ~a(ul) = ~ ( u ) and ~_a(u~) --- ~_~(u'). Proof.

Without loss of generality we may assume that u r 1. First we have

u'u f~ m ( u ) T ( g ) u _ a ( K ) , since in the other case it would follow that u e u ' - l m ( u ) T ( g ) U a ( K ) and from this ~a(u) + ~_~(u') = 0 by (7.4). Contradiction. By (0.18) we know that

L~ = U~(K)m(u)T( K)U_~( K) U U~(K)T( K)U_a( K) = m ( u ) T ( g ) v _ ~ ( g ) U U~(K)T(K)U_~(K) , if L~ denotes the subgroup of G(K) generated by U~(K), U_~(K) and T(K). Hence we obtain the existence, and the uniqueness also follows, since we have U~(K) N T(K)U_~(K) -- {1} according to (0.18). Let us show the supplement: Without loss of generality we may assume that u, u' r 1. Then ul r 1 and by (7.4) there exist u'2, u'3 e U_a(K) with

ul = ut2m(ul)u~3 and ~ _ ~ ( u ~ ) - - - - - ~ a ( U l ) . Hence u = (u'-lu~)m(ul)t(t-lu~tu~) e U_a(g)m(ul)tU~(g), , = - ~ _ ~ ( u , - i u2) according to (7.4). Since

> o we get

_

/ul--l~

t \

<

=

and therefore By using

this and the previous remark we obtain ~_~(u ,-1 u2) , = ~_~(u~) and ~ ( u )

=

=

One can show in a similar way t h a t ~_~(u~) = ~_~(u'). Finally, because of ~ ( u ) = ~ ( u l ) the maps u(m(ul))and u(m(ul)t) = u(m(u)) are b o t h reflections at the same hypersurface (see (7.8)). Hence

u(t) = u(m(ut)-l)u(m(ul)t) = id , from which t 9 Tb(K) follows.

[] 76

P r o p o s i t i o n 8.2. Let a E ~, let ~,[ E ~:~ with ~ + [ > 0 and let La be the subgroup of G(K) generated by U-~,e and U~,~. If we let Ha = Tb( K ) M L~, then L~ = U~,~. U-~,e " H , P r o o f . Obviously, U ~ j . U-~,e. Ha C_ L,, and by (7.8) we know that Tb(K) (in particular H~) normalizes the groups Ua,~ und U~,e. Then the assertion follows from (8.1). [] C o r o l l a r y 8.3.

We get that Ua,~ 9U-a,e" Tb(K) is a group, if we make the same assumptions as in (S.2).

[]

P r o p o s i t i o n 8.4.

Let a E r

let ~ E Fa and let m = re(u) for an element u E U~,(K) with

= e.

Further let L~ be the subgroup of G(K) generated by U~,e and U-a,-e. If we let H~ = L~, 7) Tb(K), then L~ = (U~,eH~U_~,_e+ ) U (U~,emH~U~,e) . P r o o f . Since we have m E U-~,-eU~,eU-a,-e by (7.4), we obtain B := (U~,eHaU_~,_e+) U (U~,emH~U~,e) C_ L~ . From (7.8) it follows that mU~,em -1 = U-~,-e. Hence L~ is generated by U~,e and m, and we have to show that B is a group. According to (7.8) and (8.2) we have BH~ = BU~,,e = B. Therefore it suffices to show that B m C_ B and B m -1 C_ B. We have (U~,,eHaU_~,_e)m = U~,emH~U~,e C_ B and

U~,emH~,Ua,em = U~,eH~mU~,em = Ua,eHaU-~,_em 2 By the remark following (7.1) the element m 2 acts trivial on A, hence it lies in H~. Thus we have shown that 5~,emH~U~,e C_ U~,eH~U_~,_e. By (7.4) we obviously have

U-~,-e C_ U-a,-e+ U U~,emTb( K)Ua,e 77

and by definition of H~ even

U-~,-e C_U_~,,_~+ U Ua,emHaU~,e and we obtain U~,eHaU-~,-e C_B. One can show B m -1 C B in a similar way.

[]

Corollary 8.5.

With the same notations and assumptions as in (8.4) we obtain: (i) The subgroup L~ of G( K) generated by U_~,_~ and U~,~ equals U_~,~U~,t.N~, if we let N~ = N ( K ) n L~. (ii) The subgroup of G(K) generated by U_~,_e, U~,e and Tb(K) equals V--a,--eUa,i" {1, m}" Tb(K). (i) follows from (8.4). Since Tb(K) normalizes the groups U~,t and U-~,-e (see (7.8)), the assertion (ii) follows. []

Proof.

Now let us put together both propositions.

Proposition 8.6.

Let f~ C A be non-empty subset and let a E ~. If L~ denotes the subgroup of G(K) generated by U_~,$a(_a) and U~,fa(a) and if we let N~ := N(K)NL~, then La = U_~,.fn(_,~) 9U~,fn(a) 9N~ Here we have (i) N= g Tb(K), if In(a) + ff~(-a) > 0 or f~(a) = -:l~(-a) • F~ and (ii) Y~ C_ {1, m}Tb(g) and N~ g Tb(K), if fa(a) = - f a ( - a ) E r~ and m = m(u) for u e U~(K) with ~ ( u ) = fn(a). P r o o f . Both assertions follow directly from (8.2)-(8.5) and (0.18).

[]

In order to describe the structure for the case that rank(q?) > 1, we will give some notations first. Furthermore, we "extend" the functions f~ in such a way that ~ C_ A is possible. 78

N o t a t i o n s 8.7.

For ~2 _C A, let 9.1(fl) denote the topological closure of ~ in A. If 0 : A ~ ~ is an affine function with linear part a 9 @ (see (7.1)), then we let (i) Ao = {x 9 A : O(x) > 0} and Ae denotes the topological closure of Ae in A. (ii) For a non-empty subset ~ C A, let fn : ~ --+ ~ U {•162 be defined by fa(a) = inf{l 9 ~ : fl C_ Aa( ..... )+~}. Here we let i n f M = + ~ (resp. = -cr if M = 0 (resp. not bounded-below). Obviously, for f} C_ A, the definition of the function fn coincides with the definition in (7.1). Furthermore, it is clear that /n(2a) = 2fn(a), if 2a 9 ~ and if we let 2. oe = oe and 2. (-oo) = - o c . N o t a t i o n s 8.8.

For a non-empty subset Q C A, we let (i)

Nn = {n 9 N ( K ) : v(n)(x) = x for all x 9 Q} (2.12);

(ii) U~,n = U~,fn(~) for all a 9 @; (iii) Un = (U~,n : a E @) C_ G(K); (iv) u 3 = u • n (v) Pn = (Un,Nn) C_ G(K). If fl = {x}, i.e. it consists of a single element, then we will simply write N=,P~,U= etc. For a non-empty subset f} _C A, the groups Tb(K) and Nfi normalize the group U~ (see (7.8)). Hence P~ = UaN~ and U~Tb(K) are groups. (In [BT 1] w our group P~ is denoted by/5~ and UnTb(K) is denoted by Pfi). P r o p o s i t i o n 8.9. (see [BT 1] 6.4.9)

Let ~ C A be a non-empty subset and fix an order on ~. Then: (i)

U~(K) M Ufl = U~,~ for all a E @,

(ii) the product map

11

aE~+~ed

Ua,~ --+ U~ is a bijection for any ordering of ~2+,

(iii) we have U~ = U~ . U~ . ( N ( K ) M U~), and (iv) N(K)MUfl is generated by all N(K)MLa, if La denotes the group generated by U~,a and U-~,a. 79

P r o o f . Let Y be the subgroup of N ( K ) generated by all N ( K ) ML~ (a e r By (5.6) and (6.1) we know that the image of 11 U~,n in G(K) under the aE~-4-~ed

product map is a subgroup which is independent of the ordering of the factors. We will denote it by D • = (Ur177 First we will show that the set U - 0 + Y is independent of the choice of the vector chamber D defining the ordering of ~. In order to do this, it is sufficient to show that the product does not change under D ~ %(D), if a is a simple root with respect to the order defined by D. Let U" (resp. U--a) be the image in G(K) of

H

Ub,a

(resp. I~ ub,~)

bE~+re~

bE(i, - r e d

b~a

b:~ - a

under the product map. For the same reasons as above, these groups do not depend on the choice of the ordering of the product. Moreover, these groups will be normalized by Ua,n and U-a,n (see (5.3) and (5.6)) and hence the subgroup U-a,nUa,n(N(K) M La) generated by them (see (8.6)), too. Then

U - ( y + y = U~aU_a,nUa,nU'aY = U'_aU~U_a,nUa,~Y = U~U:U~,aU_a,a(N(K) M La)V

= U'_aUa,~U'aU-a,~V

9

Hence U - U + Y is invariant under multiplication from the left by all Ua,a (a C (~+~ed), since for a suitable choice of an order on ~, we know that a is a negative root, and from what has been shown previously it follows that Ua(K)~f-(Z+Y C_ U - U + Y . Furthermore, U - U + Y is closed under multiplication from the left by Y. Hence Ua = U - U + Y . Now let u + E U+, u - E U - and y C Y. If u-u+y E U-, then u+y E ~]-. Since Y C N ( K ) , we obtain y = 1 by (0.18). H e n c e u + E U+(K) M U - ( K ) = {1} (see [BoTi] 5.13). So we have U~ = U - and one can show in the same way that U+ = U+. Therefore the assertions (i), (ii) and (iii) are proved. If u-u+y E N(K), then u-u+y = y by (0.18), and assertion (iv) follows.

[]

If ~ C A is a non-empty subset and if we fix an order on r associated with a vector chamber D, then it follows obviously that Un+~ = U~. The propositions (8.6) and (8.9) give a complete description of the structure of the groups U~.

Corollary 8.10. Let ~ C_-A be a non-empty subset. Then: (i)

If n E N ( K ) , then uPon -1 = P~(~)(~). 80

If we fix an order on q~, then = V

Vf

=

5ii) Pn M U + ( K ) = U+, P~ M U - ( K ) = U~ and P~ N N ( K ) = N~. P r o o f . In order to prove assertion (i), it is sufficient to show that nUa,s~(a)n -1 = U.~,(a),S~,~(a) by definition of P~. In case that f~(a) ~ • this follows from (7.8). In the other cases there is nothing to do because of the classical theory. In order to prove (ii), it suffices to show that N ( K ) N U~ C_ N~ by (8.9) (iii). But this follows directly from (8.6) and (8.9)(iv). By using the definition of U~ and by (0.18) the assertion (iii) follows from (ii). []

In the following we will describe the connection between Pn and P~ for x E ~. Since we will give an interpretation of Pz as the stabilizer of x in the Bruhat-Tits building in (9.1), the proposition (8.12) shows that Pn is the stabilizer of f~ in G(K).

L e m m a 8.11. For x , y E A, there is an order on q~ such that U+ C U~. P r o o f . We distinguish two cases: 1. C a s e : y E A; by (2.7) there exists a vector chamber D such that x lies in the topological closure f / o f y + D in A. Hence if we use the order on ~ defined by D (see (8.8)), then we obtain

u+

cut y+D

2. C a s e : y E A \ A ; let (yi) be a sequence in A which converges to y and has the property that r(yi) = y for all i (y E A / ( F ) and 7r: A --+ A / ( F ) denotes the canonical projection). As in the first case there are vector chambers Di for all i such that x lies in the topological closure of Yi + D~ in A. Since there are only finitely many vector chambers in V, we may assume without loss of generality by passing to a subsequence that D := D~ for all i. By use of (8.9) (ii) it follows that U+ C U u + = U U + _ c v$ [] i

i

y~+D

--

Proposition 8.12. (Generalization of [BT 1] 7.1.11) Forty,C-A,

we have P~ = ~ P~. xE~

81

Proof.

First let us show t h a t we have Pan

(*)

P~ = P a ~ { , }

for a n o n - e m p t y subset ft C_ A and x E A. T h e inclusion Pnu{~} C_ P a N P , is clear by definition. Let y E ft. B y (8.11) we can define an order on 9 such t h a t U + C_ U + C_ U +. Next let g E P a N P~, say g = nvu with n E N a , v E U~ and u E U + (see (8.10)(ii)). Since u E U + C U + , we obtain gu -1 e P~. Hence there exist n' E N~, u' E U + and v' e U~- with nv = n'u'v' ((8.10)(ii) again). We obtain n ' - l n = u'(v'v -1) E U + ( K ) U - ( K ) N N ( K ) and therefore n' = n by (0.18). Hence v' = v and it follows t h a t n E N~ N Nn = Nau{~}, v E U~ N U~- and u E U + N P , . By (8.9)(ii) and (8.10)(iii) we finally get U~NPx=U(~}uW

and U n N P ~ = Uiau{~}) +

,

from which ( , ) follows. So we get by induction t h a t we have P.[,,, ...... ,,} = P , , n . . . n P,,~

for all k E IN and all x l , . . . , xk E f~. T h e Coxeter complex E in V defined by is finite. Hence we can find a finite subset M _C fl such t h a t fl n (A/(F)) r 0 implies M n ( A / ( E ) ) r ~ for all F E E. Now let (fli)iEX be a directed family (with respect to the inclusion ordering) of finite sets f~i _C ft such that M C_ n fti iEI

and U f~i = f~. iEI

T h e n TM := N M n T ( K ) is the m a x i m a l subgroup of N ( K ) acting by translations stabilizing ft pointwise. Moreover, NM/TM is finite. Let g E n P a , . For an a r b i t r a r y order on ~, we can find elements ni E Nf~ iEI

,ul E U+a, and v~ E Uf~ with g = niuiv~, for all i E I, according to (8.10)(ii). Since N M / T M is finite, we m a y assume without loss of generality by passing to a cofinal s u b s y s t e m t h a t there is an element n E NM such t h a t for every i E I there is an element zi E TM with ni = nzi. So we obtain ziui•i

_7_ Z j U j V j

for all i , j E I. Now U+(K) is normalized by T ( K ) and by (0.18) we obtain

u~-lz(lzjuj = viv~ 1 E T(K)U+(K) N U - ( K ) = {1} Hence vi = vj and therefore zi = zj and ui = uj, i.e. zi, ul and vi are independent of i, and lie in TM C Nn, n u.+ fl, = U + resp. n u~, = u ~ (see (8.9) (ii)). _

iEI

iEl

F u r t h e r m o r e , nl = nz~ is i n d e p e n d e n t of i and thus lies in A Nn~ = ?Ca, from which g E P a follows.

~eI 82

[]

C o r o l l a r y 8.13.

For 0 ~ f~, fl I C -A, we have Pr~ N P~, = P a u l ' -

[]

To complete this paragraph, we will prove the generalized versions of the Iwasawa- and Bruhat decomposition. The groups considered here have to be neither compact nor parabolic but of a mixed type. Thus we will call the decompositions "the mixed Iwasawa decomposition" and " the mixed Bruhat decomposition". Since in the following we are only interested in the decompositions but not in the uniqueness of suitable double cosets, we formulate and prove the decomposition properties only. Lemma

8.14.

Let a e o~ and let G~ be the subgroup of G(K) generated by Ua(K), U_a(K) and T( K). Let ~ E R U {=Lee} and let L~ be the group generated by U~,~, U-a,-~ and Tb(K). If m := m(u') for u r e U~(K)\{1}, then Ga C_U~(K)T(K){1,m}L~. P r o o f . For l ~ JR, the assertion follows from (0.18). So let e C •. We let Ga := Ua(K)T(K){1, m}L~. By (0.18) we know that

G~ = U~(K)T( K)mU~( K) U Ua( K)T( K) Hence it suffices to show t h a t mU,~(K) C Ga, since G~ is closed under multiplication from the left by U~(K) and T(K). Let u ~ U~(K). If ~ ( u ) >_ ~, then it follows immediately that m . u E T~. So let us assume that ~ ( u ) < e. Let v', v" C U_~(K) with u = v'm(u)v" and let m' := re(u). By (7.4) we know that ~_~(v') = ~--a(V") = --~0~(U) > --e and we obtain m . u = ( m y ' m - i ) . ( m . m') . v" C U~,(K)T(K) . La C_ Ga, since m and also m r induces the reflection r~ in the Weyl group of ~. Hence

mU_~(K)m -1 = U~(K).

[]

This l e m m a gives us the existence of the Iwasawa decomposition for the case t h a t the rank equals 1. Now the general case will be reduced to this case. Proposition

8.15. (The mixed Iwasawa decomposition, see [BT 1] 7.3.1)

For x e A and any order on ~, we have G(K) = U+(K) 9N ( K ) . P~. P r o o f . Let C, = U + (K). N ( K ) . P~. Obviously, G is closed under multiplication from the left by Ua(K) (a E ~5+) and T(K). It is well known that U - ( K ) is generated by all U _ a ( K ) with a E A, if A is the basis which defines the order on ft. Since G(K) is generated by T(K), U+(K) and U - ( K ) , it suffices to show t h a t U_a(K)G C d for a e A. 83

Let a E ~+ be a simple root and let U~ = (Ub(K) : b E r b # a) C_ U+(K). By (0.17) (iii) the group U~ is normalized by U~(K) and U_~,(K), and we obtain U+(K) = U~U~,(K). Hence

U_a( K)O = U_~( K)U'~U~( K ) N ( K)P~ = U~U_~(K)U~(K)N(K)P~ C_ U~G~N(K)P~ , if G~ is the group generated by U_a(K), U~(K) and T ( K ) as in (8.14). If = f,(a) and if m and L~ are defined as in (8.14), then it follows from (8.6) and (8.14) that

U_=(K)O C U+(K)T(K){1, m}LaN(K)P~ C (U+(K)T(K)U=(K)U_~(K)N(K)P~) O (U+(K)T(K)mU-a(K)U~(K)N(K)P~). Since

U + (K)T(K)U~(K) = U + (K)T(K) and

m U _ ~ ( g ) u ~ ( g ) m -1 = U~(K)U_a(K)

,

we get U-a(K)G C_ U+(K)U_a(K)N(K)P~,. If n E N ( K ) and u E U-a(g), then it suffices to show that un E G. If we let b := -Vv(n-1)(a), then v := n - l u n E ub(g). By applying (8.14) to - b and g := f~(-b) we obtain Ub(K) C_ U_b(K)N(K)P~. Hence

un = nv E nU_b(K)N(K)P~ = n U _ b ( K ) n - l n N ( K ) P ~ : C_ Ua(K)N(K)P~ C &

[]

Let x E A and let D be a vector chamber with x E 92(0 + D) (=topological closure of o + D in A). If F is the vector face with x E A/(F} and if P is the parabolic K-subgroup associated with F (0.20), then R~,(P)(K) is the subgroup of P ( g ) generated by all Ua,~ with f,(a) = -co. For x E A, let .f*: ~ -+ ~ U {+oc} be defined by

fz(a) f;(a) =

inf{g E F a : g > f~(a)}

for fx(a) = -4-oc else

In the sequel we assume k to be algebraically closed. The following lemma is the heart of the proof of the mixed Bruhat decomposition. It goes back to [BT 1] 7.3.8 and is generalized to the A-case here. 84

L a m i n a 8.16.

Let x, y E -A, e E Y (see (1.~)) and let ~ E ~:[+ such that the function *

a

r ~-~ (f,+re())aE~

is constant on ]0, )~]. Then P , . N ( K ) . Py C_ P,+xr N ( K ) . Pg. P r o o f . First the assumption implies that we have P , + . e = Px+~e for all r El0, .k]. Moreover, if r ~-~ ( f ; + , e ( a ) ) a e r is constant on [0, ,~], then P~ = P,+),e and there is nothing to prove. So let us assume the converse. We let ~ = x + ) ~ e . Let g E P x ' N ( g ) ' P y , i.e. there exist u E P , , n E N(K) and fi E Py with g = unft. By (8.10)(ii) we can assume that u E U,. By passing from g, x and v to n-lg, ~,(n-1)(x) and ~v(n-1)(v) we may assume without loss of generality t h a t n = 1, since we have n - l P , n = P~(~-~)(,) according to (8.10)(i). Now let F be a vector face of V with x E A/(F) C A and let P be the parabolic K - s u b g r o u p of G associated with F (see (0.20)). Since Ru(P)(K) C_ P,, P~ by the previous remark, we can further assume without any restrictions that u lies in the subgroup generated by all Ua,/,(~)with f,(a) ~ - o o , in particular it lies in L(K), if L denotes the subgroup of G generated by all U~ with fx(a) ~ -oo and T. As the Levi subgroup L is m a p p e d isomorphically onto P/R~,(P) under the canonical projection P -~ P/R~(P), we will identify these groups in the following. Let x ~ E A be a point which is m a p p e d onto x under the canonical projection 7: : A --+ A/(F) and let D ' be a vector chamber with y E 9 . [ ( x ' + D ' ) ( = topological closure of x ' + D ' in A). Then D ' := ~r(D) is a vector chamber in A/(F). In case that F is a vector chamber itself there is nothing to prove in the following, since L is a torus and A/(F) consists of one element only. Finally, let D be a vector chamber in V with e E D. Then D := 7r(D) is a vector chamber in A/(F). In order to simplify the notations, we will write ~ for the oK-group scheme associated with L (!) and x (see (6.1)). Then the condition u E U~ implies that u lies in r Since k is algebraically closed, R~,(Ox) is a k-group, and we denote the image of u under the canonical m a p {~)ac(og) "4 ~ ) x ( k ) = ~ , ( k ) ---4 (~)ac/Ru(~)x))(k) by ~. By applying (0.18) to O,/R,~(O3x) we obtain elements

and an element ff E ~ x ( k ) normalizing ~ such that ~ = w---V--ff~'(recall that

~x(k) --+ (~,/R~(~x))(k) is surjective). We write H , + ~ for the OK subgroup scheme 11+ of q}~ with respect to the order defined by D and ~.1 ,+~, for the 85

OK-Subgroup scheme ~/x+ of @x with respect to the order defined by D ~. T h u s by (0.1) - - all o K - g r o u p schemes considered here are s m o o t h - - and by (6.10) there exist v E U + ~ ( C L), v' E U+~,(C_ L) and n E N ( K ) which are m a p p e d onto ~, ~' and ~ u n d e r the canonical projection, respectively. From (6.7) it follows t h a t P~(C_ L) is m a p p e d onto R ~ ( ~ ) ( / r under the canonical projection ~)(og) ~ $(/r Moreover, v E U + ~ C_ P~, from which u E P~ 9 n 9 v ~ follows. Let us go back to the subgroups of G (!). Now we know t h a t n E N ( K ) and t h a t P~ with respect to L lies in P~ with respect to G. Finally, we get v t E U , + ~ , , since obviously U , + ~ , contains all subgroups which generate U + ~ , . Since y E 92(x' + D ' ) and U , + ~ , = U 91(x'q-D --, ) (by definition), we obtain v' E U~(~,+~,) C Pu, hence g = unft E P~ . N ( K ) . P~. [] Proposition

8.17. ( T h e mixed B r u h a t decomposition)

For x, y E A, we have G ( K ) = P~ . N ( K ) . Py. P r o o f . Let g E G ( K ) and let D be a v e c t o r c h a m b e r with x E 9 1 ( o + D ) . We consider the order on 9 induced by D. T h e n by (8.15) there exist u E U + ( K ) , n E N ( K ) and p E Pu with g = u . n . p . If ~+~ed = { a l , . . . , a , ~ } , then according to (0.18) there exist u~ E U~,(K) with u = u l . . . u s . Now let v E D. T h e n for all z E x + ] R + v and all i, we have fz(a~) < 0 or fz(ai) = - ~ . Hence there exists a real n u m b e r A0 E ~ + with u E U++~o and therefore

g E Px+~,ovN(K)Py Since w is discrete and (I) is finite we can find a finite sequence of real numbers (Ai)/=l ..... m with the following properties: (i)

),0 > A1 > ... > A,~ < 0.

For i = 0 , . . . , m (ii)

1, we have:

T h e function r ~-~ (f*+,v(a))~e 4 is constant on ]Ai+l, A~[ and

(iii) there exists at least one a E 9 with f~+~,+,v(a) ~ f * + , v ( a ) for r E])~i+l, )~i[. We have the following geometrical interpretation: If x E A / ( F ) , then (x + .kiv)~=o ..... m is a sequence of points such t h a t any two neighbouring points but no three points lie in the closure of the image of a suitable c h a m b e r in

A/(F). Next we will show for all i E { 0 , . . . , m - 1} t h a t g E Px+~,vN(K)Py implies g E P~+;~,+~vN(K)P u. Since g E P~+;~o~N(K)Py, the assertion follows by induction. By (8.16) we know t h a t g E P~+;~vN(K)P u implies g E P,+,.~N(K)Py for all r E]s Now we fix a r E],ki+l,,ki[. Of course, we know t h a t Ua,x+,.v C_ 86

Ua,~+x,+lv for all a 9 O - , and for a 9 O+ we have: f*+,v(a) = f*+x,+~v(a). Hence we obtain P , + , ~ C_ P~+x,+I,N(K) (see (8.10) (ii)) and therefore g 9

Px+:~,+ivg( g ) P y .

w

[]

The definition of the building

In this paragraph we follow (partly word for word) [BT 1] 2.5 and 7.4. Since we will work without any valuation root data, it seems to be of advantege to do the proofs again because of completeness. L e m m a 9.1.

On G ( K ) x A there is an equivalence relation defined by: (g, x) ~ (h, y), if there is an element n 9 N ( K ) with y = g(n)(x) and g - l h n 9 U~. P r o o f . It is clear t h a t this relation is reflexive. Let g , h , i 9 G ( K ) and let x , y , z 9 A with y = u(m)(x), z = u(n)(y), g - l h m 9 Us and h - l ~ n 9 Uy. Then by (7.8) we have

h - l gm -1 = m ( m - l h - l g)m -1 9 mU~m -1 = Uy and

g-ls

= (g-lhm)(m-l(h-ttn)m)

9 U~ . m - l U y m = Us .m

In the sequel X ( G ) denotes the set (G(K) x A)/..~. Before we are able to define a topology on X(G), we have to study some settheoretical properties of X ( G ) and a G ( K ) - a c t i o n on it. L e m m a 9.2.

The map A --+ X ( G )

(x F-+ (~,x)), where (~,x) denotes the equivalence class of

(1, x), is injective. Thus we will identify A with its image under this map in the following. P r o o f . Let x, y E A with (1, x) ~ (1, y). Then there exists an element n e N ( K ) with y = u(n)(x) and n e U,. By (8.10) (iii) we h a v e n E N ( K ) NU~ C_ N~, from which y = v(n)(x) = z follows. [] 87

We have defined a N ( K ) - a c t i o n on A via v (see (1.8)). Now we will extend this to an action of G ( K ) on X ( G ) . In order to do this, let us consider the m a p

G ( K ) x (G(K) • A) ~ G ( K ) x A

(g, (h, x))

(gh, x)

By definition of ~ this induces an action of G ( K ) on X ( G ) extending the N ( K ) action on A. Proposition

9.3.

We have: (i) If f~ C_ A is a non-empty subset, then Pn = {g E G(K) : gx = x for all x E

5i) I r a E 9 and u E Ua(K)\{1}, then the set of fixed points in A under u is the a]:fine root {x E A : a(x - o) + e > 0} where ~ = qOa(U). P r o o f . For f2 = {x} the assertion (i) follows from the definition of X(G). By (8.13) we know Pn = N P , , from which the general case (i) follows. ~Ef2 By (8.9) and (8.10) we know t h a t P~ fq Ua(K) only if ~ > - a ( x ) .

= Ua-a(x)

,

hence u E P~ if and

[]

Corollary 9.4. For all non-empty subsets f~ C_ A with f2 = cl(f2) (see (7.13)), the set of fixed points of Pn in A is f~. Proof.

If f2 is an affine root, then the assertion follows from (9.3).

Since

~2 -= cl(f~) we get t h a t ft is the intersection of all affine roots containing ft, from which the assertion follows by definition. Definition

[]

9.5.

A subset A' C_ X ( G ) is called an apartment of X ( G ) , if there is an element g E G ( K ) with A ' = g . A. For g E G ( K ) , let g. : X ( G ) ~ X ( G ) be defined by x ~-~ g.x. T h e n the following proposition shows t h a t the m a p g. : X ( G ) --+ X ( G ) restricted to A M g - i . A is not a r b i t r a r y but of the form n- for a suitable n E N ( K ) . 88

Proposition 9.6.

(see [BT

1] 7.4.9)

Let g E G(K). Then A M g - l A = cl(A M g - l A ) and there exists an element n E N ( K ) such that g 9x = n. x for all x E A M g - l A . W i t h o u t loss of generality we m a y assume ~ = A M g - l A ~ O. Let By definition of,-~ we get t h a t M contains all subsets of ft consisting of one element. Now let Y, {x} E M and n y , n ~ E N ( K ) such t h a t g - l n y E Py and g-ln~ E P~. We can choose a vector c h a m b e r D such t h a t x E Y + D and hence x + D _C Y + D. If we consider the order on ~ defined by D (see r e m a r k after (8.10)), then Proof.

M = { Y C ~ : g - ~ N ( K ) M P y ~ 0}.

UWY : U +y+ -D C U:+-~ :

T h e r e f o r e according to (8.10) we obtain

nyln~ E PyP~ = N y U y U + U + U ~ N ~ = NyU~U+U~N~ = NyU~U~U+N~ c_ N y U - ( K ) U + ( K ) N ~ !

!

Hence there exist n y E N y and n~ E N~ with

ny'-lny-ln~n~' E N ( K ) M U - ( K ) U + ( K ) = {1} (see (0.18)). T h e n we have g - i n E Py M P~ = PYu{~} for n = n y n ~ = n~n~ by (8.13). Hence all finite subsets of f~ are contained in .Ad (induction). Let x0 E ~t and let (Y~)iEI be a filtration of ft by finite subsets containing x0. T h e n there exist no, ni E N ( K ) (i E I) with

g-lno E Pxo and g - i n i E Py~ C Pzo (i E I) Therefore we obtain n ( l n o E N~ o for all i E I. T h e image o f N , 0 in W (under u) is finite, i.e. we m a y assume without loss of generality by passing to a "cofinal" s u b s y s t e m t h a t all ni (i E I ) coincide (recall t h a t Zb(K) C_ Pu for all i E I). Hence g - l n i E N P~'~ = Pf~, from which the second assertion follows. Finally, iEI

there exists an element n' E N~ such t h a t g-ln~n~ E U~ = Uct(f~) (by definition of the simplicial closure) and therefore g. x E A for all x E c/(f~), which implies the first assertion. [] 89

Corollary

9.7.

Let ~t C_ A be a non-empty subset. Then: (i) Ua acts transitively on the set of all apartments containing ~. (ii) If C is a chamber and g E P c , then gx = x for all x E A M g-1 . A. P r o o f . Let A' = g . A be an a p a r t m e n t containing f~ and let n E N ( K ) with g-1 . x = n - x for all x E A N gA (9.6). Hence gn E P~ and A' = gn . A. Since P~ = U~Na, we m a y ~ s u m e without loss of generality t h a t gn E U~, from which (i) follows. If C is a c h a m b e r , then P c N N ( g ) = N c (see (8.10) (iii)) and assertion (ii) follows, too. [] Corollary

9.8.

N ( K ) is the stabilizer of A and ker(u) = {g E G ( K ) : gz = x for all x E A} (see (1.4)). Proof. Let g E G ( K ) with g . A = A. By (9.6) there exists an element n E N ( K ) with g - i n E PA. According to (9.3) we have UA = {1} and therefore PA = NA = ker(v). [] Now the notions of "face", "chamber" and "sector" should be transfered from a single a p a r t m e n t to the whole building.

D e f i n i t i o n 9.9. Let Y C_ X ( G ) . T h e set Y is called a face (resp. a chamber), if there is an element g E G ( K ) such t h a t g Y C_ A is a face (resp. c h a m b e r ) in the sense of (7.11). According to (9.6) the faces in the sense of (9.9) which lie in A are exactly the faces in the sense of (7.11). T h e following proposition will show t h a t in some sense the faces of X ( G ) are the smallest constituents of X ( G ) .

P r o p o s i t i o n 9.10. Let Y C_ X ( G ) be a face and let g E G ( K ) with Y MgA ~ O. Then there exits a face Y of A such that Y = g Y . In particular Y C_ gA. 90

Proofi Let Y ' C_ A be a face and let h E G ( K ) with h . Y ~ = Y. Hence hA M gA ~ 0 and therefore A M h - l g A ~ 0 because of Y~ M (A M h - l g A ) ~ 0. Since Y ' is a face, it follows from (9.6) t h a t Y ' C_ c/(Y') C_ A ~ h - l g A . Hence by (9.6) there exists an element n E N ( K ) such t h a t (g-lh)-iAnh-,gA = n'lAnh-lgA. T h e n g - l y = g - l h y r = n . yr = : 1~ is a face of A we are looking for. [] Corollary 9.11. Let F, F ~ be two faces of A with F ~ C_ -F and let g E G ( K ) . Then all apartments containing g F also contain g F ~. P r o o f . This follows directly from the proof of (9.10).

[]

T h e following f u n d a m e n t a l proposition shows t h a t there are enough a p a r t m e n t s in X ( G ) . It also yields the geometrical versions of the Iwasawa- and the B r u a h t decomposition (8.15) and (8.17). In the sequel we assume k to be algebraically closed. Proposition

9.12.

We have: (i)

Two chambers (resp. faces, resp. points) of X ( G ) are contained in a common apartment.

(ii) Let ~ be a chamber (resp. face, resp. point), x E A, g E G ( K ) and let D be a vector chamber. Then there exists a point y E x + D such that 12 and g. (y + D) are contained in a common apartment. P r o o f . : B y (9.10) it suffices to show (i) for two points x, y loss of generality we m a y assume x E A and y = g 9 x r with we have G ( K ) = P x N ( K ) P , , . Hence there exist p E P~, n E with g = p n q . T h e n we obtain y = g . x ~ = p n x t E p . A a n d from which (i) follows.

E X(G). Without x r E A. By (8.17) Y ( g ) and q E P,, x=p.x Ep.A,

In order to prove (ii) we m a y assume without loss of generality t h a t 12 = {z} C_ A. By (8.15) we know t h a t G ( K ) = P z N ( K ) U +. Hence there exist p E Pz, n E N ( K ) and u E UD+ with g = p . n . u . As in the proof of (8.17) one can show t h a t there is a point y E x + D such t h a t u~ = 7) for all ~) E y + D. Hence g . (y + D) = p . n . (y + D) E pA and z = pz E pA, which implies (ii). [] Before we are able to define a metric and a topology on X ( G ) we have to study the retractions of the building onto the apartments. 91

L e m m a 9.13. Let A and A r be two apartments containing a chamber C. Then there exists a unique map p = PA',A;C : A ~ --+ A which is the restriction of the map g. to A ~ for a suitable g E G ( K ) and has the property that ~ c : C --+ C is the identity. Moreover, ~AnA' is the identity map. P r o o f . T h e existence is clear by (9.7) (i). Let p' = g' -IA' be a further map with these properties. By (9.6) we know that g~-lg acts on A r as an affine map which stabilizes an open, non-empty subset pointwise. Hence p = p~. Finally, the supplement follows from (9.7) (ii). []

P r o p o s i t i o n 9.14. Let C be a chamber in X ( G ) and Iet A r be an apartment with C C_ A r. Then there exists a unique map p = PcA' : X ( G ) --+ A r such that for all g E P c , the following condition holds: P~.A' = h. ~A' for a suitable h E G( K ) . Furthermore, p satisfies: (i)

i~tA, is the identity and

(ii) if x E C, then f l - l ( x )

= {X}.

P r o o f . In case t h a t the existence is valid the uniqueness follows from the fact that X ( G ) can be covered by a p a r t m e n t s contalng C (9.12)(i). Let x E X ( G ) and let ,4 be an a p a r t m e n t with C U {x} C A (9.12). In order to prove the existence it suffices to show that p,~,A,;C(X) is independent of the choice of A. Let A " be a further a p a r t m e n t containg C U {x}. It follows from the uniqueness in (9.13) that PA",A';C = P.4,A';C o PA",f4;C Now we have x E A" N .4 from which follows pA,,,~;c(X) = x by (9.13). Then the assertion is clear.

[]

In w we have defined a metric d on A using the vW-invariant scalar product on V. Now we will transfer this to the other apartments. 9.15. Let A ~ be a further a p a r t m e n t and let g E G ( K ) with gA = A r. Then (x,y),

) d(g-lx, g-ly)

defines a metric d r on A ~. Since N ( K ) acts by isometries on A, this metric d r is independent of the choice of g by (9.8). 92

By using (9.6) and (9.12) (i) one can show immediately that there is a unique metric d : X ( G ) • X ( G ) --4 F~ such that the restriction to any apartment A' coincides with the metric d t defined above. The metric d on X ( G ) is G(g)-invariant, i.e. G ( K ) acts by isometries. In the following X ( G ) is assumed to be topologized by this metric. P r o p o s i t i o n 9.16. (see [BT 1] 7.4.20 and [Br 1] VI 3 Theorem)

We have: (i)

Let C be a chamber of X ( G ) , x 9 -C and let A' be an apartment with A' c c A'. f~rthermore, let p = Pc" Then d(p(y),p(z)) < d(y, z) for all y, z 9 X ( G ) . If C, y and z are contained in a common apartment, then d(p(y),p(z)) = d ( y , z ) . In particular, d(x, p(y)) = d(x, y) for all y e X ( G ) .

(ii) Let x, y e X ( G ) and let S = {z 9 X ( G ) : d(x, y) = d(x, z) + d(z, y)}. Then S is contained in all apartments containing x and y, and S coincides with the segment [x,y] (see (0.15)) in all these. All isometries fixing x and y also fix S pointwise. In the following we will denote S by Ix, y]. 5ii) Let x, y, z, z' 9 X ( G ) , z 9 [x, y] and ~ 9 ~t+ such that d(x, z') ~ d(x, z) + c . d(x, y) and d(y, z') ~ d(y, z) + E . d(x, y) Then d(z, z') 2 ~_ 4d(x, Z)d(y, z)~ + d(x, y)2~2. (iv) Let x, y 9 X ( G ) , t 9 [0, 1] and let tx + (1 - t)y be the uniquely determined element in [~, y] with d(y, z) = t . d(~, y). Then the map

[0, 1] • X(G) • X(G) --+ X(G) (t, ~, y) ~ t~ + (1 - t)y is continuous. Hence the topological space X ( G ) ist contractible. P r o o f . If there exists an apartment A' containing {x}, {z} and C, then we obtain d(p(x), p(z)) = d(x, z) by definition of the metric. In order to prove (i) it suffices to show the first assertion. Let A" be an apartment with x,z E A ~ and let x = x 0 , x l , . . . , x m = z be a sequence in Ix, z] C_ A" such that for all 1 ~ j _~ m there exists a face Fj with Xj-l,Xj E F j . Then we obtain from (9.12) (i) and the case above m

d(p(x),p(z)) _< ~ d ( p ( x j _ , ) , p ( ~ j ) ) j:l

=

d(xj_,,~j) = d ( ~ , z ) j:l

93

Hence (i) follows. Let A ~ be an a p a r t m e n t with x , y E A ~, let z E S and let t E [x,y] C_ A' with -A I d(x, z) = d(x,t). Let C be a c h a m b e r with t E C. If we let p = P c , then (i) implies

d(x, p(z)) = d(x, z) = d(x, t) and d(y, p(z)) = d(y, z) = d(y, t) Hence p(z) = t and according to (9.14) (ii) we get z = t. T h u s all assertions in (ii) follow. A d (iii): Let A ' be an a p a r t m e n t containing [x,y] and let C be a c h a m b e r of A' with z E C. Let z" = PC A' ( z ) . T h e n d(z, z') = d(z, z") and x, y, z, z" satisfy the same conditions as x, y, z, z ~. Hence it suffices to prove the assertion for the case t h a t x, y, z, z ~ E A. Now this and the assertion (iv) follow by explicit calculations as in the proof of the theorems in [Br 1] VI.3. []

Next we have to show t h a t the faces do not lie too "dense" locally.

Lemma

9.17.

Let F be a face of X ( G ) and let x E F. Then there is an e E F~+ such that F C_ F r for all faces F ~ with F' A B~(x) # O. After a (not necessarily) suitable shrinking of s we can assume that we have g. x = x for all g. x E B~(x) with g E G(K). P r o o f . W i t h o u t loss of generality we m a y assume t h a t F C_ A. T h e n there is an e > 0 such t h a t B~(x) M A only meets faces of A which closure contains F (7.12). Let A ' be a further a p a r t m e n t with B~(x) M A' # 0. Hence by (9.10) we obtain x E A ~ and therefore F C A ~. Since P~ acts transitively on the set of all a p a r t m e n t s containing z (9.7)(i), the first assertion follows. Now let g E G ( K ) with g - x E B~(x). T h e n there exists an a p a r t m e n t A' with x, gx E A'. B y (9.6) we can restrict ourselves to the case t h a t g N ( K ) . Since W ~ i f is of finite index in W (7.10), we only have to consider W ~ f f after a suitable shrinking of E again. If c is smaller t h a n the minimal distance of x to a wall not containing X (locally finiteness), then the assertion follows (see [Bou 1] V 1.3). []

Proposition 9.18. The topological space X ( G) is complete. 94

P r o o f . Let (y~) be a Cauchy sequence in X(G), let C be a chamber in X ( G ) and let (9n) be a sequence in G ( K ) with the property t h a t x,~ -.= g ~ l . Yn 9 C for all n 9 ~T. By passing to a subsequence we may assume without loss of generality t h a t (xn) converges to a point x 9 C. Since

d(gnx, gmx) < d(gnx, g~xn) + d(gnx~, gmXm) -t- d(gmXm, gmX) < d(y,y~) + d(yn, ym) + d(ym, y) for all n , m 9 ~

,

we can find a n u m b e r N E IN such t h a t g,~ -1 gmx E B~(x) for all n , m >_ N, if B~(x) is as in (9.17). Hence by (9.17) we obtain gnx = gmX for n , m > N. So for n > N, we get d(y,~, gn 9x) = d(x,~, x) --+ 0 for n --+ co, from which follows t h a t (Yn) converges to y = gN " X . [] As we have seen in w the pair (A, v) is unique up to a unique isomorphism. The roots, faces etc. defined in w were given canonically. But the scalar product on V, which enables us to define a metric on A, was only unique up to scalar factors. In so far the definition of X ( G ) only depends on the choice of the maximal K-split torus. Finally, we will show that X ( G ) is almost independent of this choice.

Lemma

9.19.

If f : X ( G ) --+ X ( G ) is a G(K)-isometry, then / is the identity map. D

-

-

-

-

P r o o f . Because of G ( K ) C = X ( G ) it suffices to show that f ( C ) C_ C and that f ~ : C --+ C is the identity map. Since the stabilizers of different faces are different (see (9.4) and (8.10) (iii)), we obtain f ( F ) C_ F for all faces F of X(G). Hence f ( C ) C_ C. As a (poly-) simplex C has at least dim A + 1 = dim V + 1 corners, which are m a p p e d (as faces) onto themselves. Since f is an isometry and the metric on A is defined by a scalar product, one can show easily by using methods of linear algebra t h a t f is an affine map. Therefore f~c : C --+ C is an affine m a p and has at least dim A + 1 fixed points, hence equals the identity map. [] Let S' be a further maximal K-split torus and let g E G(K) with S' = gSg -1. As in w we can define the e m p t y a p a r t m e n t A' with respect to S r and as in w we can define faces, chambers and walls in A'. As in this paragraph we can define a G ( K ) - s e t which will be denoted by X'(G). If Int(g) : G --+ G denotes the morphism h ~e ghg -1, then X , ( S ) -e X . ( S ' ) , ), ~-+ Int(g) o )~ induces an isomorphism of R-vector space X . ( S ) | I=~ -+ X . ( S ' ) | ]Ft. This one induces an affine isomorphism e : A -e A' such that for 95

all n e N ( K ) and x e A, we have e(n . x) = gng -1. e(x). Now let us consider the composition --1

A --+ A' C X ' ( G ) ~

X'(G) .

(,)

Then g-1 e ( n . x ) = g - l . ( g n g - 1 ) . e ( x ) = n . ( g - l . e ( x ) ) for all n e N ( K ) and x E A, i.e. the composition is N(K)-equivariant. Since N ( K ) acts on A by isometries and since g-1. is an isometry, we can normalize the scalar product on A' in such a way that the composition (,) becomes an isometric map, too. Note t h a t Int(g) also preserves the Weyl groups. Now let (hi,x1), (h2,x2) 9 G(K) • A with (hi, xl) ~ (h2,x2). Then there is an element n 9 N ( K ) with u(n)(xl) = x2 and h-~lh2n 9 U~1. In order to extend (*) to a m a p X(G) --+ X ' ( G ) , we only have to show according to the N(K)equivariance of (*) t h a t h~lh2n lies in the stabilizer of g - 1 . e(xl). And this stabilizer is obviously equal to g-lPr where P~(~I) is defined with respect to A ~. But from the definition in w it is immediately clear that P~(~I) = gP~lg -1, if P*I is defined with respect to A. Hence h~-1h2 n lies in the stabilizer of g-1. e(xl).

P r o p o s i t i o n 9.20.

There exists a unique G(K)-isometry X(G) --+ X'(G). Proofi Because of remark given above there exists a G(K)-isometric m a p X(G) --+ X'(G). In the same way we obtain a G(K)-isometric m a p X'(G) --~ X(G). The compositions obviously satisfy the conditions of (9.19), i.e. both m a p s are inverse to each other and therefore isometries. The uniqueness follows directly from (9.19).

[]

Definition 9.21. The G ( K ) - s e t X(G) together with the metric, the topology and the structures defined above (faces, chambers etc.) is called the Bruhat-Tits building of G. Let F, F' "C_X(G) be two faces with F C ~ . By (6.2) the identity of G can be extended to an oK-group homomorphism Res FF' : ~ f ' -+ ~ F which induces a .f

t

__

k-group h o m o m o r p h i s m Res F : ~ F ' -~ ~ F / R u ( ~ F ) 9

P r o p o s i t i o n 9.22.

Let X ( F ) be the set of all/aces F' of X(G) with F C_ -~7 equipped with the ordering F' < F ' , if F ~ C F ~'--7. Then the map p: X ( F ) F',

) (combinatorial) Tits building Y(q3F/R~(C3F)) )image ofReSFg' 96

in-~F/Ru(~F)

is a bijection which respects the orderings. P r o o f . Without loss of generality we may assume that F C A. Then the walls of A containing F are the hypersurfaces {x E A : a(x) + fF(a) = 0} for a e ~ F (see (6.9)). If we consider the order on ~ defined by a vector chamber D, then there is a unique chamber C with F C C and C C {x e A: a(x) + fF(a) > 0} for all a E 9 + = O+ N OF. Let A be the basis of 9 with respect to D and let B=AA~F. For J C B, let (J) be defined as in (0.20) and let Fj be the face defined by J withFcFj cC. T h e n we know by (6.7)-(6.9) that p(F2) is the parabolic k-subgroup of OF/Ru(OF) of type J. According to [Bo] 20.9 all parabolic k-subgroups in OF/R~,(OF) are conjugate to a standard parabolic k-subgroup of type J C_ B. Since OF is smooth, we get by (0.1) t h a t OF(OK) -+ OF(k) = -~F(k) is surjective. Moreover, -~F(k) (-~F/R~,(-~F))(k) surjective. Hence p is surjective, too. Now let F ' , F " e X ( F ) with p(F') = p(F"). By (9.12) (i) there exists an a p a r t m e n t A ~ with F ~, F " C_ A ~. Without loss of generality we may assume that A' = A, since by (9.7)(i) we know that UF C_ OF(oK) acts transitively on the set of a p a r t m e n t s containing F. Hence F ~ = F j, for a suitable J' C B' (with respect to a vector chamber D ~) and F" = Fj,, for a suitable J " C_ B " (with repect to a vector chamber D"). By (6.9) we know that ~ F is the root system of OF/R~,(OF) and because of p(F') = p(F") we obtain J ' = J " . Hence we may assume without loss of generality that B ~ = B", so F ' = F " follows. Thus we have shown t h a t p is also injective. []

97

Chapter The

IV

building

and

its compactification

In this chapter we will first construct smooth oK-models of the reductive Kgroup G in case that G is not necessarily quasi-split but OK has a finite residue class field. Then we will use these to construct the Bruhat-Tits building for the same case. Finally, the polyhedral compactification of the Bruhat-Tits building will be defined and its topological properties will be studied. The ~tale descent will be done in w If k is perfect, then G is quasi-split over the strict Henselization K ~h of K according to a theorem of Steinberg [St 2]. Thus we can make use of the results in chapter II and III (for Ggsh) to construct group schemes ~ a with generic fibre G by use of the descent theory. From these we will also obtain the "V-axioms" for the case that G is not quasi-split over K. The paragraphs 11, 12 and 13 are strictly analogous to the paragraphs 7, 8 and 9. Up to the requirement that G has to be defined over a locally compact field, the same definitions and propositions will be formulated and proved. In w the compactification X ( G ) will be defined as G(K) x A / ~-,*, where ~* is a suitably chosen equivalence relation. If we equip G(K) with the Kanalytic topology and A with the topology defined in w then X(G) carries the product-quotient topology. This makes X(G) a topological space which is Hausdorff, compact and contractible. We will examine the boundary OX(G) =

-X(G)\X(G); it

turns out that X-(G) =

U PEa

X(P/R~(P)).

An explicit

P K-parabolic

U

description of the topology on

X(P/R~(P))

will show that the

PCG P K--parabolic

topology on coincide.

w

X(P/R,(P)) induced

by X ( G ) and the canonical building-topology

The 6tale descent

In this paragraph we will show that all properties from w (in case that G is quasi-split over a strictly Henselian field K ) still hold, if G is arbitrary and k is perfect. This will be done by the fitale descent. We will follow [BT 1] 9 and [BT 2] 5. In this paragraph we assume that k is perfect. The residue class field of will be denoted by ks. 98

OK~h

Proposition 10.1. If G is a connected reductive K-group, then GK~. is quasi-split. P r o o f . By [Se] II-10 3.3 c) we know that K is a field of dimension < 1 (a Cl-field). Hence it follows from [St 2] and [BoSp] 8.6 that H I ( K , G ) = O. According to [BoSp] 8.5 the homogeneous space ~3 of the Borel subgroups is a K-variety. By [Se] III-16 Corollary 1 we obtain a K-rational point in if3. Hence there exists a Borel subgroup defined over K. [] Let S be a maximal K-split torus of G, Z = ZG(S), N = AfG(S ) and let = O(G, S , K ) be the root system of G with respect to S. Furthermore, let = Gal(Ksh/g). Let G = GKsh and let T be a maximal K~h-split torus of G (in opposition to chapter II and III not necessarily maximal) with S K~h C_ T, let ~ = r G, T, K sh) be the root system of G with respect to T and let N = AfG(T). The Bruhat-Tits building of G (9.21) will be denoted by _~(G) and the apartments of X(G) will be equipped with a "-~", too. Let A be the apartment associated with T. In the first step we will show that T can be chosen in such a way that there exists a K-torus T with Tgsh = T. In order to do this we will establish a E-action on X(G).

L e m m a 10.2. Let a E ~. Then there exists a unique isometry a: 2 ( a )

) 2(a)

such that a(g. x) = a(g). a(x) for alI g E G ( K sh) and x e f f ( V ) . This isometry preserves all faces of fC ( G). P r o o f . Obviously, T~ is a maximal KSh-split torus of G. Now a : T --+ T~, t ~-+ t ~ induces a linear map X . ( T ) | ~ X.(T~)| Moreover, Z~(T) ~ = ZO(r ~) and a induces a group isomorphism X*(T ~) --+ X*(r By using the notations in (1.4) it follows that W with respect to T is mapped isomorphically onto W with respect to 25% Finally it follows from the invariant definition of the filtrations of the root groups (see w that ~ respects all faces. In particular, a special point is mapped onto a special point. So if .4~ is the apartment associated with ~b~, then we obtain an affine map

ii

(*) 99

such that a ( n x ) ----a ( n ) a ( x ) for all n e IY(K ~h) and x E -~. Then we can extend it to at: X(G) --+ .~(G), since the relation in (9.1) is clearly respected. Now the uniqueness follows from (9.19) by using the usual universal arguments. [] Hence E acts on _~(G) by isometries. By using the following lemma we will be able to guarantee the existence of fixed points. L e m m a 10.3. (Fixed-Point-Lamina of F.Bruhat and J.Tits)

Let E be a metric space and let E ~ C_ E be a subset with the following property: For all x, y E E ~ there exists a point m C E ~ such that the following condition is valid for all z C E~ : d(x, z) 2 + d(y, z) 2 > 2d(m, z) 2 + ~d(x, y)2 If M C E j is a non-empty, bounded subset, then the group of all f E I s o m ( E ) (= group of isometries of E ) with f ( M ) C_ M admits a fixed point in the closure of E' in E. []

P r o o f . see [BT 1] 3.2.3. L a m i n a 10.4. (see [BT 1] 3.2.1)

Let x , y e ) ( ( G ) and let m e -~(G) with d ( x , m ) = d ( y , m ) = 89 (9.16)). Then we obtain

(see

1 d(x, z) 2 + d(y, z) 2 >_ 2d(m, z) 2 + ~d(x, y)2

for

z C :C(G).

P r o o f . Let A' be an apartment of .~(G) with x, y E ,4' and let C be a chamber 4' of _~' with m E C. If z' := gc (z), then, by (9.16) (i), we obtain

d(x, z) > d(x, z') , d(y, z) >_ d(y, z') and d(m, z) = d(m, z') Since, for x, y, z I, m CAf, the equality 1 d(x, z') 2 + d(y, z') 2 = 2d(m, z') 2 + 5d(x, y)2 holds (simple calculation in elementary geometry) the assertion of the lamina follows. [] 100

P r o p o s i t i o n 10.5.

E admits a fized point in f ( ( G ) . P r o o f . Let T! be a maximal K~h-split torus and let A! be the apartment associated with T!. Then there exists a finite Galois extension L / K with L C_ K sh and a L-split torus T ! in GL with T~,h = ~b. By examining the proof of (10.2) we get that ~L = Gal(K~h/L) acts trivial on .4r. Therefore all N-orbits are finite sets, from which the assertion follows by use of (10.3) and (10.4). []

In order to show that E admits a fixed point in Z(K~h).,~, we have to consider the Bruhat-Tits building X(Z). Obviously, the apartment A := A ( X ( Z ) , T , K ~h) of .~(Z) is a quotient of A. Let r : A -+ A be the canonical projection and, for fi/z := Afa(T) n Z, let ~: N z ( K ~h) ~ Aft(A) be the canonical map from (1.8). In the sequel we will make use of the notations from w (in particular, see (8.8) if). Recall, for a non-empty subset f~ C_ A, we let N~ = {n 9 2V(K ~h) : v(n)(x) = x for all x 9 fl}, let Un be the subgroup generated by all U~,fa(a) and let Pa = UaNf~ be the stabilizer of fl in the Bruhat-Tits building (see (9.3) (i)). Moreover, we know that n P ~ n -1 = P~(n)(a) for n 9 N(K~h).

Lemrna 10.6.

Let f~,f~! C A be non-empty subsets, let n , n ! 9 N ( K sh) with n 9 P~,n!Pa and suppose that f~r contains a chamber. Then n 9 (Pn, N N ( K s h ) ) n ' ( P ~ N N(K~h))

.

P r o o f . I f p 9 Pa, and q 9 Pa w i t h p n = n'q, then fl! C A N p A andpnf~ =

n' qf~ = n'f~ C C. f t n pA. Hence by (9.7) there exists an element g 9 G ( K sh) with p.4 = gA and gy = y for all y 9 ~. n p_4. Hence g 9 Pn' N n'Pfln '-1 and !

m := g - l p n = g - l n ' q 9 Pan n n Pfl, N IV(K ~h) Therefore n e ( IV ( K ~h ) n Pa, )m and n' 9 m( N ( K ~h) n Pa), which implies the assertion. [] 101

Proposition

10.7. (see [BT

1] 7.6.4)

There exists a unique map 7r : Z(Ksh)A --+ f[(Z) extending the canonical projection 7r : A --+ A such that the following condition is valid: 7r(z. x) = z. Tr(x) for all z E Z ( K ~h) and x E Z(K~h)A P r o o f . T h e uniqueness is i m m e d i a t e l y clear. Since the c h a r a c t e r s of Z are independent of considering Z as a s u b g r o u p of a greater group or not, it follows from (1.4) t h a t n ~-+ 7r o u(n) for n E N z ( K ~h) e x t e n d s the group action of Z ( K ~'h) on A. Hence we conclude from (1.8) t h a t the action p of N z ( K ~h) on A can be normalized in a unique way such t h a t for all n E 2fIz(KSh), we have: o .(n)

=

o

(,)

.

N o w let x E -A. Then we obtain U~(~) C_ Ux by definition,if we regard U~(x) as a subgroup of Z(K~h). Hence by (8.10) (ii)and (8.17) we obtain z(I

=

and therefore z(K

n F'. =

n

c

(**)

Let g E Z ( K sh) and f2 = ~ f q g - l ~ . Suppose x E i2 # 0. W e l e t f2' = f ~ M ( x + L ) where L = {v E r~ : a ( v ) = 0 for all a E ~) with his = 0} and V denotes the vector space of directions of A. By (9.6) there exists an element n E N ( K ~h) such t h a t for all y E f~, we have 9.y = n.y. Let Cz be a c h a m b e r in A and let C be a c h a m b e r in A with re(C) = Cz. For y E Cz, we obtain Z ( K ~h) = Uyfi(z(K~h)u,~(~) by applying (8.10) (ii) and (8.17) to Z. B y definition we have Uu = Ucz C Uc and moreover, U.(,) _C Pf~,, since U~(x') C U~, for all x' E -4. Therefore we obtain

g E nPa M Z ( K ~h) C_nP~ n UcNz(K~h)P~, , and hence there exists an element nl E N z ( K ~h) with n E PcnlPf~,, since ~2' C f2. According to (10.6) and (8.10) (ii) we obtain n E (1Q(K~h)MPc)nl(N(KSh)M Pa') C (f((K sh) A Pc)Hill(n,. Hence we get n E n l N a , by (9.8). Altogether, we obtain g E nP~ fq Z ( K ~h) C_ nllVa, Pa M Z ( K sh) C nlPa, M Z ( K sh) = nl(Pa, f) Z(Ksh)), since nl E Z(K~h). Hence we get

9 E nl(Pu fq z(gsh)) C nlP~r(y) = nlPTr(~) (see (**)) for all y E i2'. Now if x E A and g E Z ( K sh) with g . x E .4, then there exists an element nl E N z ( K ~h) with 7r(gx) = 7r(ntx) = nlTr(x) = gTr(x) (see (.)). []

102

C o r o l l a r y 10.8.

We have: (i) 7r-l(A) = A, and the pre-image of any apartment under zr is an apartment. (ii) E admits a fixed point in Z ( K s h ) f i .

P r o o f . Let x 9 A and g 9 Z ( K ~h) with 7r(gx) 9 -A. By (10.7) and (9.6) there exists an element n 9 fiiz(K ~h) with g 9 nP,~(~). Since P~(z) = 2Qz,,r(~)U,~(~) C_ fi/z(K*h) 9P~ (see (8.7)), it follows g. x 9 N z ( K * h ) . x C_ ft. Hence (i) follows. By (10.5) we know that E admits a fixed point in X(Z). By (i) we get that ~r-1 (x) is an affine subspace of an apartment which is E-invariant. Since E acts on it by isometries (see (10.5)), the assertion follows. []

Next let us show that the group schemes 03a (f~ C_ .4) from w and w "descent". P r o p o s i t i o n 10.9.

Let r be a smooth, affine Ogsh-group scheme of finite type with generic fibre Ggsh. Then the following statements are equivalent: (i) OK,h[r

is E-invariant and

(ii) there exists a smooth, aj2fine oK-group scheme 03 of finite type with generic fibre G such that ~ = 0 3 o ~ .

P r o o f . Because of OK,h [~] _C Ksh[G] = l~ sh O K K[G] the assertion (i) makes sense. It is clear that (i) follows from (ii). So let us show the converse. Since ~ is of finite type, there exists a finite Galois extension L / K with L C_ K sh such that ~ = 03# OK8 h where 03# is a smooth, affine oL-group scheme of finite type with generic fibre GL. Then the existence follows by 6tale (rasp. Galois) descent (e.g., see [BLR] 6.2). []

Two oK-group schemes as in (10.9) (ii) are obviously isomorphic over a finite extension L / K . Hence by descent theory ([BLR] 6.2) it follows that the group scheme 03 in (10.9) (ii) is unique up to a unique isomorphism.

In the following we will denote the group schemes On, defined in w and w for f2 C A, by 03n. 103

Corollary

10.10.

Let f~ C ft be a non-empty, bounded, E-invariant subset. Up to a unique isomorphism there exists a unique smooth, aJfine oK-group scheme ~ n of finite type with generic fibre G such that ~ a = ~)~ xoK OK~h. P r o o f . Let a E E. T h e n f ~ = Ft and we can construct with respect to T~ an o K - g r o u p scheme ~5~ as in (6.1). Because of the uniqueness in (6.1) we obtain OK~h[~5~] = OK~h[~Sa] 9 Since Ua C_ ~Sa(OK) (6.1), we can find an elem e n t g E ~Sa(OK) with g . fi~ = A (see (9.7)(i)). If we let Int(g) : ~ a --+ ~Sa, h ~ ghg -1, then it follows from the uniqueness in (6.1) (g stabilizes f~), t h a t Int(g)*(OK,~ = OK~ so t h a t OK,h [~a] is a-invariant. T h e n the assertion is clear by (10.9). [] ~

~

[~a])

O"

[~h],

For a E-invariant subset f2, we will denote the oK-group schemes which have been descended by r Now let ~7 be the canonical Og,h-grou p scheme associated with 7~ and let | the canonical o K - g r o u p scheme associated with S (see w Proposition

10.11.

be

(see [BT 2] 5.1.10)

Let F be a E-invariant face of X ( G ) with F C_ A and suppose one of the following objects is given: a)

a K-split torus $1 in G with S1 XK K ~h C_ T;

b)

a k-split torus 61 in ~)F.

Then there exists a K-torus T r such that T~K.~h is a maximal K 8h-split torus with F C ft' (the apartment associated with T'K~) and the following property:

a) ~1 C_ T' ; b)

If T ~ is the canonical oK-group scheme associated with T t, then this one contains a oK-subtorus isomorphic to (Gin/oK)~ such that its image in ( ~ F )k admits 61 as special fibre.

P r o o f . In case a) let G1 be the canonical oK-group scheme associated with S1 and let u : | • OKs, -+ T be the extension of the inclusion S1 ~ T (3.4). In case b) let ~ : | -+ OF be the identity. By (6.3) we know t h a t T is a m a x i m a l ks-split torus of (~SF)ks. Since ks is algebraically closed, 2[ is a m a x i m a l torus. Now let ~ be a m a x i m a l torus which is defined over k and contains ~ ( ~ 1 ) (see

[SGA 3] Exp. XlV 1.1), and let y ~ ~F(ks) with ~' = g-~y-x. In case a) we get t h a t ~ lies in the centralizer of 61. 104

By (0.13) there exists an OK-Subgroup scheme ~' of ~F which is OK,h-isomorphic to (lI~m/oKsh )~, where r = d i m ~ ' = d i m ~ = dimT, and has special fibre ~ . Since ~F is smooth, there exists a pre-image g of ~ in ~ f ( o K , h ) such that T.' = g,~g-1. Since "~F(Og,h) is generated by UF+, UF and T b ( g sh) (see (6.1)), we know by definition of PF in (8.8) that g C PF. Ad case a): Without loss of generality we can choose g such that | is centralized by g, since, according to [SGA 3] Exp. XI 5.3 and 5.9, the centrahzer of | can be represented by a smooth oK-group scheme ((0.1) is used again). Hence T' = gTg -1 is a maximal K'h-split torus definied over K with F = gF C gA' and S 1 ~ T t. In case b) we can extend the k-isomorphism ~ : (~m/k)" --+ ~(~1) to an OKgroup morphism v : (~m/oK)~ -+ ~' which is a closed immersion by (0.13) and [SGA 3] Exp. IX 2.5 and 6.6. The remaining assertions follow from this. []

C o r o l l a r y 10.12.

We have: (i) If F is a E-invariant face of 4, then ~ is a maximal k-split torus of

-~F"

(ii) There exists a maximal Ksh-split torus which is defined over K and contains S. P r o o f . (i) is clear by (6.3) and (10.11). Let F be a E-invariant face in Z ( K ~ h ) 4 (see (10.8)). Then it is contained in an apartment A' of X(G) with T' _D S, if T' denotes the torus associated with 4'. By using (10.11) the assertion (ii) follows. []

So in the sequel we can assume that T is of the form Tg,h where T is a K-torus. Then A := 4 f3 )((G) ~ is a non-empty affine subspace, since 4 is E-invariant and E acts as a group of isometries (cf. (10.5)). Since 4 is an affine re-space with V = (X.(T)| ~)/V0 (see (1.4)), we get that A is an affine V-space with Y = V~ = (X.(S)| (see w and [SoTi] w Definition 10.13.

Let M C -~(G) be a non-empty subset and let F be a face of )((G) with F N M # 0. F is called maximal with respect to M, if for all faces F ' # F of )((G) with F C_ ~ ' , we have F ' N M = 0.

105

L e m m a 10.14.

Let A be an apartment of f((G), let M be a convex subset of fi, let F be a face of fC(G) maximal with respect to M and let L be the affine subspace generated b y F N M . Then: (i)

M C_ L,

(ii) the interior of F N M in L is non-empty and (iii) d i m ( F N M) = dim M. (iv) If F' is a further face of f~ with F' n M # 0 and if the interior of F' N M in L is non-empty, then F' is maximal with respect to M, too. P r o o f . Let x 0 , . . . ,XdimL C F N M C_ L be in general position, i.e. Xl x 0 , . . . ,XdlmL -- x0 are linear independent. Since F and M are convex, F n M is also convex and F n M contains the convex hull of {x0,... ,XdimL}. But obviously this one contains inner points, from which (ii) follows. Obviously, (iii) follows from (i) and (ii). So let us prove (i): Let z E M with z ~ L, let x E F N M , y E F N M and let (F) be the affine subspace of A generated by F (recall F C_ 4). Hence all inner points of the triangle with the edges x, y, z lie in M. Then we distinguish two cases: 1. C a s e : z e (F); since F is open in (F), we obtain ]y,z[NF # O, and for u E]y,z[nFweobtain]x,u[C_ F N M . Since ]x, y[C_ F N M , we get z e L, in contrast to the assumption. 2. C a s e : z r (F); then obviously [y, z ] n ( F ) = {y}. Hence there exists a face F ' with F C_ ~ ' and ]y, z[NF' # 0. So we obtain ]x, t[_C M N F ' for all t e]y, z[nF', in contrast to the assumption of the lemma. Ad (iv): Let F " be a face which is maximal with respect to M and has the --t! property F ' C_ F . Then dim F = dim F " , since: Let n be the dimension of the intersection of all affine roots containing L. Because of L _C (F} and L C_ ( F " ) ((ii) for F " ) we obtain d i m F , d i m F " > n. On the other hand, F and F " lie in all affine roots containing L, hence dim F = dim F " = n. In the same way one can show that the assumption implies that dim F = dim F ' , from which dim F ' = dim F " and hence the validity of (iv) follows. [] Proposition

10.15. (see [BT 2] 5.1.14)

Let F be a face of ft with F N A # 0 (hence F is E-invariant) and the property that for all faces F' # F of ft with F C_-F', we have F ' N A = O. Then: (i)

F n f~(G) ~" = F N A is an open subset of A, and A is the smallest a]:fine subspace of f~ containing F n A. 106

(ii) F is maximal with respect to X ( G ) ~. (iii) For all apartments A' with F C__ ~i', we get that F N X ( G ) ~ is an open subset of ft' n A. (iv) For all apartments 7t' with A C_ A', we obtain fl' N f ( ( G ) ~ = A. P r o o f . (i) follows directly from the assumptions. So let us show (ii). Obviously, the parabolic k-subgroups of @g are the H-invariant parabolic kssubgroups. It follows from the way we have constructed the H-action in (10.2) that exactly these are mapped onto the H-invariant faces F ~ with F C F ~ under p-1 (9.22). By (10.12) we know that | is a maximal k-split torus. Hence all parabolic k-subgroups of r F are conjugate to a parabolic k-subgroup containing ~ (see [Bo] 20.9). All these contain the centralizer of ~ and therefore ~ (6.3). But these are exactly the images of H-invariant faces F ' in ~] with F C_T ~ under p (9.22). Now let x E F'. Then Hx C_ F ' is a finite set (see proof of (10.5)). By (10.3) there exists a fixed point in F ' under H and therefore F ~ n A ~ 0. According to the assumption we obtain F = F ~, i.e. (ii) is valid. (iii) is clear. By using M := fii'N X(G) ~, (10.14) and (iii) it follows that .4' N .~(G) z C_ .4' N A = A. Finally, the converse inclusion in (iv) is clear by the assumptions. [] Next we will show that A is invariant under Z ( K ) and that Z ( K ) acts on A in the way defined in w L e m m a 10.16.

We have Z ( K a h ) f t N X ( G ) ~ = A. P r o o f . Let 7r: Z ( K S h ) A -+ f ( ( z ) be as in (10.7). First, let us show that 7r(A) only consists of one element. In order to do this, it suffices to show by construction that

~ z C (v E ~Z : a(v) = O for a E ~ w i t h a [ s = 0 } Since 17~ can be identified with V = X , ( S ) | diately.

.

~ , the assertion follows imme-

Let z E Z ( K 8h) and x E A. Then there exists an apartment A1 of )~(Z) with re(A) C_ A1 and 7r(zx) e A1 (see (9.12)). By (10.8) we know that r - l ( A 1 ) is an apartment of -~(G) containing A and z. x. If z. x e X(G) ~, then the assertion follows by use of (10.15) (iv). [] 107

Proposition 10.17. For x E A and z E Z ( K ) , we have z . x = x + ~(z) where ~, : Z ( K ) -+ V is the map defined in (1.4). P r o o f . Let ~ : Z ( K ) x A --+ V be defined by =

z.

x -

x.

By (10.16) this definition makes sense. If z, z' E Z ( K ) and if x E A, then a simple calculation shows t h a t O(zz',x) =

+

0(z',z)

Since Z is the centralizer of S, we obtain =

+O(z,x)

for all z E Z ( K ) and s E S ( K ) . W i t h /-]1 : T ( K s h ) --~ ~z as in (1.4) we obtain 0(s, x) = ~'l(S) = 0(s, zz). Hence we get 0(z, sx) = 0(z, x) for all s E S ( K ) . Since z . [A is an affine m a p (see (9.6) and recall A C_ A A z - l A ) , it follows t h a t there exists a vector O(z) E V with 0(z) = O(z,x) for all x E Y. Now one can check by a simple calculation t h a t ~ : Z ( K ) ~ V is a group h o m o m o r p h i s m . Hence we know t h a t 0iS(K) = ~'[s(g). Furthermore, ~ ( S ( K ) ) has a finite index in v ( z ( g ) ) (see (1.3)). Hence it is sufficient to show t h a t O ( S ( K ) ) has finite index in O ( Z ( K ) ) , too. By (9.6) we know t h a t z-[A = n . [A for a suitable n E N ( K ~ h ) . So the assertion follows. [] Lemma

10.18.

For n E N ( K ) ,

we have n . A = A.

P r o o f . Obviously, we know t h a t n T n -1 is a maximal K~h-split subtorus of Z, and therefore there exists an element z E Z ( K ~h) with n ~ = z - i n E ./V'c(T)(Ksh), since Z is reductive. Hence by (10.16) we obtain

n A = (n~t) M f ( ( G ) ~ = (zn'f~) N f ( ( G ) ~ = (z f4) M f ( ( G ) ~" = A .

[]

According to (9.6) the multiplication by n E N ( K ) is an affine m a p on A. Hence by (1.8) we get t h a t A is the a p a r t m e n t A(G, S, K). 108

In order to define faces, c h a m b e r s etc. as in w we have to define filtrations of the root groups. In the sequel roots in ~ will be denoted by Greek letters and roots in 9 by Latin letters. We choose an o E A as "0-point". Note t h a t o need not be a special point, i.e. o need not have the properties discussed in w (see also the r e m a r k in [BT 2] 5.15).

T h e n all F~ ( a E ~ ) are E-invariant, i.e. F~ = I"~(~) for all a e E, since for u E U~(Ksh)\{1}, we have (see (7.8)(i)): ~(u)

o = ~(u)~

~+ o

and m ( u ~ ) 9o = ~ ( ~ ) ( ~ ) ~ ( ~ ) ~

B y (0.19) we obtain

+ o

m(u ~) = re(u) ~"and since o ~ = o, we get ~(~)

= ~(~)(~)

Therefore we also obtain H~(~),~ = a(Ha,t) for all a E E, a E ~ and e E ~ , if one makes use of the uniqueness properties stated in (4.12), (4.24) and (4.25).

N o t a t i o n s 10.19. We fix an order on (~. (i)

For~ENandaE

~,let

Ua,e = I I U~,e. I I a15 = a

U~,2e

c~[s = 2 a

and let Ua,e = Ua(K) M Ua,~. As in (4.8) and (4.14) we make the usual conventions for the case t h a t ~ = +oo. (ii)

For a E ~, let Ta :

Ua(K) -+ IR U {co} be defined by

(iii) Let Ca = {~a(u) : u E U a ( K ) \ { 1 } } C_ ~ and l e t (iv) F" = {~a(u) : u E U a ( K ) \ { 1 } and ~ a ( u ) = U2a(K) := {1}, if 2a ~ ~.

109

sup~a(uU2a(K))}, where

Lemma

10.20.

We have: (i) Every U~,, (a E ~, ~ E ~Z) is a group independent of the fixed ordering. (ii) If a, b E 9 and if ~,[ 9 ~ZU{• is contained in the group

with b ~ - R + a , then the group (U~,~, Ub,~)

{Up~+qb,pt+q~ : p, q E IN>0 and pa + qb 9 ~} P r o o f . (i) follows directly from (5.2), (5.6) and (6.1). As in (5.2) it suffices to prove the assertion for ~ , [ E ~ . Let a,/3 E ~ with a]s = a and/3is = b (in the other cases the proof can be done in a similar way). T h e n by (5.2) we obtain (U~,~, U~,i) C_ (Up~+q~,p,+q~: p, q E lN>0 and pa + q/3 E ~} =: M Since b !~ - ~ + a , N

.

we get t h a t M is contained in :=

(fffpa+qb,ps

: P, q E IN>0 and pa + qb E r

F u r t h e r m o r e , it is clear t h a t N is normalized by U~,, and UZ,i. T h e n (ii) follows i m m e d i a t e l y from the following group theoretical lemma. [] Lemma

10.21.

Let G be an abstract group, let H , [ I C_ G be subgroups and let {Hi}ie! and {/-Ij}jeJ be families of subgroups of H and [I which generate these groups. Let F C_ G be a subgroup which contains (Hi,ftj) (i E I , j E J) and is normalized by all Hi (i E I) and [-Ij (N E J). Then (H,/:/) C_ F. P r o o f . First let us show t h a t (Hi, i-I) C F for all i E I. Let a E Hi, b E /7/, say b = bl ... bn with bj E ~Iij, and let n be the smallest natural n u m b e r such t h a t this t y p e of decomposition exists. T h e n we will show by induction on n t h a t

(a, b) e F. For n = 1 the assertion is clear. So let n > 1. T h e n

aba-lb -1 = a b l . . . b n a - l b ~ l . . . b ~ l =abl...bn_la-l(abna-lb~l)b~l...b~

1

=(a, bl...b~_l)(bl...b~_l)(ab~a-lb~l)(bl...b~-l) In a similar way one can show t h a t (H,/7/) C_ F. 110

-1 E F []

Lemma 10.22.

Let L / K be a finite Galois extension with L C_ K `h, let ~L M be a free oL-module of finite type. Then:

:

G a l ( L / K ) and let

(i) For all semi-linear actions of EL on M and all i > 1 we have HI(EL, M ) = {0} and HI(EL, G L ( M ) ) = {1}. (ii) Let :E be the canonical OL-Scheme associated with M and we equip it with the canonical group scheme structure and the natural Gm-action. Let (a, x) ~4 x a be an action of ~L on ~ such that for all a, the map x ~-~ x a is a a-automorphism of ~ and the property that (tx) ~ = tax ~" for t E Gin(L) and x E ~E(L). Then there exists a free oK-module M # of finite type with 2~ ~ :E# • o~: OL where ~E# denotes the canonical oir associated with M # and equipped with the canonical group scheme structure. Moreover, this isomorphism is compatible with both actions. []

P r o o f . see [BT 2] 5.1.17.

In the following propositions we will apply this lemma to the extension K ~ h / K . In doing this a problem comes up, since all group schemes considered in chapter II are defined over K "h, but, in general, K s h / K is not finite. Since G is defined over K and since all schemes which will be considered are of finite type, we may assume without loss of generality that all group schemes and all morphisms are defined over a finite Galois extension L / K with L C_ K sh. P r o p o s i t i o n 10.23. (see [BT 2] 5.1.18) Let a E ~, ~ be the stabilizer of a in E, let K s = (Ksh) ~ valuation ring of K s . Then we obtain for l E F~ : (i)

and let og~ be the

The Og.~-group schemes Ha,t, ~2s,~ and i/s,~/:t/2s,2t arise by the base change OK~ --40K.h from OK,-schemes associated with suitable free og~ -modules of finite type equipped with a suitable group scheme structure compatible with the T.-action.

( ii) The map ~l~,~(OK~ ) --4 (~la,t / ll2s,2t ) ( ks ) is surjective (ks=residue class field

of OK.). P r o o f . Since all group schemes considered here are smooth, the existence of the schemes in (i) follows from (10.22) (ii). Since lla(s),t = a(lls,t), it is clear that the group scheme structure also descends. By (10.22) (i) we know that H 1(Ks, l/2~,2~(OK,h)) = {0} and by construction we get that the canonical map s --4 (ll~,e/U.2a,2e)(OK,h) is surjective (see (4.25)). Hence s --4 (~ls,~/~12~,2e)(OK,) is surjective. Since ila,e/~12a,2e is smooth, the assertion (ii) follows from (0.1). [] 111

The following proposition makes the behaviour of the filtrations of the U s ( K ) under 4tale descent clear. Later we will see that this proposition gives us the relationship between the faces before and after the descent.

Proposition 10.24. (see [BT 2] 5.1.19) Let a E q~ and ~ E ]Ft. Then the following statements are equivalent: (i) t E F" and 5i) there exists a root a E ~2 with his = a and ~ E F~. P r o o f . For ~ E F~, there exists an element u E U~,t with

II

II vs, ,

sE~ sls=a

(,)

sE~'~d ~ls~2a

by definition o f F ' . Therefore there exists a root a E (~ with Us,i ~ Us,e+'U2s,2t, from which t E F " follows. Conversely, let a E ~ with his = a and let t E F'~. Further let M be the free OK -module associated with ~ls,e/~12s,2t and let I be the canonical image of U~,~+U2s,2~ ___~s,~(OK~h) in (~s,t/~/2s,2e)(k~) = M | k~. By definition of F'~ we get t h a t I is a proper k~-subspace of M | k~ which is Es-invariant (see (10.23) (i)). Hence I ~ (LG,~/~12~,2e)(ks), and from (10.23) (ii) it follows that there exists an element v E U~,~(og,) := Us,t M G(K~) with v ~ Us,e+ 9 U2s,2t. Now let x =

1-I

v a (note that v ~ = v for all cr E ~ a ) and

= { a ( a ) -4- a'(a) : cr a ' E ~} M ~. Then k9 together with the constant valuation function g : a ~ 2g satisfies the conditions of (5.5), and we can consider :E := s which is associated with a free OK,h-module of finite type. Obviously', :E is invariant under ~ and | acts on X through the root 2a. By (10.22) (ii) we know that :E arises by the base change OK -+ Og,h from an OK-group scheme :E# associated with a free oK-module of finite type. Next (10.22) (i) shows t h a t HI(~,X(OK,h))= {0}. Now the c o m m u t a t o r subgroup of the group (U~(s),t : (a E ~)) is obviously contained in :E(OKsh ) (5.2) and so the m a p a ~-+ x - i x ~ is a 1-cocycle with values in ~(Og~h ). Hence there exists an element y E 2~(OKs~) with x - i x a = y - l y a for all a E ~ / ~ s and therefore u := xy -1 E G(K). So u E Ua,~ and u satisfies (.). []

Finally, we will prove the validity of the remaining "V-axioms" (see [BT 1] 6.2). 112

Lemma 10.25.

Let a E 9 and ~ E ~t. equivalent:

Then, for u E Ua(K), the following statements are

(i) ~a(U) = ! and (ii) A e X(G) ~ = a~,~ := {x E A : a(x - o) + ~ >_ 0}. P r o o f . By definition of ~ and A it follows that we have U=,t 9x = {x} for all x E a=,t (see (9.3)(ii)). This means (i) implies a~,~ C_ A n f ( ( G ) ~ and (ii) implies <

Now let x E A with u. x = x and let ~ := - a ( x D

=

o). If we let ,

aE4 ~[S ~(~

then obviously ~ := x + D C_ )((G) ~. Choose an order on ~ such that all roots a E ~' with his = a are positive. Then u E P~ N U + ( K ~h) and therefore u E U~,e (see (8.10) and (9.3)). Hence (ii) implies ~ ( u ) _> ~ and (i) implies A N f~(G) ~ C a~,~, from which the equivalence follows. L e m m a 10.26. (see [BT 1] 9.2.7)

Let F be a face of X ( G ) with F N A ~ @ which is maximal with respect to A and let C be a chamber of fi with F C -C. Then the image of X ( G ) ~ under pA : f~(G) --+ f~ is equal to A (see (9.14)). Moreover, Pci2(G)~ is independent of the choice of the chamber C. P r o o f . Let x E X(G) ~ and let A' be an apartment containing x and F (see (9.12)). Hence F N )~(G) ~ = F N _~(G) ~ N A is contained in the fixed point set of p~ (9.14). Let L be the affine subspace of A' generated by F n )~(G) ~. Then L is mapped onto the affine subspace of .4 generated by F N )~(G) ~ under pA By (10.15) (i) we know that this is exactly A. We let M = A' N .~(G) ~. Since E acts by isometries on X(G) (9.15), the segments in )((G) with endpoints in ,Y(G) ~ are fixed pointwise. Hence M is convex and from (10.14) (i) (notice (10.15) (ii))it follows that x E L and therefore pAc(x) E A. Moreover, by (10.14) (ii) the interior of F N ) ( ( G ) Z in L (and in A) is non-empty, and therefore pA is the uniquely determined affine map from L to A which equals to the identity on F n )((G) ~. [] ~

113

Proposition 10.27. Let a E ~, u t E U ~ ( K ) \ { 1 } and let m = m ( u t) (see (0.19)). Then there exists a v e c t o r t E V such that for all u E U ~ ( K ) , we have T _ ~ ( r n u m -1) = p~(u) - a(t)

.

P r o o f i B y the consequences of (10.18) we get that ~(m) is the composition of a translation t E V with the reflection r~. Therefore {x E A : a(x - o) + t > 0} is m a p p e d bijectively onto {x E A : - a ( x - o) + e - a(t) > 0} for a suitable t E V. T h e n the assertion follows from (10.25). [] Proposition

10.28.

(see [BT 1] 9.2.13)

Let a E ~, 1 ~ u E U a ( K ) and let 1 7s u ' , u t' E U - a ( K ) with u = u ' m ( u ) u 't. We let e = Ta(U), t' = T_~(ut), e" = ~ _ ~ ( u tt) and m = m ( u ) . T h e n e = - e t = - e " .

Proof. in A:

B y (10.25) we know t h a t u, u t, u" a d m i t the following fixed point sets a : = { x E A : a(x - o) + e > O}

a t := {x E A : - a ( x - o) + t t > 0} a " := {x E A :

-a(x

o) +

-

0}

e t >

O

and t h a t a M a

Hence it suffices to show t h a t a M a t 7s

~'= 0.

O r

Suppose there exists a point x E a N a ?. T h e n U tt 9 X

=

m-lut--lu

9 x

=

?n -1

9x

E

A

.

F u r t h e r we have d ( u " . x, y) = d ( u " . x, u " . y) = d(x, y) for all y E a " . O

Since a " contains an open half space of A, we obtain u " . x = x. Since a M a C A is a n o n - e m p t y , open subset, the multiplication by m is the identity on A, in contrast to the consequences of (10.18). So let us assume t h a t a M a t =!~. Let x E a " and let x t = u x = u t m x . T h e n x t E u ( A ) M u t ( A ) C_ X ( G ) ~. On the other hand, Y := A \ ( a U a t) is a non-empty, open subset of A, i.e there exists a face F of 2i such t h a t the interior of F N Y in A is n o n - e m p t y (10.18). Hence by (10.14) (iv) we know t h a t F is m a x i m a l with respect to A. For a c h a m b e r C of .4 with F C_ C and p := p~ : ) ( ( G ) -~ A (see (9.14)), we know by (10.26) t h a t x" := p ( x ' ) E A. Let A' be an a p a r t m e n t containing x' 114

o

and C (see (9.12)). Since F _C_/i' and ~ ~ ~, it follows t h a t the interior of A~ M Y in A is non-empty. So we can find a point y E .~ N Y with y ~ x" and the p r o p e r t y t h a t [y, x ' ] is not parallel to 0 a and c~a ~. Hence the line containing [y, x ' ] has a n o n - e m p t y intersection with a and a ~. T h e r e f o r e there exists a point z E a with y E [ z , x ' ] or a point z ~ E a ~ with

y e [z', x"]. In the first case we obtain (recall y E A)

d(z, x') > d(z, x") = d(z, y) + d(y, x")

= d(z, y) + d(y, x') >

x') (see (9.1S)).

Hence d(z,x') = d(z,y) + d(y, x') and y 9 [z,x'] (9.16). Since z and x' lie in u(d), we obtain y 9 u(m) because of the convexity of u(d). B u t d(u. y,t) = d(u. y , u . t) = d(y,t) for all t 9 a = u(a) C u(A). Therefore u.y=y, incontrasttoyCaUa ~. T h e second case can be proved in a similar way. [] Finally, we will e x a m i n e the unipotent split case. Let ~2 C A be a n o n - e m p t y let f a ( a ) := f a ( a ) , if a E ~ with air t h a t f a is a well-defined function ~ -~

s u b g r o u p schemes of ~ a as in the quasib o u n d e d subset and let a E ~. T h e n we = a. Since t2 is a E-invariant, we know 1R.

Proposition 10.29. Leta E 9 and let f~ C A C_A be a non-empty, bounded subset. We Iet~ = f~(a). Then:

(i)

The product morphism induces an isomorphism from the scheme H

~lc,,S"

aE~ ~ls=a

rI ~E~ ~d ~ls=2a

onto a closed subgroup scheme ~la,~ of ~a which is independent of the given ordering. (ii)

Up to a unique isomorphism there exists a unique smooth, affine oK-group scheme ila,~ of finite type with generic fibre Ua and (l.la,~)Og~h = ~,e.

(iii) Let ~ C_ ~ be positively closed. Then the product morphism induces an isomorphism from the scheme I1 {la,]~(~) onto a closed subgroup scheme aEff2red

~1~,~ of ~

which is independent of the given ordering.

(iv) Up to a unique isomorphism there exists a unique smooth, affine oK-group scheme llr a of finite type with generic fibre U~ and (lJ~Jl)OK, ~ = ~IV,~. 115

P r o o f . The assertions (i) and (iii) follow directly from (5.6) and (6.1). Because of (10.9) the assertions (ii) and (iv) are also clear. [] With the notations of the last proposition we obtain t.l~,e(OK) = U,~,e.

Corollary 10.30. With the same assumptions and notations as in (10.29) we get that the canonical action of ~o~:,h on ~la,e through the root a descends to an action of G on 12~,~. P r o o f . This follows immediately from (10.29) and (10.22).

[]

Proposition 10.31. Let 3~ be the OK,h-grou p scheme associated with Zgsh (see (6.1)). Up to a unique isomorphism there exists a unique smooth, affine oK-group scheme 3f~ with generic fibre Z and (3a)oK~h = 3~. []

P r o o f . This is clear by (10.9).

According to (0.3) and (0.9) we get that 3fl can be identified with a closed subgroup scheme of ~n- Hence we can regard 3a as a closed OK-Subgroup scheme of @ft. As in (3.1) one can show that 3n(OK) C_ Zb(K).

Proposition 10.32. The product morphism ~lo-,n • 3f~ x ~1r

-+ ~

zs an open immersion.

P r o o f . Since open immersions descend (see [Mi] 1.2.2.4), it suffices to show that the product morphism ~1r x 3n x ~ + , a -+ ~ a is an open immersion. Let T' be a maximal torus of G which is defined o v e r K sh and let ~' be the canonical Ogh-grou p scheme associated with T'. By (6.1) the product morphism t2~_,a x ~' x ~ + , n --+ s x 3~ x ~l~+,a is an open immersion. By using (6.1)(v) again one obtains the validity of the assertion. []

Corollary 10.33. The centralizer of | in r

is 3a and the centralizer of-~ in - ~ is 3a.

P r o o f . By using (10.29)-(10.32) this can be shown as in (6.3). 116

[]

Corollary 10.34. We have: (i)

~l~,a is a maximal closed, connected, unipotent subgroup of r acts on its Lie algebra through the character a.

such that

(ii) For a positively closed subset 9 C_ O, we get that ~l~,a is a maximal closed, connected, unipotent subgroup of r such that | acts on its Lie algebra through the characters in q2. (iii) The unipotent radical of-3~ is a maximal closed, connected, unipotent subgroup of ~ such that | acts on its Lie algebra trivial. P r o o f . The proof can be done as in (6.4).

[]

10.35. For a non-empty, bounded subset ~ C A let

O~ = {a E O : f~(a) + f ~ ( - a ) = O and f~(a) E F t } ={aEO:a

is constant on ~ with value in F~} .

Proposition 10.36 Oa is the root system o f - ~ / R ~ , ( ~ q ) . P r o o f . By (6.9) we know that ( ~ is the root system of (~fl/R,(-~fl))k~. Then the assertion follows from (10.24) and (10.34). [] L e m m a 10.37.

Let p : ~ --+ r be the canonical projection. If Q is the normalizer of p(-~) in - ~ / R u ( - ~ n ) , then the canonical map N ( K ) N ~n(OK) -4 Q(k)/(p(3~)(k)) is surjective.

Proof. By (10.33) it suffices to show that for all a E On C_ 9 there exists a n E N ( K ) N O~(og), which is mapped onto the reflection r~. But this follows directly from the definition of On and (10.28). []

Proposition 10.38. Let ~ C ~ C_ A be non-empty subsets. Then the canonical morphism ~q, --+ ~ from (6.2) descends to an oK-group homomorphism ~n' ~ ~)~.

Proof. This assertion follows directly from the descent theory (e.g., see [BLR] 6.2).

[]

117

w

The full apartment

in t h e general c a s e

In this p a r a g r a p h we will f o r m u l a t e t h e s a m e s t a t e m e n t s as in w T h i s will be easier b y now b e c a u s e we have p r o v e d some s t a t e m e n t s which will be n e e d e d by d o i n g t h e 6tale descent. Let t h e a s s u m p t i o n s a n d n o t a t i o n s be as in w F u r t h e r m o r e , we choose a basis A of O. W e fix a p o i n t o E A w i t h t h e p r o p e r t y t h a t for all a E A, t h e r e exists an e l e m e n t 1 r u E Ua(K) w i t h u(m(u))(o) = o. Since for all a E O "~d, 1 o ... o r~ 1 , t h e r e e x i s t r o o t s bl,.. . ,bn,c E A w i t h r~ = r b l o . . . O r b ~ O r c o r -b~ it follows t h a t for all a E O "~d, t h e r e exists an e l e m e n t 1 # u E U=(K) with u(m(u))(o) = o. N o t e t h a t for a E 9 a n d 1 7~ u E Ua(K) t h e affine bijection ~,(m(u)) is a reflection a c c o r d i n g to (10.25) a n d (10.28).

Definition 11.1. Let a E 0 . A n affine f u n c t i o n 0 : A -+ IR is called an affine root a), if t h e r e exists a real n u m b e r f E I ~ with 0 = a ( . . . - o) + f.

(with direction

We will often i d e n t i f y an affine r o o t 0 with t h e half space 0-1([0, e c[). If 0 : A --+ IR is an affine function, t h e n t h e linear form a : V --+ IR with 0 = a ( . . . - o) + f (for e E IR) is obviously i n d e p e n d e n t of t h e choice of o. We will call a t h e linear part of O.

N o t a t i o n s 11.2. (i)

For a E 9 and u E G(K)\{1}, w i t h l i n e a r p a r t ra a n d

let

O(a,u): A --+ ~ be t h e affine function

=

A:

,(m(u))(x)=

.

(ii) If 0 : A ~ N is an affine function w i t h linear p a r t ra, t h e n we let

Uo = {u E U a ( K ) \ { 1 } : O(a,u) _> 0} U {1}

The canonical homomorphism vu as in w

N(K) -+ v W ( = N(K)/Z(K)) will be d e n o t e d by

118

L e m r n a 11.3. Let ~ : A --4 ~Z be an affine function with linear part a E ~ and let n E N ( K ) . Then nUen -1 = Ueo~(n-1 ).

P r o o f . The proof can be copied from (7.3).

[]

Now we will compare the two filtrations (U~) and (U~,t). If a E ~, then we abbreviate U-a( K ) ( Ua( K ) \ {1} )U-,~( K ) M N ( K ) as M~. L e m m a 11.4. Let a E ~. Then: (i)

r o t each m e M~, the function u ~ ~_~(u) - ~ o ( m u m -1) is constant on

U_o(K)\{1}. (ii) If u E U~(K)\{1} and if u ' , u " E U_~(K) with u'uu" E N ( K ) , ~_o(u') = - ~ o ( u ) = ~_o(u").

then

5ii) For u E U~(K)\{1}, we have ~ ( u ) = ~ ( u - 1 ) .

P r o o f . (i) follows from (10.27), (ii) from (10.28) and (iii) from (10.20) (i).

[]

P r o p o s i t i o n 11.5. If a E 9 and ~ E ~ and if we let 6 = a(.. . - o) + ~, then Ue = U~,,.

P r o o f . This assertion follows directly from the definition of Ue by using (10.25) and (10.28). [] Therefore we have obtained an invariant description of the filtrations of the root groups U~(K). Corollar 11.6. If a E 9 and ~ E ]R, then: (i)

For I # u E U~(K), we have u(m(u)) : r~(... - o) + ~(u)a ~.

5i)

For z E & ( K ) ,

we have z U o / z -~ = U~,~,

(iii) For n E N ( K ) , we have nU~,in -1 -= U~.(n)(a),~+a(~) where we let v =

,(n-~)(o)

-

o

P r o o f . The assertion (i) follows directly from (10.25) and (10.28). (ii) follows from (10.17) and (10.25) (note (1.1)). Finally, the assertion (iii) can be proved as in (7.8), but the validity of the assertion for n E Z ( K ) (see (10.17) and 10.25)) has to be taken into account. [] 119

Now we are able to define the affine root system in A. In order to do this, let ( , ) be a "W-invariant scalar produkt on V* (vW denotes the Weyl group of ~, see (1.3)). Such a scalar product exists by [Bou] VI w Prop. 3 and is uniquely determined on each irreducible component of V (with respect to vW) up to a scalar factor (see [Bou] VI w Prop. 7). By use of the canonical pairing ( , ) : V • V* --4 ~ we also obtain a ~W-invariant scalar product on V.

N o t a t i o n s 11.7. (i) Let 7 / b e the set of all hypersurfaces H in A such that there exists an affine root 0 of A with H = 0-1({0}). (ii) Let W a / f C Aft(A) be the subgroup generated by all orthogonal reflections SH with H e 7t (see [Boa] V w

Proposition 11.8. We have: (i) Waff is a normal subgroup of finite index in the image of W in Aft(A) and

(ii) w

7/ c_ 7/.

P r o o f . The proof can be copied from (7.10).

[]

D e f i n i t i o n 11.9. (i)

The hypersurfaces H E 7-/are called walls in A.

(ii) The connected components of A \

~J H are called chambers in A. HEN

(iii) Two points x, y E A are called equivalent, if for all affine roots a we have: a(x) and a(y) have the same sign or are both equal to zero. This defines an equivalence relation on A. The equivalence classes are called faces in A. Note that according to (10.24) the faces of A are the intersections of the ~invariant faces of A with A. Hence we have obtained a (poly-) simplicial complex in A which is a simplicial complex in every irreducible component (with respect to vW). In this context "poly" intends to make clear that it is a product of simplicial complexes. Further properties can be found in [Bou] V w and VI w 120

L e m m a 11.10.

Let x E A. Then there exists a real number ~ for all faces F in A with F n Be(x) r r

E

~+ such that we have x E F

P r o o f . Since A locally compact, this follows immediately from [Bob] V 3.1 L e m m a 1. [] D e f i n i t i o n 11.11.

(i)

Let 12 C_ A. Then cl(12) =

N

{xeA:a(x)+l>_O}

aE~5,~EFa,g.>_frl(a)

=

N

{x e A : - o ( x ) _ < e }

aeO,gEF.,flC{~eA:-a(z) <_s is called the simplicial closure of 12 in A. (ii) The empty apartment A together with the affine root system and the faces, chambers etc. (hence the (poly-) simplicial complex) is called the full apartment. In the notation we will suppress the dependence on the choice of the scalar product on V (see above). Let A C_ ~5 be a basis and let 0 C_ A. I f P = Po is the standard parabolic K-subgroup (see (0.20)), then the empty apartment of P/R,~(P) is A/(Fo) by (1.11). Now we choose a point o E A/(Fo) such that the o E A is mapped onto it under the canonical projection A --+ A/(Fo). Moreover, we assume that the scalar product on V/(Fo) is induced by that one on V. P r o p o s i t i o n 11.12.

With the notations given above we can also identify the full apartment of Ps with A/(Fo). Let ~r : P --+ P / R ~ ( P ) be the canonical projection. If Ua C_ P with Ua ~ Ru(P), then U-a C_ P. Hence we can define the element m(u) with respect to u e U~(K) in P/R~(P), too. By the uniqueness statement in (0.19) this is our well known element m(u), and we get that the filtrations coincide. [] Proof.

Whenever we consider the apartment of G and the apartment of a parabolic K-subgroup simultaneously, we assume the scalar products to be compatible in the sense give above. 121

w

T h e g r o u p s Un a n d P a in t h e g e n e r a l c a s e

In this paragraph we will generalize the most important results from w to the case t h a t k is perfect. In particular, we will give the Bruhat- and Iwasawa decomposition. We will make the same assumptions as in w and w but we will consider Z instead of T.

Proposition 12.1. Let f~ C A be a non-empty subset and let a E ~. If La denotes the subgroup of G ( K ) generated by U_a,fn(_a) and U~,fa(~ ) and if we let N~ := N ( K ) N L ~ , then L,~ = U_a,.fn(_a )

9

Ua,ff~(a)

9 N~

.

Here we have (i) N~ C ZD(K), if f a ( a ) + f a ( - a ) > 0 or fa(a) = - f a ( - a )

~ r a and

(ii) Na C {1, m } Z b ( g ) and N~ ~ Z b ( g ) , if f a ( a ) = - f a ( - a ) E F~ and m = re(u) for an element u E Ua(K) with 9%(u) = f a ( a ) . P r o o f . By using the corresponding statements from w as in (8.1)-(8.6).

the proof can be done []

C o r o l l a r 12.2.

If we consider the subgroup of G( K ) generated by g_a,fn(_a) , Ua,ffl(a ) and Zb( K ) instead of na in (12.1), then Na = Zb(K) in case (i) and Na = {1, m } Z b ( K ) in case (ii). [] In order to describe the structure in case that rank(B) > 1, we introduce some notations first. Furthermore, we "extend" the functions ffl such that fl C_ A is possible.

N o t a t i o n s 12.3. For fi C_ A, let P2(~) be the topological closure of fl in A. If 0 : A -+ IR is an affine function with linear part a E r (see (11.1)), then we let (i) Ao = {x E A : O(x) >_ 0} and let A0 be the topological closure of As in A. (ii) For a non-empty subset fl G A, let f a : ~ --+ ]it U {4-00} be defined by ffl(a) = inf{t E ~t : fl G Aa( ..... )+e}. Here we let i n f M = + e c (resp. = - o o ) , if M = 0 (resp. not bounded-below). 122

Obviously, for ~ C A the definition of f~ coincides with the definition in (10.29). Moreover, it can easily be seen that f~(2a) -- 2f~(a), if 2a E @ and if we let 2. ~ = cc and 2- ( - c o ) = - c o .

N o t a t i o n s 12.4. If f~ C_ A is a non-empty subset, then we let: (i)

N~ = {n e N ( K ) : u(n)(x) = x for all x e 12} see (2.12);

(ii) U~,~ -- Ua,f~(a) for all a E ~; (iii) U~ = (Ua,~ : a e ~) C_ G(K); (iv) V~ = V + M U~; (v) P~ -- C_ G(K). If f~ = {x} consist of one element, then we will write simply N,, P,, U~ etc. For a non-empty subset f~ C_ A, we know that Zb(K) and Na normalize the groups Ua (see (11.6)). Hence P~ = U~N~ and UaZb(K) are groups. (In [BT 1] w our group Pa is denoted by/Sa and UaZb(K) is denoted by Pa). P r o p o s i t i o n 12.5. (see [BT 1] 6.4.9)

Let f~ C_ A be a non-empty subset and fix an order on (~. Then: (i)

Ua(K) M U~ -- Ua,a for all a E ~,

5i)

the product map

1-I

ua,~ -+ U~ is a bijection for an arbitrary ordering

nEd•

of ~ ~:, (iii) Un = U~ . U + . ( N ( K ) M U~) and (iv) N ( K ) M U~ is generated by the N ( K ) N L~, where L~ is the subgroup generated by Ua,~ and U_~,~. P r o o f . With (10.20) and (10.29) instead of (5.3), (5.6) and (6.1) the proof can be done as in (8.9). [] m

Let 12 C_ A be a non-empty subset and fix an order on (]}. If D denotes the vector chamber defining the order, then U~+~ = U~. 123

C o r o l l a r y 12.6. Let ~ C A be a non-empty subset. Then: (i)

If n E N ( K ) , then u p o n - I -- P~(~)(~).

For a fixed order on q~, we obtain (ii) Pf~ = U~ U+ N~ = Nf~U+ U~ and (iii) P~ N U + ( K ) = U+, P~ n U - ( K ) = U~ and P~ N N ( K ) = N~.

P r o o f . In order to prove assertion (i), it suffices to show that nU~,$n(a)n -1 = U~(~),$~(n)(a) by definition of Pn. In case that fn(a) ~ • this follows from (11.6). In the other cases there is nothing to do because of the classical theory. For (ii) it suffices to prove N ( K ) NUn C_ Nn according to (12.5) (iii). This is clear by (12.1) and (12.5) (iv). By definition of U~ and by (0.18) the assertion (iii) follows from (ii). [] In the following we will make clear the connection between Pn and P~ for x E ft.

L e m m a 12.7. Let x, y E A. Then there exists an order on ~ such that U+ C_ U~.

P r o o f . By using (12.4) instead of (8.8) and (12.5) instead of (8.9) one can copy t h e proof from (8.11). [] P r o p o s i t i o n 12.8. (Generalization of [BT 1] 7.1.11) For O ~ Et C_ A, we have Pn = n P~. ~Efl

P r o o f . The proof can be done as in (8.12). Here we make use of (12.5) resp. (12.6) resp. (12.7) instead of (8.9) resp. (8.10) resp. (8.11). [] Corollary 12.8. For {~ ~ ~, ~' C -A, we have Pn

D

P~, =

P~u~'

.

[]

Finally in this chapter we will give analogous versions of the Iwasawa- and Bruhat decomposition in case that k is only perfect. 124

P r o p o s i t i o n 12.9. (The mixed Iwasawa decomposition, see [BT 1] 7.3.1) Let x 9 A and fix an order on ~. Then G ( K ) = U+(K) 9N ( K ) . Px. P r o o f . The proof can be done as in (8.14) and (8.15). Here we make use of (11.4) instead of (7.4) and (12.1) instead of (8.6). []

P r o p o s i t i o n 12.10. (The mixed Bruhat decomposition) For x, y e A, we have G ( K ) = P~ . Y ( g ) . Py. P r o o f i The proof can be done as in (8.16) and (8.17). But one should notice the following: Since k is perfect, the unipotent radical of a k-group is a k-split subgroup (see [BT 1] 1.1.11). Furthermore, we have to make use of (10.29)(10.37) instead of (6.3)-(6.9) and propositions in this paragraph instead of the propositions in w [] For the rest of this chapter we assume k to be finite.

Proposition 12.11. The group U~,t is a compact, open subset of U~(K). P r o o f . By (10.29) we know that ~la,e(ok) = U~,e and (Ha,e)g = U~. Since s is a smooth scheme, both assertions follow from the fact that K is locally compact. [] C o r o l l a r y 12.12. Let f~ C_ A be a non-empty, bounded subset. Then: (i)

The group UaZb(K) is a compact, open subset of G(K).

5i) The group Pa is an open subset of G(K). P r o o f . First by (1.2), (12.5) and (12.11) the compactness of UnZb(K) is clear, since N a / Z b ( K ) is finite. According to (1.2), (12.5) and (12.11) we know that UI~Zb(K) N U + ( K ) U - ( K ) Z ( K ) is an open subset of U + ( K ) U - ( K ) Z ( K ) for an arbitrary order on ~. Hence (i) follows. Therefore we know by (0.21) that U + ( K ) U - ( K ) Z ( K ) = U + ( K ) Z ( K ) U - ( K ) is an open subset of G ( K ) with respect to the K-analytic topology. The assertion (ii) follows from (i), since UnZb(K) C_ Pa. []

125

w

T h e building in t h e general case

In this p a r a g r a p h we will construct the B r u h a t - T i t s building for the case t h a t k is perfect.

L e m m a 13.1.

On G(K) • A there is an equivalence relation defined by: (g, x) ~ (h, y), if there is an element n 9 N ( g ) with y = L,(n)(x) and g-lhn 9 U,. Proof. (9.1).

If one uses (11.6) instead of (7.8), then the proof can be copied from []

Let X(G) be the set (G(K) • A)/ ~. Before we are able to define a t o p o l o g y on X(G) we will examine some settheoretical properties of X(G). In particular, we will define a G ( K ) - a c t i o n on

x(a). L e m m a 13.2.

The map A -+ X(G) (x ~+ (~x)), where (~,x) denotes the equivalence class o/ (1, x), is injective.

Therefore we will identify A with its image under this map. P r o o f . T h e proof can be copied word for word from (9.2). But one has to make use of (12.6) instead of (8.10). []

We have defined a N ( K ) - a c t i o n on A t h r o u g h u (see (1.8)). Now we will extend this to an action of G(K) on X(G). Let us consider the m a p

G(K) • (G(K) x A) ~ G(K) • A

(g, (h, x)) . (gh, x) B y definition of ~ this m a p induces an action of G(K) on X(G) extending the N ( K ) - a c t i o n on A. 126

Proposition 13.3. We have: (i) If it C_ A is a non-empty subset, then Pa = {g E G ( K ) : gx = x for all x E ~}

.

(ii) If a E 9 and u E U ~ ( K ) \ { 1 } , then the set of fixed points of u in A is the a2~ne ~oot {x e A : a(x - o) + ~ > 0} where ~ = ~o(u). P r o o f . For it = {x}, the assertion (i) is clear by definition of X ( G ) . By (12.11) we have Pa = N P~, from which the general case (i) follows immediately. By :cEil

(12.8) and (12.9) we know that P~ NU~(K) = Ua,--a(~), hence u E P~ if and only if ~ > - a ( x ) . [] C o r o l l a r y 13.4. For all non-empty subsets it C A with it = cl(it) (see (11.10)), the set of fixed points of P~ in A equals ~. P r o o f . If ~ is an affine root, then the assertion follows from (13.3). Since it = cl(it), we get that ~ is the intersection of all affine roots containing it. Hence the assertion follows from the definition of P~. []

Definition 13.5. A subset A' C_ X ( G ) is called an apartment of X(G), if there is an element g E G ( K ) with A' = g. A. For g E G ( K ) , we let g. : X ( G ) -+ X ( G ) be the map x ~-~ g . x . Then the following proposition makes clear that the map g. : X ( G ) --+ X ( G ) restricted to A M g-1 . A is not arbitrary but of the form n. for a suitable n E N ( K ) .

Proposition 13.6. (see [BT 1] 7.4.9) Let g E G ( K ) . Then A M g - l A = cl(A M g - i A ) and there exists an element n E N ( K ) such that g. x = n . x for all x E A N g - I A . P r o o f . The proof can be copied from (9.6). But here we have to consider (12.6) resp. (12.8) instead of (8.10) resp. (8.13). [] 127

Corollary 13.7. Let fi C_ A be a non-empty subset. Then: (i) Un acts transitively on the set of all apartments containing ft. (ii) If C is a chamber and g E Pc, then gx = x for all x E A M g-1 . A .

Proofi Let A' = g 9 A be an a p a r t m e n t containing ft and let n E N ( K ) with g-1 .x = n . x for all x E AMgA (13.6). Hence gn E Pa and A' = gn.A. W i t h o u t loss of generality we m a y assume t h a t gn E Ufl, since Pfl = U~Na. T h u s we obtain (i). If C is a c h a m b e r , then P c M N ( K ) = N c (see (12.6) (iii)), from which (ii) follows. [] Corollary 13.8. N ( K ) is the stabilizer of A and ker(u) = {g E G(K) : gx = x for all x E A} (see (1.,~)). Let g E G ( K ) with g - A = A. By (13.6) there exists an element n E N ( K ) with g - i n E PA. According to (13.3) we have UA = {1} and therefore PA = NA = ker(u)). [] Proof.

T h e notions of "face", " c h a m b e r " , "sector" etc. will be transferred from the a p a r t m e n t to the whole building.

Definition 13.9. Let Y C X ( G ) . T h e set Y is called a face (rasp. a chamber), if there is an element g E G ( K ) such t h a t g Y C A is a face (resp. c h a m b e r ) in the sense of (11.10). By (13.6)'*he faces in the sense of (13.9) contained in A are exactly the faces in the sense of (11.9). T h e following proposition will show t h a t the faces of X ( G ) are in some sense the smallest constituents of X(G).

Proposition 13.10. Let Y C_ X ( G ) b e e face and let g E G(K) with Y MgA ~ O. Then there exists a face Y of A such that Y = gY. In particular, Y C_gA.

Proof. W i t h (13.6) instead of (9.6) the proof can be done as in (9.10). 128

[]

Corollary 13.11. Let F and F' be two faces of A with F' C -F and let g E G(K). apartment containing g F also contains gF'.

Then evew

[]

P r o o f . This follows directly from the proof in (13.10).

The following proposition will show that there are enough apartments in X ( G ) . It is in some sense the geometrical version of the Iwasawa- and Bruhat decomposition (12.9) and (12.10).

Proposition 13.12. We have: (i)

Two chambers (resp. faces, rasp. points) of X ( G ) are contained in a common apartment:

(ii) Let f~ be a chamber (rasp. face, rasp. point), let x e A, g E G ( K ) and let D be a vector chamber. Then there is a point y E x + D such that f~ and g. (y + D) are contained in a common apartment.

Proof. By using (13.10) rasp. (12.9) resp. (12.10) instead of (9.10) rasp. resp. (8.17) the proof can be copied from (9.12).

(8.15) []

Before we are able to define a metric and a toplogy on X ( G ) we have to study the restrictions of the building onto the apartments.

Proposition 13.13. Let C be a chamber in X ( G ) and let A ~ be an apartment with C C_ A ~. Then A' : X ( G ) ~ A t such that for aIl g C P c , we there exists a unique map p = Pc obtain fltq.A' = h. ~A' for a suitable h C G ( K ) . Moreover, p satisfies: (i) f~A' is the identity, and 5i) if x e -C, then p - l ( x ) = {x}. P r o o f . The proof can be copied from (9.13) and (9.14).

[]

In w we have introduced a metric d on A by using the vW-invariant scalar product on V. This will be transferred to the other apartments. 13.14. Let A' be a further apartment and let g E G ( K ) with g A = A'. Then (x,y) , - ) d ( g - l x , g - l y ) 129

defines a metric d t on A ~. Since N ( K ) acts on A by isometries, this metric d t is independent of the choice of g by (13.8). By use of (13.6) and (13.12) (i) one can show immediately that there exists a unique metric d : X ( G ) • X ( G ) -4 1R such that the restriction to any a p a r t m e n t A t coincides with the metric d t given above. Obviously, the metric d on X ( G ) is G(K)-invariant, i.e. G ( K ) acts by isometries. In the sequel we assume X ( G ) to be topologized by this metric. Proposition

13.15. (see [BT 1] 7.4.20 and [Br] VI 3 Theorem)

We have: (i)

Let C be a chamber of X ( G ) , let x E C and let A' be an apartment with C C A' Then d(p(y), p(z)) <_ d(y, z) for all y, z e X ( G ) . A'. Further let p : Pc" If C, y and z are contained in a common apartment, then d(p(y), p(z)) = d(y, z). In particular, d(x, p(y)) = d(x, y) for all y e X ( G ) .

(ii) L e t x , y e X ( G ) and l e t S = {z 9 Then S is contained in every apartment containing x and y, and coincides with the segment Ix, y] (see (0.15)) in these apartments. All isometrics fixing x and y also fix S pointwise. In the following S will be denoted by [x, y]. (iii) Let x, y, z, z' 9 X ( G ) , let z 9 [x, y] and let ~ 9 F~+ such that d(x, z') < d(x, z) + r d(x,y) and d(y, z') <_ d(y,z) + ~ . d(x,y)

.

Then d(z, z') 2 <_ 4d(x, z)d(y, z)c + d(x, y)2~2. (iv) Let x , y E X ( G ) , t E [0,1] and tx + (1 - t)y be the uniquely determined element in [x, y] with d(y, z) = t. d(x, y). Then the map

[0, 1] • X(G) • X(G) --+ X(G) (t, x, y)

tx + (1 - t)y

is continuous. In particular, the topological space X ( G ) is contractible. P r o o f . By use of (13.12) instead of (9.12) and (13.13) instead of (9.14) these assertions follow as in (9.16). [] Before we are able to prove that X ( G ) is complete have to show that the faces do not lie too dense locally. 130

L e m m a 13.16. Let F be a face of X ( G ) and let x E F. Then there exists an E E K:[+ such that for all faces F I with F t N B~(x) ~ O, we have F C_ F I. After a suitable shrinking of ~ we can assume that we have g 9 x = x for all g. x e B~(x) with g e a ( g ) . P r o o f . The proof can be copied from (9.17).

[]

Proposition 13.17. The topological space X ( G ) is complete. P r o o f . By using 13.16) instead of (9.17) the proof can be done as in (9.18). [] As we have seen m w the pair (A, u) is unique up to a unique isomorphism. The affine roots, faces, etc. defined in w were given canonically, too. But the scalar product on V defining the metric on A was unique up to scalar factors, only. Apart from this the definition of X ( G ) only depends on the choice of the maximal K-split torus. Finally, we will show that X ( G ) is almost independent of this as well. Let S ~ be a further maximal K-split torus and let g E G ( K ) with S t = gSg -1. As in w we can define the empty apartment A' with respect to S ~ and as in w we can defined faces, chambers and walls in A I. A G(K)-set which will be denoted by X~(G) can be constructed in the very same way as in this paragraph.

Proposition 13.18. There exists a unique G(K)-isometry X ( G ) ~ X ' ( G ) . P r o o f . The proof is analogous to (9.19) and (9.20). Here one has to make use of (13.4) and (12.6) instead of (9.4) and (8.10). []

Definition 13.19. The G(K)-set X ( G ) together with the metric topology and the structures defined above (faces, chambers etc. ) is called the Bruhat-Tits building of G. Let F , F ' C_ X ( G ) be two faces with F C_ F'. By (10.38) the identity on G extends to an oK-group homomorphism Res F' F : OF, -+ OF which induces a ,F !

__

k-group homomorphism Res F : OF, --+ O F / R u ( - O f ). 131

Proposition 13.20. Let X ( F ) be the set of all faces F r of X(G) with F C F' equipped with the ordering F I < F ' , if F I C F ' . Then the map p: X ( F )

F',

) (combinatorial) Tits building Y(ObF/Ru(C~F))

)image of Rest' in

is a bijection which preserves the orderings. P r o o f . The E-invariant parabolic K~h-subgroups of G are exactly the Kparabolic subgroups of G. By making use of (10.29)-(10.37) the proof can be done in complete analogy to (9.22). [] By using the local description in (13.20) we will give another characterization of the topology on X(G).

Proposition 13.21. The following topologies on X(G) coincide: (i)

the metric topology,

(ii) the product-quotient topology on X(G) = (G(K) x A ) / ~, if G(K) carries the discrete topology and (iii) the product-quotient topology on X ( G ) = (G(K) • A ) / ~ , if G(K) carries the K-analytic topology. P r o o f . First let us show that the topologies in (ii) and (iii) coincide. Obviously, the topology in (ii) is finer than the topology in (iii). Thus let U C_ X(G) be an open subset with respect to the topology in (iii) and let ~ : G(K) x A ~ X ( G ) be the canonical projection. Hence if (g, x) E Ir -1 (U), then there exists a bounded open subset ~t C_ A with x C 12 and g. ~ C_ U. Then obviously 7r(gPn • ~t) C_ U and therefore U is an open subset with respect to the topology in (iii), since Pn C_ G ( K ) is an open subset according to (12.12). Now let x E X(G), ~ E ~:r let y E B~(x) be arbitrary and let 5 E ~ + such that B,(y) C_ B~(x) and B~(y) only meets faces F with y e F (13.16). Since G(K) acts on X ( G ) by isometries, we get for (g, z) e ~r-l(y) that g. (B,(z) n A) C_ Bz(y) C B~(x) and therefore that the topology in (ii) is finer than the metric topology. Conversely, let U C X ( G ) be an open subset with repect to the topology in (ii). Hence for (g,x) E ~ - I ( u ) , there exists an open subset ~ C A with z e and g~ C_ U. Without loss of generality we may assume that g = 1 and that = B~(x) with r as in (13.16). 132

Since k is finite, Y ( ~ / R u ( ~ ) ) is finite and therefore X ( F ) is finite, too, if F is the face of X ( G ) with x 9 F. Say X ( F ) = {F1...F,~}. For Fi 9 X(F), there exists an apartment A~ with F, F~ C_ Ai by (13.12)(i). Let g~ 9 G(K) with giA = Ai. Then ({gi} x A)NTr-I(U) # 0, and hence there exists an open subset f~i C_ A~ with x 9 giQi C U. Without loss of generality we may assume that f~i = B ~ ( x ) M A for a suitable ei 9 ~;[+. By using 5 = min{e,si : Fi 9 X ( F ) } we obtain B6(x) C_ U. [] If k is not finite, then the topologies in (ii) and (iii) are strictly finer than the metric topology. This can easily be seen by having a glance on the example

SL2(Q;h).

Proposition 13.22. If G( K) is equipped with the K-analytic topology, then G( K) acts continuously on X(G). P r o o f . By (13.14) every element of G(K) acts isometrically, in particular continuously on X(G). Since the stabilizer P~ of a point x is an open subset (12.12) the assertion follows. []

w

The compactification of the building

We will make use of the results of w to construct a compactification X(G) of X(G) which "looks" like the compactification A of the apartment A. Again we assume k to be finite. L e m m a 14.1.

On G(K) • -A there is an equivalence relation defined by." (g,x) ,.~* (h,y), if there is an element n 9 N ( K ) with y = u(n)(x) and g-lhn 9 U~. P r o o f . The proof can be done as in (13.1).

[]

Notations 14.2. Let X(G) denote the set G(K) • A / ~* equipped with the product-quotient topology, where G(K) carries the K-anMytic topology and A the topology defined in w 133

By definition of ~ and ~* it is clear that there exists a canonical injective map The topology on X ( G ) induced by this inclusion coincides with the metric topology by (13.21). Thus we will simply write -,~ for ,-* (note that A C A is an open subset (2.10)(iv)).

X(G) --+ X ( G ) .

As in (13.3) one can show that the m a p A ~ X ( G ) , x ~ denotes the equivalence class of (1, x), is injective.

(1--~), where (1-~)

We will identify A with its image under this map.

L e m m a 14.3.

The map G ( K ) • (G(K) • -A) -~ G(K) • G(K)-action on X ( G ) .

(g, (h, x))

(gh, x) induces a

P r o o f . Let hi,h2 E G ( K ) and let xl,x2 E A with (hi,x1) "~ (h2,x2). From the definition of ~ it follows that for all g E G(K), we have (gh~,xl) ~ (gh2,x2). Hence the m a p given above induces a G ( K ) - a c t i o n on X. [] 14.4. As in (13.3) one can prove the following assertions: (i) If ~t C_ A is a non-empty subset, then the stabilizer of gt in G(K) is P~. (ii) If a E 9 and u E U~(K)\{1}, then the set of fixed points of u in A equals {x E A : - f~(a) + ~ >_ 0}, if g = ~a~(u). For non-empty subsets f~ C_ A, we will write 9./(f~) for the topological closure of ft in A and cl(f~) for the set n

P2({x E A : a(x) + ~ >_ 0})

aEr Proposition

14.5.

Let g E G ( K ) . Then A N g - l ~ = cl(-A N g-l-~) and there exists an element n E N ( K ) such that g. x = n. x for all x E -A N g-1-~. P r o o f . Without loss of generality we may assume that ~ = A N g - l A ~ 0. Let 2~4 = { Y C_ ~2: g - l Y ( K ) N P y ~ 0}. As in (13.6) one can show that Ad contains all finite subsets of ~. As in (12.8) resp. (8.12) we can find a finite subset R C_ ft such that for all vector faces E with f l n A / ( E ) ~ O, we have R n A / ( E ) ~ 9. We let Zn = Z ( K ) n NR. 134

Now if (Yi)~eI is a filtration of ~ by finite sets with U Y~ = f/ and R C_ Y~ iEI

for all i E I, then there exist n0,ni E N ( K ) (i E I) with g - i n o E PR and g-ln~ E Py~ C PR (i E I). Hence n~lno E NR for all i E I. Since NIt/ZR is finite, we may assume without loss of generality by passing to a cofinal subsystem that all n~ (i E I) are equal (note that ZR C_ Py~ for all i E I). The second assertion follows from this, and the first one can be shown as in (13.6). []

14.6. As in (13.7) and (13.8) we conclude: m

(i) Let ~ C A be a non-empty subset. Then U~ acts transitively on the set of all gA with f~ C_ gA. (ii) N ( K ) is the stabilizer of A and ker(u) = {g E G ( K ) : gx = x for all x E A}.

Proposition 14.7. Let x, y E -X(G). Then there exists an element g E G(K) such that x, y E g" -A. []

Proof. This can be shown as in (13.12) (i). Proposition 14.8. We have: (i)

Uo" A = X ( G ) ,

5i)

Uo92(o + D) = X ( G ) for any vector chamber D in V and

(iii) 92(0 + D) is a fundamental domain for the action of Po on -X(G).

Proof. (i)is clear by (14.6) (i) and (14.7) (i). (ii) follows from (i) and the way the point o is chosen. m

Finally, let x , y E 92(o + D) and let g E Po with gx = y. Then o , x , y E A and o, y E gA. Hence by (14.6) there exists a n E N ( K ) with n "~ng~ = g-1 "~ng~, i.e. u(n)(y) = x and u(n)(o) = o. Therefore n E N ( K ) A Po = No. But since 92(D) is a fundamental domain for the vW-action on V ~ (see w in particular (2.8)), we obtain x = y. []

In the sequel we will examine the topological properties of X. We will show that X ( G ) is Hausdorff, compact and contractible. 135

L e m m a 14.9. Let (xi) be a sequence in A with lim xi = x and let (ni) be a sequence in N ( K ) i --+ o o

with lim ni = n. T h e n lim L,(ni)(xi) = ~,(n)(x). i---* c ~

i-+ c~

P r o o f . T h e assertion is clear, since N ( K ) H a u s d o r f f (2.10).

acts continuously on A and A is []

L e m m a 14.10. Let (x~) be a sequence in A with lim x~ = x, let a E 9 and let (u~) be a sequence i---+ o o

in U ~ ( K ) with lim ui = u and u~ E - -

,~

for all i. Then u E U~,f~(~).

i---+ o o

Proof.

We distinguish three cases:

1. C a s e : f~(a) = - o c ; there is nothing to show because of U~,_~ = u a ( g ) . 2. C a s e : f~(a) = + ~ ; first the sequence (f~,(a)) converges to ]~(a) = +oc, since we get t h a t y ~-+ f y ( a ) is a continuous m a p from A to ~ U { • by definition of the topology on A (see (2.4)). Now it follows from the construction in (4.8), (4.14) and (10.19) t h a t ~ : U ~ ( K ) ~ IR U { + o c } ( = ~ ) is continuous, too (see (2.1)), and we obtain ~ a ( u ) = oo, since ~a(U~) --+ oc. Hence u = 1. 3. C a s e : f~(a) E JR; without loss of generality we m a y assume t h a t (xi) is a sequence in A. Since U ~ ( K ) acts continuously on X ( G ) (13.22), the assertion follows. []

P r o p o s i t i o n 14.11. Let (gi) be a sequence in G ( K ) with lim gi : g and let (xi) be a sequence in -A i--+ o o

with lim xi = x. If gi E P ~ for all i, then g E P~.

Proof.

We fix an order on q. By (12.5) we obtain

f

= e:, v :

for all i. Hence we can choose elements ui E U ~ , vi E U + and ni E N ~ with gi = uivin~ for all i. Since N ( K ) / Z ( K ) is finite, we m a y assume without loss of generality by passing to a subsequence t h a t there exists an element m E N ( K ) with nirn E Z ( K ) for all i. Hence (gim) = ( u i v l n l m ) is a converging sequence in U - ( K ) U + ( K ) Z ( K ) , which carries the topology induced by G ( K ) . Therefore the sequences (u~), (v~) and ( n i m ) converge in U - ( K ) , U + ( K ) and Z ( K ) , respectively, and hence (n~) converges in N ( K ) , too. 136

Let u = lim ui, v = lim vl and n = lim nl. Since G ( K ) is a topological group, i--+oo

i-+oo

i-+oo

we obtain g = uvn. By (14.9) we know t h a t n E N~. According to (12.5)(ii) and (14.10) we obtain u ~ U~- and v E U +, and since P~ = U~U+Nz ((12.6) again), we conclude g E P~. []

Let Uo = UoZb(K). By (14.8) (i) the m a p ~7o x X --+ X ( G ) , (u,x) ~-+ ux is surjective. Hence we can consider the quotient-topology on X ( G ) defined by it. Let -xP(G) be the set X ( G ) equipped with this topology.

Proposition 14.12. The topological space -xP ( G) is Hausdorff.

Proof. B y (12.12) and (2.10) we know t h a t Uo x A is c o m p a c t . Hence according to [Bou] I 10.4 Proposition 8 it suffices to show t h a t the equivalence relation in

(Do x X) x (gro x ~) defining the set -xP(G) is closed. Let (ui), (u~) be sequences in Do with lim ui = u and lim ul' = u' and let (x~), i---+ c ~

i---* oo

(x~) be sequences in A with lim xi = x and lira x ,i = x ~. Furthermore, we i---+ oo

i--+oo

for all i. Hence we have to show t h a t ux = utx ~. Since t h e r e are only finitely m a n y vector c h a m b e r s in V, we m a y assume without loss of generality by passing to subsequence t h a t there exist vector c h a m b e r s D and D ' with zi e 9 1 ( o + D ) and x~ e 9 1 ( o + D ' ) for a l l i . Since 9 1 ( o + D ) is a f u n d a m e n t a l d o m a i n for the Po-action on X ( G ) (14.8), there exists an element m E N ( K ) such t h a t we have u ( m ) ( z i ) = x i for all i. assume

u i x i ~ u i x 'i

Since A is Hausdorff and N ( K ) acts continuously on A ((2.9) and (2.11)), we obtain z~(m)(x) = x'. Now we have u~-lu~m E Pa~i for all i and we have to show t h a t u-lu'm E P~. This follows from the previous propositions. []

Corollary 14.13. The topological space -xP ( G) is compact.

Proof.

Since Do and A are c o m p a c t (see (12.12) and (2.10)), the assertion follows from (14.12). []

Proposition 14.14. The topological space -xP (G) is contractible. 137

Proof.

First let us consider the continuous m a p (see (2.8))

R: Lro x 92(0 + D) x [0,1 1 ---+ Uo x 92(o+ D) (g,=,t) ~ (g,~o(~,t)) We have to show t h a t R induces a m a p R : -xP(G) • [0,1] -4 -xP(G). If g, h E ~'o and x, y E 92(0 + D) with (g, x) ~ (h, y), then we have to prove t h a t (g, x t) := R(g, x, t) ..~ R(h, y, t) =: (h, yt) for all t E [0, 1]. By (14.8) we obtain x = y and therefore g - l h E Uo n P~ C_ Po n P~ = P{o,~) (see (12.8)). Hence it is sufficient to show t h a t g - l h E Pon P , , = P{o,~,) for all t E [0, 1]. In the following we will always assume r to be equipped with the order defined by D. B y (12.6) we obtain P{o,z} = N{o,~}U[o,~}U{o,~} and according to (14.4) we get t h a t U-~o,~} C_ U+ acts trivial on 92(o + D). T h e n it follows from the construction of the retraction r n in (2.2) t h a t for all simple positive roots a E we have f~,(a) E [0, f , ( a ) ] . Hence we obtain f~,(a) E [0, f~(a)] for all a E ~ + , and U(o,~ } fixes all x t. Since N{o,~} C_ No we can identify N{o,~} with a subgroup of the Weyl group of q>. From the construction of r D we know t h a t all x t with t < 1 lie in the closure of o + F in A. As in (2.8) one can show t h a t all x t are fixed by N{o,~,}, from which follows g - l h E P{o,~} for all t. Now we have to show t h a t the m a p R : X P ( G ) x [0, 1] --+ -xP(G) is continuous. In order to do this, let us consider the c o m m u t a t i v e diagram

Uo x92(0+D) x [0,1] py_~d XP(G) x [0,1]

50 • 92(0 + D)

P"~

--xP(G) .

All topological spaces considered here are c o m p a c t and Hausdorff. Moreover, we know the continuity of all maps, possibly up to R, from which the continuity of R follows, since pr x id is surjective. []

In the next step we will show t h a t '-xP(G) and X ( G ) carry the same topology. Obviously, the identity 'xP(G) -+ -X(G) is continuous.

In order to prove t h a t

the identity X ( G ) -4 x P ( G ) is continuous, too, it suffices to prove the following proposition:

Proposition 14.15.

The map G(K) • A ~ -xP (G), (g, x) ~ gx is continuous. 138

Proof.

Let (gl) be a sequence in G ( K ) with liln g~ = g and let (xl) be a i--+oo

sequence in A with lim xi = x. So we have to show t h a t lim gixl = gx. i-+oo

i--+ c ~

B y (12.6) and (12.10) we know t h a t G(K) = P o N ( K ) P , , = (]oN(K)U+U~ for all i and an arbitrary order on (I). Hence we can find elements ui E Uo, ni E N ( K ) , vl E U+~, and v i' E U~, with gi = uiniviv~ for all i. Since (7o is c o m p a c t , we m a y assume w i t h o u t loss of generality by passing to a subsequence t h a t there exists an element u E Uo with lim ui = u. Hence the sequences i---+ o o

(u~-lgl) = (niviv~) also converge.

By passing to a subsequence one obtains elements v 9 U + , v' C U~- and n 9 N ( K ) with lim vl = v, lim v il ~ v l i--+r

and lim nl = n as in (14.11). i----~o o

~-+oo

Hence g = unvv t and we have to show t h a t

(glxi) = (ulv(nl)(xi)) converges to gz = uv(n)(x). Now the m a p N ( K ) • -A ~ -xP (G), (n, x) ~ nx factors t h r o u g h N ( K ) • -A -+ 2. (n, x) ~-~ ~(n)(x) which is continuous. Moreover, the inclusion A -+ -xP(G) is continuous. Hence (v(ni)(xi)) converges to , ( n ) ( x ) , since d is Hausdorff (2.10), and we obtain lim (u, . v(ni)(x,) ) = u. v(n)(z) , i -4. o o

since X P ( G ) is Hausdorff, too (14.12). Corollary

[]

14.16.

The topological space X ( G ) is Hausdorff, compact and contractible. Proposition

[]

14.17.

The C(~:)-actio~ on X ( G ) is co~ti~uous. Proof.

Let us show first t h a t the open c o m p a c t subgroup Uo (12.12) acts

continuously on -xP(G). In order to do this, let us consider the diagram

Oo•

--~ Do•

l 0o x 2

l~ ---+

X~(c)

Again, all topological space considered here are Hausdorff and c o m p a c t and all maps, possibly up to a, are continuous. Hence we obtain the continuity of a. Since Uo is an open subgroup (12.12), it suffices to show t h a t every element of G ( K ) acts continuously. Let g E G ( K ) and let us consider the diagram

G(K) • ~

X(a)

(g)--5id C(K) • 2.

-% 139

X(a).

By the definition of the topology on X(G) the continuity of the m a p []

X(G) g') X(G) follows.

Let us show that X(G) is "independent" of the choice of a maximal K-split torus, too. Lemma

14.18.

Let f : X(G) --+ X(G) be a G(K)-equivariant, continuous map such that fix(a): X(G) ~ X(G) is the identity. Then f is the identity map. m

P r o o f . If we can show that f(A) C_ A, then we are done, since flA : A ~ A is the identity and by (2.10) we know that A is Hausdorff and that A _C A is a dense subset. Let A' be an a p a r t m e n t with x, f(x) E A'. According to the G(K)-equivariance of f we obtain P~ C_ Pf(~), from which f(x) E cl({x}) C_ A' follows, since we obtain the analogous version of (13.4) by (14.4). Then according to (14.5) it follows that f(x) E A, hence the assertion is proved. [] Proposition

14.19.

Let S ~ be a maximal K-split torus, let A' be the associated apartment and let A~ be its compactification. By using the constructions in this paragraph we obtain in complete analogy to -R(G) a topological space X ' (G) on which C(K) acts. Then there exists a unique G( K)-equivariant, continuous extension f : -X( G) --+ -X' ( G) of the uniquely determined map f : X(G) -+ X~(G) from (13.20). P r o o f . The existence can be shown as in (13.20) and the uniqueness follows from (14.18). [] D e f i n i t i o n 14.20. The topological space X(G) together with the G ( K ) - a c t i o n is called the polyhedral compactification of the Bruhat-Tits building X(G). This terminology will be justified in the following. The topology of X(G) contains three constituents: the real-metric one of A, the combinatorial one of the Coxeter complex of ~ and the K-analytic one of G(K). Till now we have combined the real-metric and the combinatorial part in the topology of A. In the following we will combine the combinatorial and the K-analytic part to study the boundary OX(G) = X ( G ) \ X ( G ) . 140

Let P be a parabolic K - s u b g r o u p of G and let S t C_ P torus. If ~t denotes the compaetified a p a r t m e n t of then by (11.12) there exists a unique vector face / ; ) a p a r t m e n t of P with respect to S t. Hence we obtain

C_ G be a maximal K-split X ( G ) with respect to S r, such that A'/(Ftp) is the a canonical m a p

qOp : P ( K ) X A'/(F~p) ----+-X(G) = (G(K) x -A')/ (p,x)

~ - ~ p . :c

.

Since ( P / R ~ ( P ) ) ( K ) = P ( K ) / R ~ ( P ) ( K ) , the m a p ~ p induces a m a p

~p : ( P / R ~ ( P ) ) ( K ) • A'/(F~p) ---+ -X(G) by definition of ,~ and by a statement completely analogous to (8.16) Then it follows from (11.12) that the m a p ~ p : X ( P / R ~ ( P ) ) --+ -X(G) induced by qop is injective and P(K)-equivariant. Now let us consider the commutative diagram

~oA' :

X(P/Ru(P))

~A":

Xt(p/Ru(P))

+

--~

-X(G)

l

~

-Xt(G).

It follows that ~ p is independent of the choice of the apartment A t since the vertical arrows have been constructed in the same way (see (13.20) and (14.19)). By the "disjoint union" of the maps qOp we obtain a map

~ : U X(P/Ru(P))

> -X(G) ,

PCq3

if ~ denotes the set of all parabolic K-subgroups.

Proposition 14.21. The map qo : (J X ( P / R u ( P ) ) --4 X ( G ) is a bijection. PEq3

P r o o f . Let P , P ' E ~3. Then by [BoTi] 4.18 there exists a maximal K-split torus S t of G with S t C P, pt. Without loss of generality we may assume that S = S t. Now let g e P ( K ) , let g' e P ' ( K ) and let x e A/(Fp), y e A/(Fp,) with ~o(gz) = T(g'y). Hence there exists an element n e N ( K ) with v(n)(x) = y and therefore Vv(n)Fp = Fp,, i.e. pt = nPn-1. If P = pt, then it follows 141

from the injectivity of (~p that gx = g~y. So let us assume that P # P~, and let g' = n[Tn-1 for a suitable ~ E P(K). Then P~ ~ g-lgtn = g-ln~, i.e. P ( K ) . n . P ( K ) n P, ~ 0. Because of a statement completely analogous to the preceeding remark of (8.16) we get P~ C_P(K). 1 . P ( g ) , hence P ( K ) . n . P ( K ) A P ( K ) . 1. P ( K ) 7~ 0, from which we obtain a contradiction according to [BoTi] 5.15. Hence ~ is injective. Let us show the surjectivity. Let x E X(G) and let A ~ be an a p a r t m e n t containing x. Without loss of generality we may assume that A = A ~. Then x E A/(F) for a vector face F of V. Let P be the parabolic K-subgroup of G with S C P and Fp = F. Then we have W ( 1 . x ) = x, where 1 . x denotes the element in

X(P/R~(P)).

[]

In the sequel we will define a topology

0

X(P/R~,(P)) such that the bijection

PEt43

becomes a homeomorphism. Let P a r G be the set of all parabolic K-subgroups of G and for a fixed basis A of r let Par0 G be the set of all parabolic K-subgroups of G of type 0 C_ A (these are exactly the conjugates of the standard parabolic K-subgroup of type 0). 14.22. In the sequel we will need the following well-known facts about parabolic subgroups (see [BoTi] 4 and (0.6)): Let P be a K-parabolic subgroup of type 0 C_ A. Then (i)

G / P is a projective K-variety,

(ii)

( G / P ) ( K ) = G ( K ) / P ( K ) as K-analytic spaces,

(iii) ( G / P ) ( K ) is compact with respect to the K-analytic topology, (iv) G(K) -+ (G/P)(K) is surjective and (v)

there exists a canonical bijection (G/P)(K) --+ Par0 G.

By (ii) it is clear that the m a p G(K) --+ (G/P)(K) is open with respect to the K - a n a l y t i c topology. We fix a basis A of r We equip gl(A) (=set of all subsets of A) with the following topology: 0 C_ ~ ( A ) is called open, if we have for all cr E 0: If T _D Cr, then T E 0. The canonical projection Par 0 G --+ Parr G (see (v)) will be denoted by pr~. This m a p is open, too, by the previous remark. Hence we can define the surjective m a p 7r : Par 0 G x ~3(A) ~

(P,r), 142

Par G

>pr,(P) .

Here p r , ( P ) denotes the parabolic K-subgroup Q E Par, G with P C_ Q. We will equip Par a with the quotient topology under 7r. If we only consider the set of parabolic K-subgroups, then we will denote it by 9 .

Lemma

14.23.

The map 77 :Parr G x g3(A) -+ Par a is open. P r o o f . Let Y x 0 C_ Par 0 G x ~ ( A ) be an open subset. Then

-1(77(y • o)) = [.J pT;I(pT,(Y)) • = U Pr;l(Pr'(Y)) • {a e A : a D_T}

[]

r60

P r o p o s i t i o n 14.24.

We have: (i)

The topology of Par G is independent of the choice of S and A,

(ii) G(K) acts continuously on Par G via conjugation and (iii) Par G is quasi-compact. (iv) If'r C_ A, then the topology on Par, G induced by Par G coincides with the canonical K-analytic topology. (v)

The closure of P a r , G in P a r G equals U Par, G. o'C'r

P r o o f . (i) and (ii) are clear by definition. (iii) follows from (14.22) (iii). In order to prove (iv) we consider the commutative diagram

Par0(G) •

Par 0 xg3(A)

P'# Par.(a)

-~

Par(G)

of continuous maps, where Par,(G) carries the K-analytic topology. By (14.22) (iii) we know that pr, is a topological quotient map. Hence the inclusion P a r , ( G ) C Par(G) is continuous. If U, C P a r , ( G ) is an open subset and if we let U x {T} := pr~l(U,), then 7r(U • {a e A : (r D T}) is an open subset of Par(G) with pre-image U, in Par,(G) (14.23), since r ( U x {T}) M Pard(G) c_ Pard(G) n Par,(G) -- 0 for

trOT. 143

Obviously, we have Par G \ P a r , G = U P a r , G, and by definition of the topo1a#r 0

ogy on ~3(A) we obtain (]~ar G \ P a r , G') =

U P a r , G, from which (v) follows. tr(Zr

[]

Now a subbasis of the topology on 0 X(P/R~,(P)) will be defined in two steps: PEq3 First we will define the skeleton of the open subsets by use of Par G and then we will make open sets out of t h e m by an e-blow up. Let S t be a maximal K-split torus, let A' be the apartment associated with S ~ and let Q E Par G with S t _C Q. The space of directions of A t will be denoted by V'. We will denote the face F in V' associated with Q by F ~ ' . Furthermore, the image of f~ C_ A t in A'/IF~' } under the canonical projection will be denoted by (a)Q. Notations

14.25.

(i) For x E X(G) and for P , Q E ~3 with P C_ Q, we let

(c[)Q =

U

--A

(x + rp)Q

apartrnents A SAC=P,z E A

where SA is the maximal K-split torus associated with the a p a r t m e n t A. (ii) For ~t C_ Par G and z E X(G), let

PEfl QCf~

Q2P

L e m m a 14.26.

Let x E X ( G ) , let P,Q E ~3 with P C Q and let A be an apartment with x E A --A andSA C P. Then(CP)Q = (x + Fp)Q. P r o o f . Let A ~ be another a p a r t m e n t with these properties. Then by (13.7) there exists an element g E Px with gA = A t. Hence SA, = gSAg -1 C_ P and we will show first t h a t g E P ( K ) . Then we may assume without loss of generality that g E UZU + (12.6), where the order on • considered here is chosen in such a way t h a t P is a standard parabolic K-subgroup. Let us assume t h a t g ~ P ( K ) . Let ~ be the root system of G with respect to S, let ~-~ed = { a l , . . . , a n } and let gi E Ua~,~ for i = 1 , . . . , n with g l " ' ' g n = g 144

(see (12.5)). T h e n we can assume that there exists a number ~ > 1 with U~, C P if and only if i > L Furthermore, we may assume without loss of generality that g~m...mgn:l. Now

there exists an element

s 9

S(K)

such that gsg-ls -I is a non-trivial

element in the subgroup generated by Ual,..., U~_I, hence cannot lie in P(K). This contradicts to the assumption. To be more precise, we have shown that g 9 (U~- fl P(K))U + can be assumed. --A

By (13.3) we know that U+ fixes every element of x + Fp. Finally, it follows --A

--A

from the definition of Fp that UZ M P(K) fixes all elements of x + F p , too. Therefore the assertion is proved. []

D e f i n i t i o n 14.27. (i) Let M be a metric space and let fi C_ M. For c 9 ~ + , the set B~(F/) = { x 9 M :

there e x i s t s y 9 Q w i t h d ( x , y ) <e}

is called the e-blow up of ft. (ii) For f~ C_ 0

X(P/R~,(P)), the set

PEq3

B~(fl) = U B~(fl N X(P/R~,(P))) PEg~ is called the r

up o.ffl in 0 X(P/R~(P)). PEN

The set B~(C~) we will be denoted by C •

Lemma

14.28.

Let x, y 9 X(P/R~(P)) and g 9 G(K). Then d(x, y) = d(gz, gy). m

P r o o f . By (14.7) and (14.13) there exists an a p a r t m e n t A with x,y 6 A. Let F be a vector face with x, y e A/(F). Since the metric on each X(P/R,,(P)) is compatible with the metric chosen on X(G) (see w the assertion follows easily. [] 145

Proposition 14.29.

{C a~,s : x E X(G), f~ C P a r G open and E > 0} is ~he subbasis of a topology on U X ( P / R ~ ( P ) ) for which ~ becomes a homeomorphism. PE~3

P r o o f . In order to prove the assertion, it suffices to show t h a t (a) all ~(C~,e) a C_ X ( G ) are open and t h a t (b) any open subset of X ( G ) is a union of suitable ~(C~fl,~).

fl Ad (a): Let y E ~(C~,e). I f y E X(G), then because of the e-blow up process there exists an open n e i g h b o u r h o o d of y contained in ~(C~,e).fl So let y E -z(a)\x(a) and let ~ = ~ - l ( y ) . T h e n there exists a point z E C a --A

w i t h ~ E B , ( { z } ) and there exist P,Q E f~ with P C_ Q and z E ( x + F p ) o . Here A is an a p a r t m e n t with x E A and SA C_ P. W i t h o u t loss of generality we m a y assume t h a t P = Q. T h e n ~ E Be((x+--PA)p) and therefore y E ~(z+ ~vFr -B,(0) ~ J C -A (see w Let fl~ be an open subset of P in P a r G with fl~ C_ ft. W i t h o u t loss of generality we m a y assume t h a t fY = ~r(Y x 0), where Y _C Par 0 G and 0 C_ V ( A ) are open subsets (14.23)9 If Pl : G(K) ~ (G/P)(K) denotes the canonical projection, then we let G = P~ n p~-l(Y). Since P~ is open (12.12), we get that G C_ G(K) is open. FTF~ ~ C _ ~(C~,z). Now we claim t h a t G 9 (x + ~B~(0)J

In order to prove this, it is

r'P~ J C sufficient to show t h a t G . (x + ~{0} - ~ ( C a ) according to (14.28). Let g E G and ~: E x + ~ {~vF~ 0}.

T h e n gx = x and there exists a parabolic K -

s u b g r o u p /5 with ~ E (x + FpA)p and P C_ /5. Hence A' := gA is an a p a r t m e n t with x E A' and SA, C_ g/59-~ (by construction of G). T h e n according to the definition of C a we obtain g37 E ~ ( C a ) . Ad (b): Let Y C X ( G ) be an open subset and let x E Y. I f z E X(G), then there exists a real n u m b e r s > 0 with Be(x) C Y and we can consider the set fl qo(C,,e) with f~ = {G}. So we can assume t h a t x E X(G)\X(G). Let A be an a p a r t m e n t with x E A. Hence there exists an open n e i g h b o u r h o o d G C_ G(K) of 1 and an open subset C_ A with (1,x) E 0 x f~ C_ p r - l ( Y ) . Here we let pr: G(K) x A --+ X ( G ) be the canonical projection. W i t h o u t loss of generality we m a y assume t h a t f~ is of the form y + c F ( 0 ) for y E A and a vector face F of V. Say a = A ( F ) . T h e n we let 0 = {-r C_ A : r _D or}, and Y' = p l ( G ) C_ F a r 0 G is an open subset (14.22), if p~ : G(K) -+ Faro G denotes the canonical map. Finally, we know t h a t f~' := ~r(Y' x 0) C_ P a r G is open (14.23) and we claim fl' ) C Y. t h a t x E qo(Cy,e 146

F n' Since f~ = y + CB~(0), we obtain x E p(Cv,~). Now let P E f'~'. Then there exists

a minimal parabolic K - s u b g r o u p Q with SA C_ Q and an element g E G with F gQg-i c P. Hence B,((y + Fp)c)) C g. (y + CB~(O)) C_G. a for all (~ _D P. So (a) and (b) are proved.

[]

C o r o l l a r y 14.30.

We have: (i) The metric topology on X(P/R~,(P)) (see (13.15)) coincides with the topology induced by X(G) (for (P E ~3)). (ii) The closure of X(P/Ru(P)) in-X(G) equals U X(Q/R,,(Q)). QEN QCP

P r o o f . Ad (i): Without loss of generality we may assume that P = P r for a ~- C_ A. Obviously, the canonical inclusion X(P/R~,(P) -+ X(G) is continuous. Let 5 E X(P/Ru(P)), let A be an a p a r t m e n t with 5 E A/(F) C_ A and let x E A be an element which is m a p p e d onto ~ under the canonical projection A --+ A/(F). Let s > 0. Then C~n,r M X(P/Ru(P)) is the s-blow up of {x} in X(P/R~(P)), if we let a = 7r(Par0(G ) x {a E A : ~ D ~-}) and if one recalls that 7r is an open m a p (14.23). O

h a (ii): It suffices to prove that

~-X(G)\X(P/R~,(P)) = U X(Q/R~,(Q)). We QEN QgP

0

will show first t h a t P a r G\{P}" = Par G\{Q E ~ : Q c_ P}. The inclusion "C_" is clear, and we only have to show that the right hand side is an open set. Let P ' E P a r ( G ) \ { Q ~ g3 : Q c_ P}. Since all projections pr, are open, it suffices to consider the case in which P ' is a minimal parabolic K-subgroup. If, for example, P is of type % then according to the fact that {P} is a closed subset of Par~-(a) its pre-image under pr, is a closed subset of Par0(G ). Hence there exists an open neighbourhood of P in Par0(G ) which does not meet prjt(P). Since 7r is open, the assertion follows. Hence O

~-X(G)\X(P/R~,(P)) D U X(Q/R~,(Q)) . QE~ Q(ZP

Suppose that the interior of X(G)\X(P/R~,(P)) is strictly bigger. Then by (14.29) there exist a point x E X(G), an open subset f~ C_ Par(G) with P E gt O

and a real number e > 0 with C~a,~ C ~-R(G)\X(P/R~(P)). Since Q is open, it follows from the statements proved above that P E f~, and therefore C~a,r N X(P/R~(P)) # O, in contrast to the assumption. [] 147

w

Finally, let us summarize all properties of X(G): Theorem

14.31.

The topological space X ( G) satisfies: (i)

X ( G ) is Hausdorff, compact and contractible,

(ii) -X(G) =

0 X ( P / R u ( P ) ) , where the induce topology on X ( P / R ~ ( P ) ) PeV coincides with the metric topology (13.15) and

(iii) the continuous a ( K ) - a c t i o n on x ( a ) G ( g ) - a c t i o n on X ( G ) .

w

Example:

can be extended to a continuous []

SL~

In this last paragraph we will give some details for the SLy-case. More information about it and about the other classical groups can be found in [BT 1] 10. Let G = SL~ with n _> 2 and denote by S the maximal (K-split) torus of G corresponding to the diagonal matrices. Since SLn is split, we have S = Z. Moreover Zb(K) is the group of diagonal matrices with entries in o~c.x Let q~ be the root system of SL~ with respect to S and denote by Xi : S ~ ~,~ the character defined by x i ( d i a g ( s t , . . . , sn)) = si. Then A := { X 2 - X l , . . . , X , ~ - X~-I} is the basis of 9 with respect to the upper unipotent matrices. Hence we can identify 9 with the set of pairs (i,j) with i,j ~ { 1 , . . . , n } and i r j. So the root subgroup U~ consists of matrices of the form 1 + (aij) with alj = 0 if i r l and j r m if a corresponds to (l, m). We can ic[entify V ( = X . ( S ) | with P ~ / < ( 1 , . . . , 1) >. Let us assume that w is normalized in such a way that w(K*) = 2 . Then the walls in A = V are the affine hyperplanes of the form

H(ij),n := {v E V :< v, Xi - Xj > = m} where i,j E { 1 , . . . , n} with i r j and m E 2~. Since N ( K ) ~- S ( K ) x vW, where vW denotes the Weylgroup of ~, the action of N ( K ) on V is clear. Let x E V be a point. The structure of U~,~, Us and P~ was discussed in w and w explicitly. So we omit this discussion here. 148

Now let x E V be a special point. Then O~ is isomorphic to SLn/oK and ~x is isomorphic to SL,/k (in particular R , ( ~ ) is trivial). Hence the combinatorial structure of the faces near any given special point can be described in terms of the combinatorial Tits building of SLn/k. In order to get a better understanding of the compactification, we restrict ourselves to the set of special points in X(G). These correspond to full OK-lattices in K " up to homothety. Here, a full oK-lattice in K '~ is a free OK-SUbmodule of K ~ of rank n which contains a basis of K ~. By (14.31) (ii) we know that

-X(G) = U X(P/R~(P)) PE~3

So we have to generalize the notion of a full oK-lattice to obtain a correspondence between these and all special points of all X(P/R~(P)). D e f i n i t i o n 15.1

A generalized oK-lattice in K n consists of the following data: (i)

A decomposition K n = V1 | ... G V~ by subspaces V~.

(ii)

Anumber0


dimV/ > l f o r i < i .

(iii) For each 1 < i < e a full oK-lattice Li in V/. We call two generalized lattices equivalent, if the decompositions coincide (up to numbering), if the numbers l are equal and if the lattices in the same V/ are equal up to homothety. Then it is clear that the set of all special points in X(G) (in the above sense) can be identified with the set of all generalized oK-lattices.

149

References [AMRT] Ash, A.; Mumford, D.; Rapoport, M.; Tai, Y.S.: Smooth Compactification of Locally Symmetric Varieties. Math. Sci. Press, Brookline, Massachusetts (1975). [BLR]

Bosch, S.; Liitkebohmert, W.; Raynaud, M.: N~ron Models. Springer, Berlin (Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Band 21) (1990).

[Bo]

Borel, A.: Linear Algebraic Groups; Second Enlarged Edition. Springer, New York (GTM 126) (1991).

[BoSe 1] Borel, A.; Serre, J-P.: Corners and arithmetic groups. Comm. Math. Helv. 48 (1973), 463-491. [BoSe 2] Borel, A.; Serre, J-P.: Cohomologie d' immeubles et de groupes Sarithm~tiques. Topologie 15 (1976), 211-232. [BoSp]

Borel, A.; Springer, T.A.: Rationality properties of linear algebraic groups II. Tohoku Math. J. 20 (1968), 443-497.

[BoTi]

Borel, A.; Tits, J.: Groupes r~ductifs. Publ. Math. (IHES) 27 (1965), 55-152.

[Bou 1] Bourbaki, N.: Groupes et alg~bres de Lie. IV, V, VI. Hermann, Paris (1968).

[Bou 2] Bourbaki, N.: Topologie G~n~rale. I. Hermann, Paris (1966). [Br 1] Brown, K.S.: Buildings. Springer, New York (1989). [Br 2] Brown, K.S.: Cohomology of Groups. Springer, New York (GTM 87) (1982). [BT 1,2] Bruhat, F.; Tits, J.: Groupes %ductifs sur un corps local I, II. Publ. Math. (IHES) 41, 60 (1972, 84).

[CaFr]

Cassels, J.W.; Fr5hlich, A.: Algebraic Number Theory. Press, London (1967).

[EGA I] Grothendieck, A.:t~l~ments de G~om~trie Alg~brique. Math. (IHES) 4 (1960).

Academic

Paris, Publ.

ICe]

G~rardin, P.: Harmonic functions on buildings of reductive split groups. In: Operator Algebras and Groups Representations. Pitman (1984), 208-221.

[Hu]

Humphreys, J.E.: Linear Algebraic Groups. Springer, New York (GTM 21) (1981).

[Ja]

Jantzen, J.C.: Representations of Algebraic Groups. Academic Press, Boston (1987). 150

[KKMS] Kempf, G.; Knudsen, F.; Mumford, D.; Saint-Donat, B.: Toroidal Embeddings I. Springer, Berlin (Lecture Notes in Mathematics 339) (1973).

[Mi]

Milne, J.S.: ]~tale Cohomology. Princeton University Press, Princeton, New Jersey (1980).

[La]

Lang, S.: Algebraic Number Theory. Springer, New York (GTM 110) 1986.

[Pr]

Prasad, G.: Elementary proof of a theorem of Bruhat-Tits and Rousseau and of a theorem of Tits. Bull. Soc. Math. Fr. 110 (1982), 197-202.

[s 1]

Satake, I.: On representations and compactifications of symmetric Riemannian spaces. Ann. Math. 71 (1960), 77-110.

[s 2]

Satake, I.: On compactifications of the quotient spaces for arithmetically defined discontinuous groups. Ann. Math. 72 (1960), 555-580.

[Se]

Serre, J-P.: Cohomologie Galoisienne. Springer, Berlin (Lecture Notes in Mathematics 5) (1964; 5~me 6dition, r@vis@eet compl@t~e, 1994).

[SGA 3] Demazure, M.; Grothendieck, A.: Sch@masen Groupes I, II, III. Springer, Berlin (Lecture Notes in Mathematics 151, 152, 153) (1970).

[st 1]

Steinberg, R.: Variations on a Theme of Chevalley. Pac. J. of Math. IX (1959), 875-891.

[st 2]

Steinberg, R.: Regular elements of semi-simple algebraic groups. Publ. Math. (IHES) 25 (1965), 49-80.

[Ti 1]

Tits, J.: Classification of algebraic semisimple groups. Proc. Symp. Pure Math. 9 (1966), 33-62.

[Ti 2]

Tits, J.: On buildings and their applications. Proc. Int. Conv. Math., Vancouver (1975), 209-220.

[Ti 3]

Tits, J.: Reductive groups over local fields. Proc. Symp. Pure Math. 33 (1979), 29-69.

[We]

Weil, A.: Foundations of Algebraic Geometry. Am. Math. Soc., Colloquium Publ. XXIX, Rhode Island (1962).

151

Index affine root, 68, 118 apartment, 14, 67 empty, 20 full, 75,121 big cell, 50 birational group law, 59 Borel-Serre topology, 26 Bruhat decomposition, 86, 125 Bruhat-Tits building, 96, 131 canonical OK-group scheme, 33 canonical R-scheme, 9 chamber, 74, 90, 120, 128 vector, 10 Chevalley system, 38 Chevalley Steinberg system, 38 closed subset, 10 positively, 10 combinatorial Tits building, 96, 132 compactification, 22, 133 of A, 29 of V, 22 of X(G), 133 corner, 22 direction, 68, 118 distribution module, 5 divisible root, 10 ~-blow up, 145 ~pinglage, 37 Steinberg, 37 extension principle, 4 extreme root, 10 face, 74, 90, 120, 128 vector, 10 fixed-point-lemma, 100 Iwasawa decomposition, 83, 125 maximal subset, 111 ordering, 10 good, 10 polyhedral compactification, 1 quasi-split, 11 oK-rational map, 59

relative root system, ii root subgroup, 11, 12 sector, 90, 128 simplicial closure, 75, 121 smooth model, 50 special point, 109 split, 11 standard parabolic subgroup, 12 standard topology, 23 V-axioms, 67 walls, 74, 120 Weil restriction, 5, 6

152