Introduction to Arithmetic Groups 2019033024, 9781470452315, 9781470454319

Table of contents :
Cover
Title page
Preface to the English Translation
Introduction
Notation
Chapter I. Some Classical Groups
1. Siegel sets and reduction in ??(?,ℝ)
2. Reduction of positive-definite quadratic forms
3. Bruhat decomposition of ??(?,?)
4. The Siegel property in ??_{?}
5. Reduction of indefinite quadratic forms
6. A finiteness lemma
Chapter II. Algebraic Groups
7. A review of algebraic groups. Arithmetic groups
8. Compactness criterion
9. Fundamental sets (first type)
Chapter III. Fundamental Sets with Cusps
10. Algebraic tori
11. Parabolic subgroups. Bruhat decomposition
12. Siegel sets
13. Fundamental sets (second type)
14. Fundamental representations. Associated functions
15. The Siegel property
16. Fundamental sets and minima
17. Groups with rational rank one
Bibliography
Index
Back Cover

Citation preview

UNIVERSITY LECTURE SERIES VOLUME 73

Introduction to Arithmetic Groups Armand Borel

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

10.1090/ulect/073

Introduction to Arithmetic Groups

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

UNIVERSITY LECTURE SERIES VOLUME 73

Introduction to Arithmetic Groups Armand Borel Translated by Lam Laurent Pham Translation edited by Dave Witte Morris

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EDITORIAL COMMITTEE Robert Guralnick Emily Riehl

William P. Minicozzi II (Chair) Tatiana Toro

This work was originally published in French by HERMANN under the title: Introduction c HERMANN, www.editions-hermann.fr, aux groupes arithm´etiques, Armand BOREL,  Paris 1969.

2010 Mathematics Subject Classification. Primary 22E40; Secondary 11F06, 20G30.

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Library of Congress Cataloging-in-Publication Data Names: Borel, Armand, author. | Pham, Lam Laurent, translator. | Morris, Dave Witte, translator. Title: Introduction to arithmetic groups / Armand Borel ; translated by Lam Laurent Pham ; translation edited by Dave Witte Morris. Other titles: Introduction aux groupes arithm´ etiques. English Description: Providence, Rhode Island : American Mathematical Society, [2019] | Series: University lecture series, 1047-3998 ; volume 73 | Originally published in French: Introduction aux groupes arithm´ etiques / Armand Borel (Paris : Hermann, 1969). | Includes bibliographical references and index. Identifiers: LCCN 2019033024 | ISBN 9781470452315 (paperback) | ISBN 9781470454319 (ebook) Subjects: LCSH: Linear algebraic groups. | Group theory. | Set theory. | AMS: Number theory – Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45} – Structure of modular gr | Topological groups, Lie groups {For transformation groups, see 54H15, 57Sxx, 58-XX. For abstract harmonic analysis, see 43-XX} – Lie groups {For the topology of Lie groups and homogeneous spaces} Classification: LCC QA179 .B6713 2019 | DDC 512/.2–dc23 LC record available at https://lccn.loc.gov/2019033024

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Contents Preface to the English Translation

vii

Introduction

ix

Notation

xi

Chapter I. Some Classical Groups 1. Siegel sets and reduction in GL(n, R) 2. Reduction of positive-definite quadratic forms 3. Bruhat decomposition of GL(n, k) 4. The Siegel property in GLn 5. Reduction of indefinite quadratic forms 6. A finiteness lemma

1 1 7 12 17 23 27

Chapter II. Algebraic Groups 7. A review of algebraic groups. Arithmetic groups 8. Compactness criterion 9. Fundamental sets (first type)

31 31 41 49

Chapter III. Fundamental Sets with Cusps 10. Algebraic tori 11. Parabolic subgroups. Bruhat decomposition 12. Siegel sets 13. Fundamental sets (second type) 14. Fundamental representations. Associated functions 15. The Siegel property 16. Fundamental sets and minima 17. Groups with rational rank one

55 55 60 74 79 84 88 98 108

Bibliography

115

Index

117

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Preface to the English Translation Fifty years after it made the transition from mimeographed lecture notes to a published book, the late Armand Borel’s Introduction aux groupes arithm´etiques is still very important for the theory of arithmetic groups. Chapter III remains the standard reference for fundamental results on reduction theory. Before presenting a few suggestions for further reading, we first note that reduction theory is crucial for our understanding of the structure of the spaces GR /Γ and K\GR /Γ. An example of this is given in the book’s final section (§17), which describes a compactification of K\GR /Γ in the special case where the Q-rank of G is equal to 1. This compactification is smooth, but, in higher rank, the boundary of a compactification usually has corners, rather than being a smooth manifold. For a discussion of later work that uses the “Siegel sets” of this book to construct several different compactifications of K\GR /Γ, see: [BJ] A. Borel and L. Ji, Compactifications of symmetric and locally symmetric spaces, Birkh¨ auser, Boston, MA, 2006. MR2189882 For the general theory of arithmetic groups, three books (other than Introduction aux groupes arithm´etiques) are often listed as essential reading: [Ma] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Springer, Berlin, 1991. MR1090825 [PR] V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Academic Press, Boston, 1994. MR1278263 [Ra] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer, New York, 1972. MR0507234 Overviews that are more recent (but without detailed proofs) include: [Ji] L. Ji, Arithmetic groups and their generalizations, American Mathematical Society, Providence, RI, 2008. MR2410298 [Mo] D. W. Morris, Introduction to arithmetic groups, Deductive Press, 2015. MR3307755 In this translation, the numbering of theorems, equations, and other material in the main text of the original French manuscript has been reproduced exactly (although page numbers may have changed). However, the same cannot be said of the bibliography, partly because it includes the references that Borel inserted in the main text, rather than in his bibliography. (We also note that items in the bibliography have been updated to their latest version.) The Math Review of the original French manuscript observes that “the style is concise and the proofs (in later sections) are often demanding of the reader.” To make the translation more approachable, numerous footnotes provide comments that are intended to be helpful. (All of the footnotes are new; the original book vii Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

viii

PREFACE TO THE ENGLISH TRANSLATION

had no footnotes at all.) Unless marked Translator’s note, they were added by the editor. Typographical errors and other minor issues have been silently corrected, but significant deviations from the original French manuscript (including changes of notation) are described in footnotes. I apologize for any errors that remain (and, even more, for any new errors that I introduced!). I would like to thank Lam Pham and the staff of the AMS Book Program for making this classic monograph accessible to a wider audience. Dave Morris Lethbridge, July 2019

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Introduction This book is based on the first part of a graduate course that was given at the Institut Henri Poincar´e in 1964. It focuses on the so-called “reduction theory” in a real algebraic group GR , with respect to an arithmetic group Γ. The proofs of the general theorems make extensive use of the theory of linear algebraic groups. However, since the students in the course were not necessarily assumed to be acquainted with this, we first gave direct proofs of some classical cases (after all, these are where the general theory originates from), and we summarized the necessary notions and results on linear algebraic groups, as we needed them. Most of these summaries are in three sections (§7, §10, §11), which also contain some examples and proofs and thus provide, to a certain extent, an introduction to some aspects of the theory of algebraic groups. By reduction in GR , with respect to Γ, we mean, roughly speaking, the search for (open or closed) subsets that meet each orbit of Γ (acting by translations on the right) in at least one point, but never in more than a finite number of points. Such sets are called fundamental sets. (Actually, we will impose precise conditions that are more restrictive (cf. 5.6, 9.6, 15.13).) Alternatively, but equivalently, the problem can be considered in the space X = K\GR of right cosets of GR modulo a maximal compact subgroup K. When G is a classical group, we encounter, as a special case, the reduction theory of quadratic forms (and of hermitian forms). The book has three parts. The first (§1 to §6) is mainly devoted to the reduction of quadratic forms, using methods that will be generalized in a natural way in later sections. We first consider the case where G = GL(n, R), Γ = GL(n, Z), and K = O(n), that is, where X is the space of positive-definite quadratic forms on Rn . We show that every orbit of Γ in G meets a suitable Siegel set (1.4) and we deduce a few consequences, including Mahler’s criterion for the relative compactness of a subset of the space GL(n, R)/ GL(n, Z) of lattices of Rn , and the finiteness of the volume of SL(n, R)/ SL(n, Z). §2 translates these results into the language of quadratic forms, and establishes some links with Minkowski reduction. §4 shows that a Siegel set only meets a finite number of its translates by elements of x . Γ, for each x ∈ GL(n, Q), (4.6). §5 is devoted to the reduction of indefinite quadratic forms, using Hermite’s method. Its key point is a finiteness property of “reduced integral” forms, which we will deduce from a more general lemma that is proved in § 6. The second part (§7, §8, §9) is devoted to two general theorems on arithmetic groups, whose proofs require only a fairly restricted collection of results on algebraic groups (which will be recalled or proved in §7). The two main theorems are: a compactness criterion for the quotient GR /Γ (§8), and a first construction of fundamental sets, which generalizes the method of Hermite. Furthermore, §8 shows that the image of an arithmetic group by an isogeny is also an arithmetic group, ix Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

x

INTRODUCTION

and §9 establishes a finiteness theorem for the number of orbits of Γ in the set of integer points of a closed orbit of G, in the space of a linear representation of G. This generalizes the finiteness of the number of classes of quadratic forms of given non-zero determinant (6.4), and also generalizes results of Jordan on the classes of homogeneous forms of degree ≥ 3, (6.5). The third part (§10 to §17) is devoted to fundamental sets that are usually better than those of §9. Their existence is proved in two very different ways: in §13, where we rely on §9, and in §16, where we apply an extremum principle to a type of function that is studied in §14, and which generalizes, among others, the function |cz + d| of the Poincar´e upper half-plane. These sets are the union of a finite number of translates (by elements of GQ ) of a set of a simple form, which is called a Siegel set. Finally, §17 describes, in a particular case, the space K\GR /Γ as the interior of a compact manifold with boundary. The intention to make the first part self-contained, and to appeal to the theory of algebraic groups only when necessary, has led to some redundancies and inconsistencies. For example, §4 is a particular case of §15, but is not assumed in the latter, and the existence of a Bruhat decomposition is proved in §3 for GL(n, k), while it is assumed without proof in a much more general setting from §11 onward. Consequently, this exposition, which, grosso modo, follows the chronological order, and contains some quite extended “reviews,” is not the most efficient possible, and reading a particular section does not necessarily require reading all of the preceding ones. Here are a few additional remarks on the interdependence of the various sections, which may serve as a “Leitfaden”: §1, up to 1.11, is fundamental for the entire book, but the remainder of that section, and §2 to §5, are not used later, other than for providing concrete examples of the theory; readers who wish to reach the general theorems as quickly as possible may focus their attention on §§1, 8, 12, 14, 15, 16, if they are willing to assume a certain finiteness property whose proof here relies on §13, but which can be established more directly by using the adelic analogue of §§1 and 8 (cf. introduction to §16); finally, §6 plays a crucial role in §5 and §9, and this latter is used in §13, but nowhere else. A first draft of the course notes (duplicated and distributed by the Institut Henri Poincar´e) was written by H. Jacquet, J.-J. Sansuc and J.-P. Jouanolou. It was very useful to me, and I warmly thank the authors. I would also like to thank J. E. Humphreys, who read the manuscript, pointed out a considerable number of “misprints” and suggested some improvements in the exposition, and also A. Robert and J. Joel, for having helped me to correct the proofs. Armand Borel Princeton, November 1968

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Notation 0.1. Z is the ring of integers,1 Q, R, C denote the fields of rational, real, and complex numbers respectively, and N is the set of integers that are ≥ 0. If A is a ring with unit, A∗ denotes the multiplicative group of invertible elements of A. If A is a commutative ring, M(n, A) is the ring of square matrices of order n, with entries in A, GL(n, A) or GLn (A) is the group of square matrices of order n with entries in A, whose determinant is a unit of A, and SLn (A) or SL(n, A) is the subgroup consisting of the elements of GLn (A) whose determinant is 1. is the subgroup of GL(n, R) that leaves invariant the quadratic form  O(n) 2 i xi and SO(n) = O(n) ∩ SL(n, R). If p and q are integers that are ≥ 0 and n = p + q, then O(p, q) is the subgroup of GL(n, R) that leaves invariant the quadratic form (x21 + . . . + x2p ) − (x2p+1 + . . . + x2p+q ), and SO(p, q) = O(p, q) ∩ SL(n, R). 0.2. Let G be a group and α be a homomorphism of G into C∗ . The value of α at g ∈ G will be denoted by α(g) or, alternatively, g α . The latter notation implies that the pointwise product of homomorphisms of G into C∗ will be written in additive notation.2 0.3. Let G be a group. If g ∈ G, we denote by int(g) the inner automorphism x → g . x . g −1 of G. If A and H are subsets of G, then A H denotes the union of the sets a . H . a−1 (a ∈ A). Assume that Vi (1 ≤ i ≤ n) are sets and that fi : Vi → G are functions. The function f : V1 × . . . × Vn → G that is defined by (v1 , . . . , vn ) → f1 (v1 ) · . . . · fn (vn ) is called the product map of the fi . 0.4. Let G be a group and Gi (1 ≤ i ≤ n) be normal subgroups of G. We say that G is an almost direct product of the Gi if the product map of the natural inclusions of the Gi into G is surjective, with finite kernel.3 0.5. A function taking values in a topological space is said to be bounded if its image is relatively compact. Let X be a topological space, and let f , g be real-valued functions on X whose values are ≥ 0. We write: f ≺ g, original French manuscript uses Z, Q, R, C, N, rather than Z, Q, R, C, N. means that if α and β are homomorphisms from G to C∗ , then the homomorphism α + β is defined by g α+β = g α. g β . 3 The subgroups G are also required to centralize each other, so the product map is a homoi morphism. This requirement follows from the other assumptions if each Gi is connected, but not in general. 1 The

2 This

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xii

NOTATION

if there exists a constant c > 0 such that f (x) ≤ c . g(x) for all x ∈ X. Also, we write f g if g ≺ f , and write f  g if we simultaneously have f ≺ g and f g. The relation f  g therefore means that there exist constants c, d > 0 such that: c . f (x) ≤ g(x) ≤ d . f (x)

(x ∈ X).

If f ≺ g (resp. f g, resp. f  g), we may sometimes say that f essentially bounds g below (resp. essentially bounds g above, resp. is comparable to g). 0.6.4 The normalizer of a subgroup H of a group G is denoted5 N(H). The centralizer of H in G is denoted6 Z(H) or ZG (H). In particular, Z(G) is the center of G. The transpose of a matrix g is denoted tg.

4 This

paragraph was added by the editor. It is not in the original French manuscript. original French manuscript uses N(H) (or sometimes Norm(H)) to denote the normalizer of a subgroup H, but the symbol N has another important role as a factor in the Iwasawa decomposition G = K . A . N. 6 The original French manuscript uses an ordinary Z instead of Z. 5 The

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10.1090/ulect/073/01

CHAPTER I

Some Classical Groups 1. Siegel sets and reduction in GL(n, R) We will denote by G the group GL(n, R) of invertible square matrices of order n with real entries and by Γ the subgroup GL(n, Z) of G consisting of the matrices of determinant ±1 with integer entries. G is a real Lie group, and Γ is a discrete subgroup of G. Our goal is to obtain a good approximation of a system of representatives of the left cosets of Γ in G. To facilitate this, let us first recall the Iwasawa decomposition of GL(n, R) [10, Chap. VII, §3, Proposition 7]: 1.1. Proposition. Let A be the group of diagonal matrices with strictly positive entries. If K and N, respectively, denote the orthogonal group and the “strictly upper triangular group,” which consists of the upper triangular matrices with all eigenvalues equal to one, then the map: (k, a, n) → k . a . n is a homeomorphism from K × A × N onto GL(n, R). For g ∈ G, we will use the notation g = kg . ag . ng for its Iwasawa decomposition. We then pose: 1.2. Definition. A Siegel set 7 of GL(n, R) is any set of the form St,u = K . At . Nu (t, u ∈ R∗+ ), where: At = {a ∈ A | aii ≤ t . ai+1,i+1

(i = 1, . . . , n − 1)}

and Nu = {n ∈ N | |nij | ≤ u

(1 ≤ i < j ≤ n)}.

It is well known that N is a closed subgroup of G, and that it is homeomorphic to Rm (m = n(n − 1)/2) under the mapping θ : n → (nij )1≤i 0) and that S . h is contained in a Siegel set if h ∈ A . N. The following result establishes a fundamental property of the sets At :  1.3. Lemma. If ω is relatively compact in N, then a∈At a . ω . a−1 is also relatively compact in N. Proof. Indeed, if n = (nij ) ∈ ω, then (a . n . a−1 )ij = (aii /ajj ) . nij , whence: |(a . n . a−1 )ij | ≤ tj−i. |nij | 7 This

if i < j.

will sometimes also be called a Siegel domain. 1

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2

I. SOME CLASSICAL GROUPS

Let us note (we will need this later on) that the Haar measure on N is the image under θ−1 of Lebesgue measure on Rm , so the modulus of the automorphism: int(a) : n → a . n . a−1 of N is |detRm (int a)| =

 aii . a i 0 on Γ. Indeed, we have g . Γ. e1 ⊂ g(Zn  {0}), so g . Γ. e1 consists of the non-zero elements of a lattice of Rn . 1.5. Lemma. Let g ∈ G and let g = k . a . n be √ its Iwasawa decomposition. Assume that Φ(g) ≤ Φ(g . γ) (γ ∈ Γ). Then a11 ≤ (2/ 3) . a22 . Proof. If u ∈ NZ , then Φ(g . u) = Φ(g) and ag . u = ag . Therefore, in view of (1), we may assume that |n12 | ≤ 1/2. Let z ∈ Γ be the element that interchanges e1 and e2 , and fixes ei (3 ≤ i ≤ n). We then have: g . z(e1 ) = g(e2 ) = k . a . n(e2 ) = k . a . (e2 + n12 . e1 ) = k(a22 . e2 + a11 . n12 . e1 ), so Φ(g . z)2 = a222 + a211 . n212 ≤ a222 +

a211 . 4

Since Φ(g) = a11 , the assumption implies: a211 , 4 whence the lemma follows. a211 ≤ a222 +

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1. SIEGEL SETS AND REDUCTION IN GL(n, R)

3

Theorem 1.4 is a consequence of the following more precise result: 1.6. Theorem. Let g ∈ G. The minimum of Φ on g . Γ is attained at a point of g . Γ ∩ S2/√3,1/2 . Proof. We write S0 for S2/√3,1/2 . The proof proceeds by induction on n. For n = 1, G = S0 , so there is nothing to prove. Let x ∈ G. We can find y ∈ x . Γ such that Φ(y) ≤ Φ(x . γ) (γ ∈ Γ), whence also Φ(y) ≤ Φ(y . γ) (γ ∈ Γ). We can write:   a11 ∗ ky−1. y = , b ∈ GL(n − 1, R). 0 b By the induction hypothesis, there exists z  ∈ GL(n − 1, Z) (n−1)

where S0

such that

b . z  ∈ S0

(n−1)

,

denotes the Siegel domain S2/√3,1/2 of GL(n − 1, R). Let:

b . z  = k. a. n be the Iwasawa decomposition of b . z  . Then:   a11 ∗ −1 ky . y . z = = k. a. n , 0 k. a. n with

   a11 0 1 ∗  = ∈ A, n ∈ N. 0 n 0 a √ By construction, we have (a )ii ≤ (2/ 3)ai+1,i+1 (2 ≤ i < n). But z fixes e1 , so Φ(y . z) = Φ(y), and therefore: k =

  1 0 ∈ K, 0 k

  1 0 z= , 0 z

a =

Φ(y . z) ≤ Φ(y . z . γ) Lemma 1.5 then shows that (a )11



√ ≤ (2/ 3) . (a )22 . Thus,

(γ ∈ Γ).

y . z ∈ K . A2/√3 . N, so, using 1.4 (1): x ∈ y . Γ ⊂ K . A2/√3 . N . Γ = S0 . Γ.



1.7. Corollary (Hermite). Let g ∈ G. Then: √ min

g(x) ≤ (2/ 3)(n−1)/2 . |det g|1/n . n z∈Z {0}

Proof. We can find an element g  ∈ g . Γ ∩ S2/√3,1/2 that satisfies (1.6). Since Γ consists of elements with determinant ±1, we have |det g| = |det g  |. On the other hand: min

x∈Zn {0}

g(x) ≤ min g . γ(e1 ) = g  (e1 ) = a11 , γ∈Γ

where a is the component in A of g  . Since a ∈ A2/√3 , we get √ √ (a11 )n ≤ (2/ 3)n(n−1)/2 . a11 · . . . · ann = (2/ 3)n(n−1)/2 |det g|.

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4

I. SOME CLASSICAL GROUPS

1.8. Remark. Let 1 be a norm on Rn , i.e., a continuous function that is strictly positive other than at the origin, such that r . x 1 = |r| . x 1 (r ∈ R, x ∈ Rn ). Then there exist two constants d, d > 0 such that: d x ≤ x 1 ≤ d x

(x ∈ Rn ).

(This is clear on the unit sphere, so it is then true on Rn by homogeneity.) The corollary then implies the existence of a constant C > 0 such that: (1)

min

x∈Zn {0}

g(x) 1 ≤ C . |det g|1/n

(g ∈ G).

We now move on to an application that will play an important role in the sequel. Let R be the set of lattices of Rn . It can be identified with G/Γ, which provides it with a topology. We will use Δ to denote the function on R that maps a lattice L to the euclidean volume of the parallelepiped defined by a basis of L. If L = g(L0 ) where L0 = Zn , we therefore have Δ(L) = |det g|. 1.9. Corollary (Mahler’s criterion). Let M ⊂ R. Then the following two conditions are equivalent: (a) M is relatively compact; (b) Δ is bounded on M and there exists a neighborhood U of the origin in Rn such that L ∩ U = {0} for every L ∈ M. Proof. Let S be a Siegel set that is mapped onto R by the map g → g(L0 ) (cf. 1.4). It is clear that (a) is equivalent to the existence of a relatively compact subset M ⊂ S such that M (L0 ) = M. On the other hand, M ⊂ S is relatively compact if, and only if, the components ax (x ∈ M ) form a relatively compact set in A, hence, if, and only if, there exist two constants α, β > 0 such that (1)

(g ∈ M ; i = 1, . . . , n).

α ≤ (ag )ii ≤ β

Therefore, we must show that (1) is equivalent to: |det g| is bounded on M , and there exists c > 0 such that g(x) ≥ c for every x ∈ Zn  {0}, g ∈ M .  (1) ⇒ (2). We have |det g| = (ag )ii , so |det g| is bounded. Let x ∈ Zn  {0}. k We can write x = i=1 mi . ei with mi ∈ Z and mk = 0. Then, g(x) = ag . ng (x) and the k-th coordinate of ag . ng (x) is (ag )k,k mk , so g(x) ≥ α. (2)

(2) ⇒ (1). We have g(e1 ) = (ag )11 ≥ c. Since the ag (g ∈ M ) belong to a set At , this inequality implies the existence of a constant α > 0 such that (ag )ii ≥ α  for all i. Since the product of the (ag )ii is bounded, this implies (1). 1.10. Siegel sets and reduction in SL(n, R). The above-described Iwasawa decomposition in GL(n, R) induces a decomposition in SL(n, R): SL(n, R) = SO(n) . A∗ . N,

(A∗ = SL(n, R) ∩ A).

We define the Siegel sets S∗t,u in SL(n, R) exactly as in GL(n, R). Thus, we have S∗t,u = SL(n, R) ∩ St,u . Theorems 1.4 and 1.6 remain valid. The only new point

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1. SIEGEL SETS AND REDUCTION IN GL(n, R)

5

is that the invariant volume of the quotient SL(n, R)/ SL(n, Z) is finite. In view of 1.4, this follows from: 1.11. Lemma. The volume of every Siegel set S∗t,u of SL(n, R) is finite (if the volume is calculated with respect to a Haar measure). Proof. Let K∗ = SO(n) and B∗ = A∗. N. Let dk, da, dn be Haar measures on K∗ , A∗ , N. These measures are necessarily bi-invariant, since the groups are unimodular. Let us show that the homeomorphism provided by the Iwasawa decomposition maps a Haar measure dg of G to   aii /ajj . (1) ρ(a) . dk . da . dn a ∈ A∗ ; ρ(a) = i 0. But we can write: n−1  bri i , (bi = aii /ai+1,i+1 ; 1 ≤ i < n) ρ(a) = i=1

where each ri is a strictly positive integer. The bi form a coordinate system on A∗ . On the other hand, the map (yi )1≤i 0 such that we can find i linearly independent elements xj ∈ L such that F(xj ) ≤ a, (1 ≤ j ≤ i; i = 1, . . . , n). Then: (1)

mi (F) ≤ F(ui ) ≤ (3/2)2(i−1). mi (F)

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(1 ≤ i ≤ n).

10

I. SOME CLASSICAL GROUPS

Proof. We clearly have: (2)

m1 (F) ≤ . . . ≤ mn (F),

m1 (F) = F(u1 ),

F(ui ) ≥ mi (F)

(1 ≤ i ≤ n).

Since the lemma is obvious for i = 1, we assume that i > 1, and that(1) is true for j < i. There exist n linearly independent elements tk ∈ L, such that: (1 ≤ k ≤ n),

F(tk ) = mk (F)

and an index j ≤ i such that (tj , u1 , . . . , ui−1 ) are linearly independent. Denote by L and M the Z-modules generated respectively by u1 , . . . , ui−1 and u1 , . . . , ui−1 , tj , and let: M = L ∩ (Ru1 + . . . + Rui−1 + Rtj ). We then have M ⊃ M ⊃ L , and M is clearly13 a direct factor in L. Hence, it suffices to prove the existence of an element t of M such that (t , u1 , . . . , ui−1 ) is a basis of M and (3)

F(t ) ≤ (3/2)2(i−1) . mi (F).

The group L is a direct factor in L, so also in M and M. It follows that M /L is a free module of rank one, and M/L is of finite index in M /L . If t is an element of M whose class modulo L is a generator of M /L , we see that (t , u1 , . . . , ui−1 ) is a basis of M and that tj ≡ c . t mod L , (c ∈ Z, c = 0). This can be written: t = c−1. tj + c1 . u1 + . . . + ci−1 . ui−1 

(ck ∈ Q; 1 ≤ k ≤ i − 1).



By adding an appropriate element of L to t , we may assume |ck | ≤ 1/2. We then have:14  1 F(t )1/2 ≤ |c|−1 F(tj )1/2 + F(u1 )1/2 + . . . + F(ui−1 )1/2 . 2 Taking into account the induction hypothesis, we obtain:  i−1 1   1/2 1/2 k−1 1/2 F(t ) ≤ mj (F) + (3/2) . mk (F) , 2 k=1

15

from which (3) follows, since the sequence of mk (F) is increasing.



2.6. Theorem (Minkowski). Let n > 0 be an integer. There exists a constant Cn such that we have: F(e1 ) · . . . · F(en ) ≤ Cn . det F, for every positive-definite quadratic form on Rn that is M-reduced. Proof. We keep the above notation, in particular for H, A, N. Let F ∈ H. It is of the form F = t(a . n) . a . n = tn . a2. n. Denoting by ai the diagonal terms of a, we therefore have: n  (1) F(x) = a2i . (xi + ni,i+1 . xi+1 + . . . + ni,n . xn )2 , i=1

finitely generated abelian group is a direct product of cyclic groups. Since L/M is torsion free, this implies that L/M is a free abelian group. Hence, the identity map L/M → L/M lifts to a homomorphism L/M → L. So M is a direct factor of L. 14 F(x)1/2 is a norm on Rn , so it satisfies the triangle inequality. i−1 15 1 + 1 k−1 = (3/2)i−1 . k=1 (3/2) 2 13 Every

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2. REDUCTION OF POSITIVE-DEFINITE QUADRATIC FORMS

11

and, in particular: F(ei ) = a2i + a2i−1 . n2i−1,i + . . . + a21 . n21,i ,

(2) so also:

a2i ≤ F(ei )

(3)

(1 ≤ i ≤ n). 

If F remains in a Siegel set S , we see from (2) that: (F ∈ S )

a2i  F(ei )

(4)

(cf. 0.5 for the notation , ≺, ). By 2.2, there exists g ∈ GL(n, Z) such that F = F[g] ∈ S0 , where we write  S0 for S4/3,1/2 . We have: F = t(a. n ) . a. n

(5)

(a ∈ A2/√3 , n ∈ N1/2 ).

It is clear that   (6) a2i = det F = det F = ai 2 , i

i

and that, in the notation of 2.5, mi (F) = mi (F )

(7)

In view of (4) and (7), we have: mi (F) ≺ ai 2 ,

(8)

(1 ≤ i ≤ n). 16

(1 ≤ i ≤ n;

F ∈ H,

F ∈ S0 ∩ F[GL(n, Z)]).

Let us now assume that F is M-reduced. Then, (3), (8) and 2.5 imply: a2i ≤ F(ei ) ≺ ai 2

(9)

(1 ≤ i ≤ n;

F ∈ M), 

and the conclusion therefore follows from (6).17

2.7. Corollary. The set M of M-reduced forms is contained in a Siegel set.  2  ai F(ei ) show that we have in Proof. The relations a2i ≤ F(ei ) and fact: a2i  F(ei )

(1)

(1 ≤ i ≤ n;

F ∈ M).

Since the F(ei ) form an increasing sequence, this implies the existence of a constant t > 0 such that a ∈ At for all F ∈ M. It remains to show that n ranges over a bounded subset of N. Fix i ≥ 2. We have: F(ei ) ≤ F(ei + ui−1 . ei−1 + . . . + u1 . e1 ), for all integers ui . Fix k (1 ≤ k < i), and set uj = 0 for k < j < i. Then we see from 2.6 (1) that the coefficient of a2j in the right-hand side is equal to n2j,i (for k < j < i), and it is easy to see that by suitably choosing the uj (j ≤ k), we can arrange that the coefficient of each a2j is ≤ 1/4 for j ≤ k. We therefore obtain, in view of 2.6 (2):18 1 a21 . n21,i + . . . + a2k . n2k,i ≤ (a21 + . . . + a2k ). 4 the obvious inequality F (ei ) ≥ mi (F ). inequality a2i ≤ F(ei ) is not needed here, but it yields the stronger conclusion that F(e1 ) · . . . · F(en )  det F, and is used in the proof of Corollary 2.7.    18 a2 + 2 2 2 2 2 2 j λ and σ(b) ≤ λ (and a < b), there is some i (with a ≤ i < b), such that σ(i + 1) ≤ λ < σ(i). So ni,λ (w) = −1. This is a contradiction, so the lemma is proved.  Let us now choose a representative sw of each w ∈ W. We then have:  N(D) = sw . D, w∈W

and for each w the set Gw = N . sw . B depends only on w. 3.3. Theorem. The sets Gw (w ∈ W) form a partition of GL(n, k). Proof. To prove this, it will be convenient to use the notion of flag: a flag in an n-dimensional vector space is an increasing sequence of subspaces: V1 ⊂ V2 ⊂ . . . ⊂ Vn , such that dim Vi = i. In particular, in kn the canonical flag F0 is the flag such that Vi is generated by the first i vectors in the standard basis. The group G acts transitively on the set F of flags and B is the isotropy group of the flag F0 , so F is canonically identified with the homogeneous space G/B. With this in mind, the theorem can also be stated as follows: for each flag F, there exist w ∈ W and n ∈ N, such that F = n . sw (F0 ); furthermore, w is uniquely determined by F. On the other hand, to prove the theorem, it suffices to consider the case where we have chosen sw to be a permutation matrix. In this proof, we also use w to denote the permutation of {1, . . . , n} associated to w ∈ W. 22 This

corrects a mistake in the original French manuscript, which erroneously states that n−1 na,b,j (w) with exponents na,b,j ≥ 0 whenever a < b. j=1 αj

the assumption implies w(ta /tb ) =

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14

I. SOME CLASSICAL GROUPS

To say that F = (V1 , V2 , . . . , Vn ) is written as n . sw (F0 ) means that the subspace Vi is spanned by the vectors v1 , v2 , . . . , vi , where: 

w(i)−1

vi = ew(i) +

nk,w(i) ek .

k=1

Thus, for each i ≥ 1, w(i) is the largest index distinct from w(1), . . . , w(i − 1) such that there exists a vector vi in Vi  Vi−1 whose w(i)th coordinate is non-zero. (We put V0 = (0).) So w is determined by F. Conversely, given F, we can find a sequence of vectors (vi ), and a permutation w satisfying the preceding condition, with vi in the form: 

w(i)−1

vi = ew(i) +

nk,w(i) ek .

k=1

We then have F = n . sw (F0 ), where n ∈ N has entries nk,w(i) .



We will now write the sets Gw in a different form. This requires the proof of a few lemmas. 3.4. Lemma. For each square matrix g of order n, let Δi (g) be the determinant of the matrix (gk )1≤k,≤i . Then the product map defines a bijection ϕ of N− × B (resp. N− × N) onto the set of g ∈ G such that Δi (g) = 0 (resp. Δi (g) = 1) for i = 1, 2, . . . , n − 1. The matrix entries of ϕ−1 (g) are polynomials, with integer coefficients, in the entries of g and the reciprocals of the Δi (g). Proof. Since Δi (n−. b) = b11 . . . bii for n− ∈ N− and b ∈ B, it is clear that the image of N− × B (resp. N− × N) is contained in the set of g such that Δi (g) = 0 (resp. Δi (g) = 1). For the rest, it suffices23 to prove the assertion about N− × B. This assertion is obvious for n = 1. We may assume that it holds for n − 1 (n > 1) and argue by induction. Let n− ∈ N− and b ∈ B. Let us write them in the form:       n 0 b b n− = , b = , 0 bn,n n 1 where n and b are square of order (n − 1), n is a matrix with 1 row and (n − 1) columns, and b is a matrix with 1 column and (n − 1) rows. Then,     n .b n. b n−. b = . n. b bn,n + n. b Let g be a matrix such that Δi (g) = 0 for all i. Write it as:    g q g= , where g  is square of order n − 1. p gn,n With this notation, the relation g = n−. b is equivalent to: g  = n. b ,

p = n. b ,

q = n. b ,

gn,n = bn,n + n. b .

By the induction hypothesis, there exists a unique unipotent lower triangular n of order (n − 1) and a unique upper triangular b , such that g  = n. b . Since n and b are invertible, the relations p = n. b , q = n. b uniquely determine n and b . 23 Surjectivity of the map defined on N− × N is obtained by combining the bijectivity of the map on N− × B with the observation that if b ∈ B  N, then Δi (n−. b) = 1 for some i.

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3. BRUHAT DECOMPOSITION OF GL(n, k)

15

Then, bn,n is determined by the final equation. The existence and uniqueness of n− and b are therefore proved. The last statement immediately follows from the above argument.24  For each pair of indices (i, j), (i < j), denote by Nij the subgroup of matrices n of N such that all the non-diagonal entries other than nij are zero.    3.5. Lemma. (i) The product defines a bijection of Nij onto N. 1≤i≤n−1

i+1≤j≤n

(ii) For every ordering of the set of pairs (i, j) with i < j, the product defines a surjection of the product Nij onto N, where the factors in the product are taken in the order of the pairs (i, j). Proof. Let us prove (i) by induction on n (there is nothing to prove for n = 1). Let25 n be a matrix of N. Write:   1 a n= , 0 n where n is a unipotent matrix of order (n − 1) . We then have:    1 0 1 a n= . 0 n 0 In−1 In other words, if we identify the group N of unipotent upper triangular matrices of order n − 1 with the subgroup of N consisting of the matrices in N satisfying n12 = n13 = . . . = n1n = 0, and if we denote by A the matrices in N such that all entries, other than the diagonal entries and those in the first row, are zero, then the product defines a bijection of N × A onto N. But by the induction hypothesis the product defines a bijection of    onto N . Nij 2≤i≤n−1

i+1≤j≤n

On the other hand, each element a of A can be written uniquelyas a product of elements belonging to the N1j (2 ≤ j ≤ n). More precisely, a = aj where aj is the element of N1j such that the matrix entry with index (1, j) is a1j . It follows from this that every element of N can be written uniquely as the product of elements belonging to the Nij taken in the lexicographic order.26 Statement (ii) immediately follows27 from (i) and the two following remarks: (a) Nij and Ni j  centralize each other unless i = j or j  = i. 24 Cramer’s Rule implies that the matrix entries of the inverse of a matrix are polynomials, with integer coefficients, of the entries of the matrix and the reciprocal of its determinant. 25 The proof uses n for the size of the matrix and also for an element of N, but the intended meaning should always be clear from the context. 26 Actually, this proof does not establish the conclusion for the lexicographic order, because N12 . N13 . . . Nnn is at the end of the product, rather than at the beginning. To obtain the  lexicographic order,   show that  the product map A × N → N is also a bijection, by writing 1 a . n −1 1 0 n= . . 0 In−1 0 n 27 For each j, we have N . N   ⊂ N   . N   1j 1,j . N1,j+1 . . . N1,n for all i , j . Therefore, by i j i j downward induction on j, we may assume that the last factors in the product are N12 . N13 . . . N1n (in that order). The product of these factors (call it A) is a normal subgroup of N, and N/A is naturally isomorphic to the subgroup called N for the group GL(n − 1, k), so we conclude by induction on n that the product in any order contains the product in the same order as in the proof of (i).

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16

I. SOME CLASSICAL GROUPS

(b) If i = j, we have: Nij . Ni j  ⊂ Ni j . Nij . Nij  .



Remark. For every orderingof the set of pairs (i, j), it can be shown that the product defines a bijection28 of Nij onto N (where the factors in the product are taken in the increasing order of the pairs (i, j)). For each w ∈ W, the set s−1 w . N . sw depends only on w; we will denote it by Nw .  −1 3.6. Lemma. Let Nw = sw . (Nw ∩ N− ) . s−1 w , Nw = sw . (Nw ∩ N) . sw . Then the   product defines a bijection of Nw × Nw onto N.

Proof. Indeed, by 3.5 (i):    Nw = 1≤i≤n−1

 Nw−1 (i),w−1 (j) .

i+1≤j≤n

But by (ii), we can also write:   Nw = Nij . Nij , (i,j)∈L, i>j

(i,j)∈L, i 0 such that: (1)

α ≤ Φi (c) ≤ β

(c ∈ C;

Finally, if we use the Bruhat decomposition: g = kg . ag . ng = ug . sw . tg . vg , we have (by 3.8): ag = a(s−1 w . ug . sw ) . a(tg ), so (2)

Φi (g) = Φi (ag ) = Φi (s−1 w . ug . sw ) . Φi (tg ).

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i = 1, . . . , n).

18

I. SOME CLASSICAL GROUPS

4.2. Lemma. (i) For every n ∈ N− , we have Φi (n) ≥ 1 (ii) For every g ∈ G, we have Φi (ag ) ≥ Φi (tg ).

(i = 1, 2, . . . , n);

Proof. (i) Indeed, n(e1 ∧ e2 ∧ . . . ∧ ei ) = e1 ∧ e2 ∧ . . . ∧ ei + a, where a is a linear

combination of elements of the canonical basis of i Rn other than e1 ∧ e2 ∧ . . . ∧ ei , so: Φi (n) = n(e1 ∧ e2 ∧ . . . ∧ ei ) ≥ 1. − (ii) This follows from (2) and the fact that s−1 w . ug . sw is in N .



4.3. Lemma. For each index i (1 ≤ i ≤ n), there exists a constant di > 0 such

that g(v) ≥ di . v . Φi (g) whenever g belongs to S and v belongs to i (Rn ). Proof. It is enough to prove the inequality for v = 1. Write G = K . A . N and S = K . At . ωs where ω is a relatively compact subset of N, and At is the set of diagonal matrices with entries > 0 satisfying aii ≤ t . ai+1,i+1 .

Let v ∈ i (Rn ) be a unit vector and g ∈ G. Then:

g(v) = kg . ag . ng (v) = ag . ng (v) . We can write: ng (v) =



βj (ng ) . fj ,

where the fj run through the canonical basis of i (Rn ). On the other hand, if fj = e1 ∧ e2 ∧ . . . ∧ ei (1 < 2 < . . . < i ), then: a  a  ag (fj ) = a1 1 . a2 2 · . . . · ai i . fj = Λi (aj ) · 1 1 · . . . · i i · fj , a11 aii m

,j

n−1 = Λi (ag ) . α1m1 ,j. α2m2 ,j . . . αn−1 . fj ,

where αi = aii /ai+1,i+1 , (i = 1, 2, . . . , n − 1) and the mi,j (i = 1, . . . , n − 1) are negative integers.30 By combining these two results, we obtain:  mn−1,j 1 ,j ag . ng (v) = Λi (ag ) . βj (ng ) αm · . . . · αm−1 . fj . 1 Now, if g ∈ S, we have 0 < αi ≤ t by assumption. Thus, there exists a constant δ > 0 such that: 

ag . ng (v) 2 ≥ δ . Λi (ag )2 . βj (ng )2 . But v = 1, and ng remains in a compact set. Therefore, we have:  βj (ng )2 = ng (v) 2 ≥ δ > 0, hence there exists a constant d > 0 such that:

ag . ng (v) ≥ d . Λi (ag )

(g ∈ S, v = 1),

which can be rewritten:

g(v) ≥ d . Φi (g). This proves the lemma. 30 More

precisely, −n2 ≤ αi ≤ 0.

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4. THE SIEGEL PROPERTY IN GLn

19

4.4. Theorem (Harish-Chandra). Let S ⊂ GL(n, R) be a Siegel domain, {sw } be a choice of representatives in N(D) ∩ K of the elements of W, and M be a subset of G satisfying: (i) M = M−1 ; and (ii) For each i (1 ≤ i ≤ n), there exists a constant Ci > 0 such that we have Φi (tm ) ≥ Ci for all m ∈ M (tm chosen with respect to sw , cf. 3.7). Then, the set MS = {m ∈ M | S . m ∩ S = ∅} is relatively compact in G. Proof. Let us first prove that when m runs through MS , its component am in the Iwasawa decomposition (m = km . am . nm ) and its components um and tm in the Bruhat decomposition (m = um . sm . tm . vm ) run through relatively compact sets. Let m ∈ MS , and let x, y ∈ S be such that xm = y. Apply 4.3 to the vector v and also to the vector m(e1 ∧ e2 ∧ . . . ∧ ei ). We obtain:

(xm) . v ≥ d . v . Φi (xm), and also: Φi (xm) ≥ d . Φi (m) . Φi (x), hence:

(xm) . v ≥ d2 . v . Φi (m) . Φi (x). In particular, taking v = (m−1 ) . (e1 ∧ e2 ∧ . . . ∧ ei ), this becomes: Φi (x) ≥ d2 . Φi (m−1 ) . Φi (m) . Φi (x), whence (1)

Φi (m−1 ) . Φi (m) ≤ d−2 .

Besides, by 4.2 (ii) and the assumption (ii): (2)

Φi (m) = Φi (am ) ≥ Φi (tm ) ≥ Ci > 0.

Since M = M−1 , (1) and (2) show that Φi (m) and Φi (tm ) are also bounded above on MS . Hence, there exist constants α, β > 0 such that: α ≤ Φi (am ) ≤ β,

α ≤ Φi (tm ) ≤ β

(m ∈ MS ).

This proves that a(MS ) and t(MS ) are relatively compact (cf. 4.1 (1)). On the other hand, by 3.8: am = a(s−1 w . um . sw ) . a(tm ), so a(s−1 w . um . sw ) remains in a relatively compact set when m runs through MS ∩Gw . − −1 Since s−1 w . um . sw ∈ N , the same holds (by 3.9) for sw . um . sw , which completes the proof of the claim. We can now prove the theorem by induction on the dimension n. It is immediate for n = 1. Suppose now that n > 1 and that the theorem holds for n − 1. It all comes down to proving that the intersection of MS with each of the sets Gw (of which there are only finitely many) is relatively compact. Fix w ∈ W. We consider two cases: (i) The element sw is not in any of the parabolic subgroups Pλ (λ = 1, . . . , n−1). A Siegel set is invariant under translation on the left by any matrix c . I (c > 0); therefore, if m satisfies a relation xm = y (x, y ∈ S), we see, by multiplying

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20

I. SOME CLASSICAL GROUPS

by |det x|−1/n. I, that it also satisfies an equality of this type with |det x| = 1. Consequently, to establish (i), it suffices to prove that the set X = {x ∈ S | |det x| = 1 and xm ∈ S for some m ∈ MS ∩ Gw } and the set Y = {y ∈ S | y = xm, x ∈ X, m ∈ MS ∩ Gw }, are relatively compact. So let x ∈ X, y ∈ Y, and m ∈ MS ∩ Gw be such that xm = y. We have: ky . ay . ny = kx . ax . nx . um . sw . tm . vm = kx . sw . kc . ac . nc . (s−1 w . ax . sw ) . tm . vm , where −1 c = s−1 w . a x . n x . um . a x . s w .

Since kx . sw . kc is an element of K, this implies: ay = ac . (s−1 w . ax . sw ) . a(tm ). We established at the start of the proof that um runs through a relatively compact set. Since nx is in a fixed compact set, this implies that ax . nx . um . a−1 x remains bounded in view of the fundamental property (1.3) of Siegel sets. So c and ac are also bounded. The proof will be complete if we prove that ax remains in a relatively compact set: indeed, we established that tm (hence also a(tm )) remains bounded, and so the same will be true of ay . Since det ax = 1, it is enough to prove that:31 αλ (ax ) ≥ cλ > 0

(λ = 1, 2, . . . , n − 1).

Since all the matrix entries of ac and a(tm ) are bounded above and below, and αi (ay ) is bounded above, we see that for i = 1, 2, . . . , n, the w(αi ) (ax ) remain bounded above. For any given λ (1 ≤ λ ≤ n − 1), there exists (3.2) at least one index i, (1 ≤ i ≤ n) such that: w(αi ) (ax ) =

n−1 

αj (ax )nij ,

with niλ < 0.

j=1

But the αj (ax ) are bounded above. Thus, αλ (ax ) cannot be arbitrarily small, which is what was to be proved. (ii) The element sw lies in one of the groups Pλ , (1 ≤ λ ≤ n − 1). If sw ∈ Pλ , we immediately verify that Gw ⊂ Pλ . Thus, it suffices to prove the following statement: For every λ (1 ≤ λ ≤ n − 1), MS ∩ Pλ is relatively compact. Let us first note that if x and y are in S, and m ∈ MS ∩ Pλ is such that xm = y, then we may assume, by changing x, that kx = e, so x ∈ A . N ⊂ Pλ . Therefore, we have y ∈ Pλ , which shows that MS ∩ Pλ is exactly the set of all m ∈ M, such that S ∩ Pλ intersects (S ∩ Pλ )m. The group Pλ is the semidirect product Pλ = S . R where S is the direct product: S = GL(λ, R) × GL(n − λ, R), 31 α

λ (a)

= aλλ /aλ+1,λ+1 , as in 3.1 and the proof of 4.3.

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4. THE SIEGEL PROPERTY IN GLn

21

and R is the normal subgroup of unipotent matrices of the form:   ∗ Iλ . 0 In−λ Let Π, Π0 , Π1 , Π2 be the projections of Pλ onto S, R, GL(λ, R), GL(n − λ, R), respectively. Our goal is to show that Πi (MS ∩ Pλ ) is relatively compact for each i = 0, 1, 2. We have O(n) ∩ Pλ = O(λ) . O(n − λ), (direct product). It follows that Π1 (resp. Π2 ) maps the Iwasawa decomposition of an element g ∈ Pλ to the Iwasawa decomposition of Π1 (g) (resp. Π2 (g)) in the group GL(λ, R) (resp. GL(n − λ, R)). More precisely, if x = kx . ax . nx is in Pλ , we have: Π(x) = kx . ax . Π(nx ),

Π0 (x) = Π0 (nx ),

and Π1 (kx ), Π1 (ax ), Π1 (nx ),

(resp. Π2 (kx ), Π2 (ax ), Π2 (nx ))

are the components of Π1 (x) (resp. Π2 (x)) in the Iwasawa decomposition. In particular, Πi (S) ⊂ Si (i = 1, 2) where Si is a Siegel domain. If we put Mi = Πi (M ∩ Pλ ), we see that Πi (MS ∩ Pλ ) ⊂ (Mi )Si . It is clear that Mi satisfies the first hypothesis of the theorem. It is immediate that Π1 (resp. Π2 ) maps the Bruhat decomposition of an element g ∈ Pλ to the Bruhat decomposition of Π1 (g) (resp. Π2 (g)) in GL(λ, R) (resp. GL(n−λ, R)); we immediately deduce that Mi also satisfies the second hypothesis.32 By the induction hypothesis, (Mi )Si is relatively compact (i = 1, 2). All that remains is to prove that Π0 (MS ∩ Pλ ) is relatively compact. But if m ∈ MS ∩ Pλ , there exist x and y in Pλ ∩ S such that xm = y, which can be written: kx . ax . nx . m = ky . ay . ny , whence: kx . ax . Π(nx ) . Π0 (nx ) . Π(m) . Π0 (m) = ky . ay . Π(ny ) . Π0 (ny ). The left hand side can also be written as:   kx . ax . Π(nx ) . Π(m) . Π(m)−1. Π0 (nx ) . Π(m) . Π0 (m).   Since kx . ax . Π(nx ) . Π(m) is in S and Π(m)−1. Π0 (nx ) . Π(m) . Π0 (m) is in R, we have: Π0 (ny ) = Π(m)−1. Π0 (nx ) . Π(m) . Π0 (m). Π0 (nx ) and Π0 (ny ) are bounded since x and y belong to a Siegel set. We have just proved that Π(m) is bounded. Thus, the same is also true of Π0 (m), and the theorem is proved.  4.5. Corollary. Let S be a Siegel domain in the space of positive-definite symmetric matrices and M be a subset of G satisfying the assumptions of Theorem 4.4. Then the set {m ∈ M | S [m] ∩ S = ∅} is relatively compact. 4.6. Theorem (Siegel). Let S be a Siegel domain in GL(n, R), and let M be a set of invertible matrices with integer entries, whose determinants satisfy |det m| ≤ c, for all m ∈ M. Then MS is finite. 32 Indeed,

the start of the proof established that tm stays in a relatively compact set.

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22

I. SOME CLASSICAL GROUPS

Proof. Since M(n, Z) is closed and discrete in M(n, C), the set MS is itself closed and discrete. It therefore suffices to prove that the set L = M ∪ M−1 satisfies hypothesis (ii) of Theorem 4.4. This is immediate from the following lemma.  4.7. Lemma. Let L be a subset of GL(n, Q) whose elements have entries whose denominators are bounded above in absolute value. Then there exists a constant c > 0 such that Φi (tx ) ≥ c for every x ∈ L and every i = 1, 2, . . . , n. Proof. Let x ∈ Gw ∩ L. It suffices to show that the products |t11 . . . tii | of the diagonal entries of tx have denominators that are bounded above. −1 − But s−1 w . x = cx . tx . vx , where cx = sw . ux . sw ∈ N . In particular, when x ∈ Gw ∩ L, the denominators of the entries of cx . tx . vx are bounded above. Thus,  the elements det (cx . tx . vx )jk 1≤j,k≤i = t11 . . . tii have the same property.  4.8. Corollary. The group Γ = GL(n, Z) is finitely generated and has only finitely many conjugacy classes of finite subgroups. Proof. We have H = S [Γ] where S is a suitable open Siegel domain (see Theorem 2.2 i). It is then immediate that Γ is generated by the set ΓS of all γ ∈ Γ, such that S [γ] intersects S , and this set is finite by Theorem 4.5. Now let L be a finite subgroup of Γ; it has a fixed point F in H. We can write F = c[γ], where c is in S and γ is in Γ. Then, γ . L . γ−1 ⊂ ΓS . The conclusion then  follows from the fact that ΓS is finite. 4.9. We conclude this section with a proposition of Harish-Chandra that plays a role in the study of automorphic functions. By the norm x of x ∈ Rn , we mean the Euclidean norm of x, whereas, for g ∈ GL(n, R), the norm g of g is the trace of tg . g, or, in other words, the sum of the squares of the matrix entries of g. Thus, we have:

g =

n 

g . ei 2 .

i=1

We easily see that if u ∈ GL(n, R), then: (1)

g  g . u  u . g

(g ∈ GL(n, R)).

This implies that if H is a subgroup of GL(n, R) and H is a finite index subgroup of H, then: (2)

inf g . u  inf  g . u

u∈H

u∈H

(g ∈ GL(n, R)).

This implies that if L is a subgroup of GL(n, R) that is commensurable with H, i.e., such that L ∩ H is of finite index in both L and H, then we have: inf g . u  inf g . u

u∈L

u∈H

(g ∈ GL(n, R)).

4.10. Proposition. Let S be a Siegel set of GL(n, R), and let Γ = GL(n, Z). Then:

g  inf g . u u∈Γ

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(g ∈ S).

5. REDUCTION OF INDEFINITE QUADRATIC FORMS

23

Proof. Obviously, the left-hand side is larger than the right-hand side. It suffices to prove that g ≺ inf u∈Γ g . u (g ∈ S). By enlarging S, we may assume, in view of 2.7, that S contains the set M of M-reduced elements of GL(n, R), in the sense of 2.8. Let C be the set of all c ∈ GL(n, Z), such that S . c ∩ S = ∅. By 4.6, C is finite. Note that g  g . c (g ∈ GL(n, R), c ∈ C). Also, since c . Γ = Γ we have33 inf g . γ = inf g . c . γ

γ∈Γ

(g ∈ GL(n, R);

γ∈Γ

c ∈ C).

Now let g ∈ S. There exists c ∈ Γ such that g = g . c (g  ∈ M). Since M ⊂ S, we then have c ∈ C. Therefore, we are reduced to proving:

g ≺ inf g . γ

(g ∈ M).

γ∈Γ

Denote by Fg the quadratic form of the matrix tg . g. We have the following relations, where mn (Fg ) is the number introduced in 2.5:

g . γ =

n  i=1

g . γ(ei ) 2 =



  Fg (γ . ei ) ≥ mn Fg [γ] = mn (Fg ),

i

where the last equality follows from the definition of mn (F). By 2.5, there exists a constant d > 0, independent of g ∈ M, such that: mn (Fg ) ≥ d . Fg (en ) = d . g . en 2 whence, since the g . ei form an increasing sequence:34 d  d

g . γ ≥ .

g . ei 2 = . g n i n

(g ∈ M),

(g ∈ M). 

Bibliographical note. Theorem 4.6 is proved in [38]. It seems Hermite realized that his reduction theory provided sets of reduced elements that only meet a finite number of their translates by Γ. At least, this is what appears to be the case from a remark [18, p. 230] that he makes about the reduction of indefinite forms, from which he specifically deduces that the group of units of such a form is finitely generated. Theorem 4.4 is in fact a special case of an unpublished theorem of HarishChandra that will be stated and proved in §15. The proof given here is modeled after that of the general case. Proposition 4.10 is essentially equivalent to [40, Satz 4, p. 34]. 5. Reduction of indefinite quadratic forms 5.1. Before treating the reduction theory of indefinite quadratic forms, we will discuss the notion of a Hermite majorizer of a quadratic form, which links this theory to the reduction theory of positive-definite quadratic forms that was presented in § 2. V denotes a real vector space with linear group G = GL(V) and H is the set of positive-definite quadratic forms on V; H is endowed with a partial order, defined 33 The original French manuscript states that, in view of 4.9, there exist constants a, b > 0 such that: a . inf γ∈Γ g . γ ≤ inf γ∈Γ g . c . γ ≤ b . inf γ∈Γ g . γ . This is true, but more complicated than necessary. 34 Because F is M-reduced. g

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24

I. SOME CLASSICAL GROUPS

by F ≤ F if F(v) ≤ F (v) for every v ∈ V. G acts on the right on the set of quadratic forms by: F → F[X],

where

(X ∈ G, v ∈ V),

F[X](v) = F[Xv],

which can be translated in a basis of V to: F[X]0 = t X0 . F0 . X0 , if we denote by X0 (resp. F0 ) the matrix of an element X of G (resp. of a quadratic form F) with respect to the chosen basis. Finally, O(F) denotes the isotropy group35 of the quadratic form F: O(F) = {X ∈ G | F[X] = F}. Definition. (1) C ∈ H is said to majorize 36 the quadratic form F if |F(v)| ≤ C(v), for every v ∈ V; (2) C ∈ H is a minimal majorizer (or Hermite majorizer ) of F, if it is minimal in the partially ordered set of all majorizers of F. The set of minimal majorizers of F is denoted by H(F): if F ∈ H (resp. −F ∈ H), then H(F) consists only of F (resp. −F). Since X ∈ G maps the set of majorizers of F to the set of majorizers of F[X], we have:   H F[X] = H(F)[X], and O(F) acts on H(F) in this way: we will see that H(F) is thereby a topological homogeneous space that is isomorphic to the space of right cosets of a maximal compact subgroup of O(F). 5.2. Proposition. If C ∈ H and if F is a non-degenerate quadratic form of signature (a, b), then the following assertions are equivalent. (i) C ∈ H(F). (ii) There exists a decomposition of V into a sum V1 ⊕ V2 that is orthogonal with respect to F and C, and such that F = C on V1 and F = −C on V2 . (iii) There exists a basis of V with respect to which F0 = Ia,b and C0 = I. 2 (iv) In every basis of V, we have (F0 . C−1 0 ) = I. (Ia,b denotes the diagonal matrix in which the first a diagonal entries are equal to 1 and the last b entries are equal to −1.) Proof. By the assumptions, we can find37 a basis of V in which: F=

a 

x2i −

i=1

C=

a+b 

a+b 

x2i ,

i=a+1

αi x2i

(αi > 0; 1 ≤ i ≤ a + b).

i=1

Then, C is a majorizer of F if and only if αi ≥ 1, (1 ≤ i ≤ a + b). It follows that: C ∈ H(F)

⇔

αi = 1

35 This

(1 ≤ i ≤ a + b).

is also called the isometry group of the quadratic form. also says that C is a majorizer or a majorant or a majorizing form of F. 37 The two forms are simultaneously diagonalizable. (To see this, first note that, since C is positive definite, we may assume (after a change of basis) that C0 = I. The symmetric matrix F0 is diagonalizable by an orthogonal matrix (which preserves C0 ).) Now that C and F are diagonal, the desired basis vectors are scalar multiples of the vectors in the standard basis. 36 One

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5. REDUCTION OF INDEFINITE QUADRATIC FORMS

25

The equivalences (i) ⇔ (ii) ⇔ (iii) are immediate from this. In the same basis, the 2 2 relation (F0 . C−1 0 ) = I reads as αi = 1 for every i, so we have (iv) ⇒ (iii). Finally, a very short calculation shows that if the equality in (iv) holds in some basis of V, then it also holds in all other bases, which proves the implication (iii) ⇒ (iv).  5.3. Corollary. Under the assumptions of the proposition, the map: C → V(C), where C ∈ H(F) and where V(C) denotes the unique a-dimensional subspace on which F = C, is a bijection of H(F) onto the set of a-dimensional subspaces on which F is positive-definite. Proof. The existence of at least one space V(C) follows from (iii), and its uniqueness follows from the fact that it is the eigenspace for the eigenvalue 1 of the self-adjoint operator associated to F by C. Finally, if V1 is a subspace of dimension a on which F is positive-definite, its orthogonal complement V2 with respect to F is a complement of V1 and we have V1 = V(C) with C = F on V1 and C = −F  on V2 . 5.4. Proposition. O(F) acts transitively on H(F). The isotropy group of a point C of H(F) is a maximal compact subgroup K ∼ = O(a) × O(b), and H(F) is homeomorphic to K\O(F). Proof. The action of the group O(F) is transitive, because if C and C belong to H(F), then the two corresponding decompositions of V provided by (ii) are orthogonal with respect to F and there exists an element of O(F) that maps one to the other, and therefore maps C to C . The isotropy group in O(F) of C ∈ H(F) is O(F) ∩ O(C), which can be identified with O(Ia,b ) ∩ O(n) in the basis of (iii). An immediate calculation shows that this is equal to the compact subgroup O(a)×O(b) of O(n). Any compact subgroup of O(F) leaves a point D ∈ H fixed [10, Chap. VII, §3, no. 1, Proposition 1], so, in order to prove that K is maximal, it suffices to verify that the inclusion K ⊂ O(F)  ∩ O(D)  implies equality; but we easily see that the Lie algebra of O(Ia,b ) ∩ O diag{αi } , where αi > 0 for all i, is contained in that of O(a) × O(b). Finally, we know that GL(n, R) acts properly on H, and the same is therefore true of O(F). One of the orbits of O(F) is precisely H(F): it follows that H(F) is homeomorphic to K\O(F).  5.5. Let F be a non-degenerate indefinite rational quadratic form on Rn , of signature (a, b), let G be the real Lie group O(F), and let Γ = G ∩ GL(n, Z). The elements of Γ are called the “units” of F. Thus, the units are the elements of O(F) that leave the lattice Zn invariant. (In a natural way, one could replace Rn with the real points of a vector space V over Q, and Zn with a lattice L of the latter.) G acts properly on H(F), so the same is true of the closed subgroup Γ and the problem of reduction theory amounts to finding a good set of representatives of the orbits of Γ. For all that follows, we fix a Siegel set S of H that contains S4/3,1/2 , so that H is equal to S. [GL(n, Z)], by 2.2. Definition. F is reduced (or, more precisely, S -reduced ) if: H(F) ∩ S = ∅.

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26

I. SOME CLASSICAL GROUPS

5.6. Definition. Ω ⊂ H(F) is a fundamental set 38 for Γ if it satisfies the conditions: (F1) Ω[Γ] = H(F); (F2) ∀b ∈ O(F)Q , {γ ∈ Γ | Ω[b] ∩ Ω[γ] = ∅} is finite. Condition (F2) is the Siegel property. Combining the following finiteness lemma (which is a special case of a result that will be proved in §6) with Siegel’s Theorem 4.6 will enable us to prove the existence of a fundamental set. 5.7. Lemma (Finiteness Lemma). The set of reduced integral quadratic forms with a given non-zero determinant39 is finite. 5.8. Theorem. There exist s1 , . . . , sm ∈ GL(n, Z) such that:  m   S [si ] ∩ H(F) Ω= i=1

is a fundamental set. Proof. From the equality H(q F) = |q| . H(F), we see that if q ∈ Q and if Ω is a fundamental set relative to F, then |q| . Ω is a fundamental set relative to q F; moreover, since S is invariant under positive homothety,40 it suffices to prove the theorem for integral F. But if F is integral, then, since all the forms in the orbit of F under GL(n, Z) have the same determinant, there are among them only a finite number of reduced −1 forms, say F[s−1 1 ], . . . , F[sm ], by the finiteness lemma. We will show that: m    Ω= S [si ] ∩ H(F) i=1

satisfies (F1) and (F2). 41 some (F1) Let C ∈ H(F); by the reduction of definite   forms (§2), we have  s ∈ GL(n, Z) such that C[s] ∈ S . But then, H F[s] = H(F)[s] meets S in C[s], which means that F[s] is reduced, therefore equal to F[s−1 i ] for some i. Note that F[s . si ] = F, so s . si ∈ Γ. We also have C[s . si ] ∈ S [si ] ∩ H(F), so the orbit of C indeed meets Ω. (F2) Given b ∈ O(F)Q , assume that γ ∈ Γ is such that Ω[b] ∩ Ω[γ] = ∅. By the definition of Ω, there exist indices i and j such that S [si . b] ∩ S [sj . γ] = ∅. But the Siegel property is known to hold for S (cf. §4); in view of the finiteness of the  set {si }, this suffices to conclude the proof. Let us also mention that the volume of such a fundamental set is finite. We will next explain how the notion of reduced form is translated as we return our attention to the groups GL(n, R) and SL(n, R). In GL(n, R), the Siegel set associated to S is S, the pre-image of S under the map X → t X . X = I[X]. In SL(n, R), it is S1 = S ∩ SL(n, R). 38 Translator’s note: This is more restrictive than the definition of a coarse fundamental domain, which requires (F2) only in the special case where b is the identity element. 39 This determinant is more commonly called the discriminant of the quadratic form. It is the determinant of the matrix corresponding to the quadratic form. 40 Positive homothety refers to multiplying by a positive scalar. 41 More precisely, this is by the choice of S .

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6. A FINITENESS LEMMA

27

5.9. Proposition. Let F be a non-degenerate quadratic form of signature (a, b). The following conditions are equivalent: (i) F is reduced; (ii) F ∈ Ia,b [S]; (iii) F ∈ α . Ia,b [S1 ], where α = |det F|1/n . Proof. (i) ⇒ (ii), because there exists C ∈ H(F) ∩ S and then, by 5.2 (iii), there exists X ∈ GL(n, R), such that Ia,b [X] = F and I[X] = C, i.e., X ∈ S. (ii) ⇒ (i), because if F = Ia,b [X] with I[X] ∈ S , we have: I[X] ∈ H(Ia,b )[X] ∩ S = H(F) ∩ S . As for the last equivalence (ii) ⇔ (iii), it follows from an obvious calculation.



This last proposition makes it possible to restate the finiteness lemma as follows: 5.10. Lemma. If S is a Siegel set of SL(n, R), then the set M(n, Z) ∩ α . Ia,b [S]

(α ∈ R; α > 0),

is finite. It is in this form that it will be proved in §6.42 6. A finiteness lemma 6.1. Definition. A subgroup G of GL(n, R) is said to be self-adjoint (or more precisely, I-self-adjoint 43 ) if: X∈G

⇒

t

X ∈ G.

6.2. Lemma. Let G be GL(n, R) or SL(n, R). We consider a Siegel set S of G, a representation of G on a real vector space V (on the right44 ), a lattice L of V, and a vector v of V. Assume that: (i) the orbit v . G of v is closed in V, (ii) the isotropy group Gv of v is self-adjoint, and (iii) if we let A be the group of diagonal matrices of G with diagonal entries that are > 0, then the operators of A are simultaneously diagonalizable; {w ∈ V | w . a = μ(a) w, ∀a ∈ A} be the furthermore, if we let Vμ =  subspace of weight μ, so V = μ Vμ is the associated eigenspace decomposition of V, then Vμ ∩ L is a lattice of Vμ for each μ. Then, the set v . S ∩ L is finite. Proof. In this proof, we denote by x = kx . ax . nx the Iwasawa decomposition of an element x of G and we put: yx = x . a−1 x

and

zx = x . a−2 x .

We begin by proving  some preliminary bounds. There is an inner product on V such that the sum Vμ is orthogonal; we denote  by Eμ the projection of V onto Vμ . Assumption (iii) implies that the subgroup μ (Vμ ∩ L) of L is a lattice of V; hence, it has finite index in L and since its projection onto Vμ is the lattice Vμ ∩ L, 42 See

6.3. denotes the identity matrix. 44 The original French manuscript refers to this as a representation of the opposite group. 43 I

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28

I. SOME CLASSICAL GROUPS

the projection Eμ (L) of L is a lattice of Vμ . This proves the existence of a real number c > 0, such that for every μ: w ∈ L and Eμ (w) = 0

⇒

Eμ (w) ≥ c.

On the other hand, the set {ax . nx . a−1 x }x∈S is relatively compact (1.3); and we have yx = kx . ax . nx . a−1 x , so the set {v . yx }x∈S is bounded; therefore there exists a constant c such that we have v . yx ≤ c for every x ∈ S. By combining these two bounds, we will show that the set: { v . zx }x∈S, v . x∈L −2 is bounded. Since yx = x . a−1 x and zx = x . ax , we have:

Eμ (v . yx ) = μ(a−1 x ) . Eμ (v . x),

Eμ (v . zx ) = μ(a−2 x ) . Eμ (v . x).

For each μ, we have either Eμ (v . x) = 0, so that Eμ (v . zx ) = 0, or Eμ (v . x) = 0, in which case we obtain from the above:

Eμ (v . zx ) = Eμ (v . yx ) 2 . Eμ (v . x) −1 ≤ c 2. c−1

(v . x ∈ L; x ∈ S).

The assumption (i) implies that the bijection Gv\G → v . G is a homeomorphism: indeed G acts continuously and transitively on v . G, which is a Baire space, since it is a closed subset of V, so we can apply [10, Chap. VII, App. 1, Lemme 2]. The bound that was found above can therefore be pulled back to G: there exists a compact set M of G, such that we have: {zx }x∈S, v . x∈L ⊂ Gv . M. −1 2 −2 2 But zx = kx . ax . nx . a−2 x = kx . ax . ax . nx . ax ; if x ∈ S, then ax ∈ At2 so (by 1.3) } is relatively compact. Hence there exists another compact the set {a2x . mx . a−2 x∈S x set M such that we have:  {kx . a−1 x }x∈S, v . x∈L ⊂ Gv . M .

By hypothesis (ii), the automorphism θ : x → tx−1 of G leaves Gv invariant, whence: −1  x . n−1 x = kx . ax = θ(kx . ax ) ∈ Gv . θ(M ).

Finally,45 this implies there exists a compact subset M of G such that the set {x}x∈S, v . x∈L is contained in Gv . M , proving the finiteness of the set {v . x}x∈S, v . x∈L , which is contained in both a compact set and a lattice of V.



6.3. Application to 5.10. It is immediate that the conclusion of the above lemma, in the special case where we let: V = M(n, R), 45 Since

L = M(n, Z),

v = α . Iab ,

nx is bounded for x ∈ S.

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6. A FINITENESS LEMMA

29

and G = SL(n, R) acting on the right on V in the usual way,46 is none other than Lemma 5.10. It then remains to see that such a choice of data satisfies the hypotheses (i), (ii) and (iii) of the lemma: (i) v . G = {α . Iab [g]}g∈SL(n,R) is the set of symmetric matrices of signature (a, b) and determinant αn (−1)b ; therefore, it is a closed subset of V. (ii) Gv = O(Ia,b ) ∩ SL(n, R); this is self-adjoint, as is O(F) in general when the symmetric matrix F is equal to its inverse. (iii) A basis of the lattice is {Eij }1≤i,j≤n , where Eij denotes the matrix whose only non-zero entry is the one in the i-th row and j-th column, which is equal to 1. But Eij is an eigenvector for every element a ∈ A, since Eij [a] = aii ajj Eij . Let us make some final remarks concerning hypothesis (iii) of the lemma: 1) If G = SL(n, R), we can simultaneously diagonalize the operators of A. 2) If the representation of the group is defined over Q, we can diagonalize the operators of A simultaneously over Q. Thus, for a representation G → GLm that is defined over Q, the last hypothesis is always satisfied when L is a lattice of Qm . 6.4. Remark. Let us assume that the hypotheses of 6.2 hold, and assume moreover that the lattice L is invariant under GZ = G ∩ GL(n, Z). Then, v . G ∩ L consists of a finite number of orbits of GZ . Indeed, we have G = S . GZ for suitable S (by 1.4 or 1.10), so if x ∈ v . G ∩ L, then x . GZ intersects the set v . S ∩ L, which is finite by 6.2. Let us return to the special case considered in 6.3. An orbit of GL(n, Z) (resp. SL(n, Z)) in the space of non-degenerate quadratic forms is a class (resp. a proper class) of quadratic forms. Thus, we see that the integral quadratic forms of given non-zero determinant are partitioned into a finite number of classes (resp. proper classes). 6.5. Homogeneous forms. We would also like to point out a generalization of the previous result to forms of degree ≥ 3, which goes back to Jordan. Let Fm,n be the space of homogeneous forms of degree m on Cn . It is defined over Q, the Q-structure being given by the forms with rational coefficients. We have a natural representation πm,n of GL(n, C) on Fm,n , which is obviously defined over Q. Let m, n ≥ 3 and let F be an element of Fm,n of non-zero discriminant. It was proved by Jordan [20] that the  isotropy  group of F in SL(n, C) is finite. It follows that the orbit X = πm,n SL(n, C) . F of F is closed. Indeed, by an elementary principle of the theory of algebraic groups [5, Proposition I.1.8, p. 53], this orbit is a Zariski-open subset of its closure, and its boundary consists of orbits of dimension strictly less than dim(X), which here is equal to that of SL(n, C) by Jordan’s theorem. But SL(n, C) leaves the discriminant invariant, so any form F in the closure of X also has non-zero discriminant, and its orbit therefore has the same dimension as X. The first part of 6.4 then applies to X and shows that the forms with integer coefficients that are images of F under SL(n, R) constitute a finite number of orbits of SL(n, Z), a result proved by Jordan and Poincar´e. In fact, Jordan’s theorem   is stronger and establishes the finiteness of the number of orbits of SL n, Z[i] (where Z[i] denotes the ring of Gaussian integers) on the set of elements of X 46 I.e.,

g

x → tg . x . g.

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30

I. SOME CLASSICAL GROUPS

whose coefficients belong to Z[i]. Clearly, it is possible to give an analogous proof of this, but,   to do so, it would be necessary to adapt §1 to consider SL(n, C) and SL n, Z[i] . This result is also encompassed in a theorem of [6], to which we will return in 9.11. Remark. Let F0 be the hypersurface in the projective space Pn−1 (C) that is defined by the vanishing of F. The form F has non-zero discriminant if and only if F0 is smooth. Jordan’s theorem cited above is equivalent to the fact that the group of projective transformations leaving F0 invariant is finite. More generally, at least in the case where n ≥ 5, let us now show that the group Aut F0 of automorphisms (i.e., homeomorphisms that are biholomorphic, or, equivalently, by Chow’s theorem, biregular) of F0 is finite. It is known that this group is a complex Lie group. By Kodaira-Spencer [21, Lemma 14.2, p. 406], it is discrete. Consequently, for every projective embedding of F0 , the group of projective transformations leaving F0 invariant, which is obviously algebraic, is finite. But, if n ≥ 5, Lefschetz’s Theorem implies that F0 is simply connected, and has second Betti number equal to one. Therefore, the group Aut F0 has a subgroup H of index ≤ 2 that acts trivially on the second integral cohomology group H2 (F0 ; Z) ∼ = Z of F0 . The group H therefore fixes each polarization of F0 , and acts by projective transformations in the embedding defined by an ample line bundle, so H is finite. The last argument also shows that the identity component of Aut F0 consists of projective transformations in a suitable embedding. For n ≥ 5, it therefore suffices to apply Jordan’s theorem. Bibliographical note. The construction of fundamental sets described in 5.8 is due to Hermite [18, pp. 122–135]. The key point there is the finiteness of the number of reduced integral forms of given non-zero determinant (5.7); however the proof provided by Hermite is incomplete, because it implicitly assumes that the form F does not represent zero rationally (i.e., it assumes F(x) = 0 if x = 0 has rational coordinates), which is a serious restriction, since A. Meyer later proved that it implies the number of variables is ≤ 4. Proofs of this lemma were subsequently given by Stouff [41], Mordell [27], Siegel [38]. On the other hand, Stouff was preceded by C. Jordan [20] and H. Poincar´e [33], who established analogues of Hermite’s lemma for forms of degree > 2, which are the topic of 6.5. Lemma 6.2 itself is proved in a more general form in [6, Lemma 5.3], but the case considered here is sufficient for the applications we have in mind.

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10.1090/ulect/073/02

CHAPTER II

Algebraic Groups 7. A review of algebraic groups. Arithmetic groups A. Generalities. 7.1. In what follows we fix an algebraically closed field Ω (in practice, it will be 2 C). Recall that the embedding of GL(n, Ω) in Ωn +1 defined by   g → gij , det(gij )−1 gives it the structure of an affine algebraic variety, whose ring of regular functions is     −1  . Ω[Xij , X0 ]/ X0 det(Xij ) − 1 = Ω Xij , det(Xij ) Definition. An algebraic group of matrices is a subgroup G of GL(n, Ω) that is an algebraic subset, i.e., the set of zeros of an ideal of polynomial functions. We say that G is defined over the subfield k of Ω, or that G is a k-group, if the ideal of all the polynomials vanishing on G admits a system of generators belonging to the subring:   −1  . k Xij , det(Xij ) If B is a subring of Ω, we put GB = G ∩ GL(n, B). More generally, we can consider a subgroup G of GL(V), where V is a finitedimensional vector space over Ω. If, by the choice of a basis of V, it can be identified with an algebraic group of matrices, then such a group G is said to be an algebraic group (more precisely, a linear algebraic group). The ring of regular functions on G is denoted by Ω[G]. Assume V has a k-structure, for some subfield k of Ω (i.e., V is obtained by extension of scalars47 from a vector space Vk over k). We say that G is defined over k if this is true of the group of matrices that is associated to G by choosing a basis of Vk . We then use k[G] to denote the subring of Ω[G] that consists of the regular functions that are defined over k. Furthermore, the Lie algebra g of G has a k-structure: g = Ω ⊗k gk , where gk denotes gl(Vk ) ∩ g. 7.2. Definition. A morphism of algebraic groups is a morphism of groups: ϕ : G → G , whose comorphism ϕ0 maps a regular function on G to a regular function on G. If G and G are defined over the subfield k of Ω, we say that the morphism ϕ is defined (or rational) over k, or is a k-morphism, if:   ϕ0 k[G ] ⊂ k[G]. 47 Translator’s

note: I.e., V = Vk ⊗k Ω. 31

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An algebraic representation of an algebraic group G on a vector space W over Ω is an algebraic morphism of G into GL(W). Such a representation gives rise to an action of G on Ω[W] (on the right) by: ϕ → ϕ . g, where this function is defined by (ϕ . g)(w) = ϕ(g . w). G acts in the same way on Ω[X] for every affine subvariety X of W that is invariant under G: if W and X are defined over k, then, since the action of G preserves the filtration48 of Ω[W], each function ϕ ∈ k[X] is contained in a finite-dimensional vector subspace of Ω[X] that is invariant under G, and defined over k. In the special case where X = G ⊂ GL(V) and W = End(V), we recover the usual action of G on Ω[G] by left translations. Definition. A character of the algebraic group G is an algebraic morphism of G into GL(1). The commutative group49 of characters is denoted by X(G). If G is defined over k, then the subgroup X(G)k is the subgroup consisting of the characters that are defined over k. 7.3. Let G ⊂ GL(n, Ω) be a k-group. The semisimple part gs and unipotent part gu of an element g ∈ G belong to G. This Jordan decomposition is preserved by every morphism. If k is perfect, and g ∈ Gk , then gs , gu ∈ Gk . Assume that k has characteristic zero. Then, the exponential map defines a bijection from the set of nilpotent elements of gk onto the set of unipotent elements of Gk , and this map is a polynomial whose inverse, defined by the logarithm, is also polynomial. In particular, Gk consists entirely of semisimple elements if and only if gk contains no nonzero nilpotent elements. Let us also note the infinitesimal analogue of the first statement of the preceding paragraph: the semisimple part xs and nilpotent part xn of x ∈ g belong to g. This decomposition is preserved by the differential of every morphism of algebraic groups. 7.4. The connected component of the identity of an algebraic group (in the Zariski topology) is irreducible and of finite index. In the case where Ω = C, that is, when G also has the structure of a complex Lie group, it can be identified with the identity component of G in the usual topology. If Ω = C, then GR is a real Lie group with a finite number of connected components (in the usual topology), but it is not necessarily connected even if G is a connected R-group. We will denote by G0 the identity component of G, and if Ω = C we will also denote by G0R the identity component of GR in the usual topology. If a surjective morphism f : G → G of R-groups is defined over R, then it induces a homomorphism GR → GR of real Lie groups, but this induced map is not necessarily surjective. However, it maps G0R onto GR0 , so f (GR ) is a finite-index subgroup of GR that is both open and closed. If H and H are R-subgroups of G and G such that f (H) ⊂ H , then the image of the map GR /HR → GR /HR induced by f is both open and closed, and GR /HR is a union of a finite number of orbits of GR , each of which is both open and closed. 48 In the natural filtration Ω[W] ⊂ Ω[W] ⊂ · · · of Ω[W], the subspace Ω[W] consists of the 0 1 i polynomials of degree ≤ i. 49 The group operation is pointwise multiplication: (α . β)(g) = α(g) . β(g). However (as mentioned on page xi), we often write this as: g α+β = g α. g β .

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The finiteness of the number of connected components of GR can be thought of as a special case of the following proposition [7, §6.4]: Let ρ be a representation of the R-group G on the vector space V, such that ρ is defined over R, and let v ∈ VR . Then the set (G . v)R of real points of the orbit G . v of v is the union of a finite number of orbits of G0R . B. Reductive groups. 7.5. Recall that a torus is a connected, commutative algebraic group that consists entirely of semisimple elements, which means that it is connected and diagonalizable, i.e., isomorphic to (Ω∗ )m for some integer m. A group whose radical consists only of {e} is said to be semisimple. (The radical is the largest closed normal subgroup that is solvable and connected.) Definition. An algebraic group G is said to be reductive if its identity component G0 is equal to the almost-direct product S . H of a central torus S with a semisimple group H. In characteristic zero, this property is equivalent to complete reducibility of all representations of G. In what follows we will restrict ourselves to characteristic zero. 7.6. Proposition. Let G be a reductive group acting on a vector space W and let X be an irreducible subvariety of W that is invariant under G. Let I be the ring Ω[X]G of regular functions on X that are invariant under G. (i) There exists a projection  : Ω[X] → I that is I-linear and leaves invariant every G-invariant subspace; (ii) I separates closed algebraic subsets of X that are invariant under G; (iii) I is a finitely generated Ω-algebra. Proof. (i) Let N be the sum of the minimal subspaces of Ω[X] on which G does not act trivially. The complete reducibility of the action of G on Ω[X] shows that Ω[X] is equal to I ⊕ N. The map  is defined to be the projection onto I associated with this decomposition. It remains to see that it is I-linear; however, we have: I ∩ I . N = {0}, which, by complete reducibility, implies the inclusion I . N ⊂ N, and therefore equality; next, if ϕ ∈ I and ψ ∈ Ω[X], we have: (ϕ . ψ) = ϕ . ψ + (ϕ . ψN ) = ϕ . ψ . Finally, let E be a subspace of Ω[X] that is invariant under G; complete reducibility implies E = (E ∩ I) ⊕ (E ∩ N), so E ⊂ E. (ii) Let A and B be two algebraic subsets of X that are invariant under G and with corresponding ideals I(A) and I(B). We assume that these are disjoint and therefore (Nullstellensatz ) I(A) + I(B) = Ω[X]. Thus, there exist α ∈ I(A) and β ∈ I(B) such that α + β = 1, whence α + β = 1. But the invariance of A and B under G implies that of I(A) and of I(B); it follows (cf. (i)) that α ∈ I(A) and β ∈ I(B), so α is zero on A, and equal to 1 on B.

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(iii) Ω[X] can be identified with the quotient of Ω[W] by the ideal of X, and the projection π : Ω[W] → Ω[X] commutes with G. Complete reducibility implies the equality:50   π Ω[W]G = I. Therefore, it suffices to prove (iii) when X = W. In this case, the functions in I that vanish at the origin generate an ideal with a finite generating set {f1 , . . . , fs }. Since the homogeneous components of an invariant polynomial are also invariant, we can assume the fi to be homogeneous, and it suffices to show that every homogeneous s element f ∈ I of degree > 0 is a polynomial in the fi . We can write f = i=1 ai . fi , where ai ∈ Ω[W] is homogeneous, and its degree d0 ai is d0 f − d0 fi . In view of (i), we may assume ai ∈ I is homogeneous of degree d0 f − d0 fi . The conclusion then follows by induction on the degree.  7.7. Proposition. Let G be a connected algebraic group, H be a reductive subgroup of G, and be k a field of definition for G and H. Then, there exists a finitedimensional vector space W that is defined over k, a representation of G on W (on the right) that is rational over k, and a point w ∈ Wk , such that the orbit X of w is closed and its isotropy group is H. In particular, this means that the set H\G of right cosets of G modulo H has the structure of an affine variety defined over k. Proof. The proof uses the conclusions of 7.6 in the following special situation: the reductive group H acts on the left on the variety G ⊂ GL(V), which we may view as a closed subset of W = End(V ⊕ Ω) via the usual embedding (cf. 7.1). Thus, the algebra I = Ω[G]H of functions that are constant on the right cosets Hx of G modulo H admits a finite generating set {w1 , . . . , ws }, which we may take to be in k[G] (7.6 (iii). These then belong to subspaces Wi of Ω[G] that are defined over k, invariant under G, and finite-dimensional (cf. §7A). We will show that s the problem given in the statement is solved by taking the vector space W = i=1 Wi , the representation of G on the right that is defined by: (v1 , . . . , vs ) → (v1 . g, . . . , vs . g), and the point w = (w1 , . . . , ws ) ∈ Wk . To that end, we must first show that the isotropy group Gw of w is H; we have Gw ⊃ H, since wi ∈ I implies that wi . h = wi and w . h = w for h ∈ H; conversely, Gw ⊂ H, because if g ∈ Gw , we have wi . g = wi , so wi (g) = wi (e) for every i; but {wi } generates I, which implies f (g) = f (e) for every f ∈ I and it follows that g ∈ H, since I separates the closed algebraic subsets of G that are invariant under H, and in particular, it separates the right cosets of G modulo H, in view of 7.6 (ii). It remains to show that the orbit X of w is closed, i.e., equal to X. For the morphism ϕ : G → W defined by g → w . g, there is a corresponding comorphism ϕ0 : A = Ω[X] → Ω[G]. 50 Complete reducibility implies that we may write Ω[W] = Ω[W]G ⊕ N , where N is the sum  simple submodules on which G acts nontrivially. Since π commutes with G, we have  of the π Ω[W]G ⊂ I and π(N ) ⊂ N. Therefore, π(N ) does not contribute anything to Ω[X]G = I.

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In fact, ϕ0 (A) is contained in I, because the inclusion Gw ⊃ H implies: [(ϕ0 f ) . h](g) = f (w . h . g) = f (w . g) = (ϕ0 f )(g). The mapping ϕ0 is injective because, if ϕ0 f is zero, then f must be zero on X, so it is also zero on X. Finally,51 im(ϕ0 ) = I; to see this, it suffices to show that each generator wi of I belongs to ϕ0 (A). But if {z1 , . . . , zn } is a basis of Wi and {a1 , . . . , an } is the dual basis, then the latter defines functions on W: uj : (v1 , . . . , vs ) → aj (vi ), such that (ϕ0 uj )(g) = aj (wi . g). It follows that:   wi (g) = (wi . g)(e) = aj (wi . g) . zj (e) = zj (e) . (ϕ0 uj )(g). j

j

Thus, ϕ is an isomorphism of A onto I. Establishing the equality X = X amounts to proving, for each x ∈ X, that the maximal ideal Ix of x has a zero that is in X or, equivalently, that ϕ0 (Ix ) has a zero that is in G. For this, it suffices to establish that ϕ0 (m) . Ω[G] is a proper ideal of Ω[G] for every proper ideal m of A; but if this werenot the case, there would exist ϕ0 (mi ) . fi = 1, which would m1 , . . . , mt ∈ m and f1 , . . . , ft ∈ Ω[G] such that 0 imply according to 7.6 (i) and the inclusion ϕ (A) ⊂ I:  (ϕ0 mi ) . fi = 1.  mi . fi = 1, with fi ∈ A, contradicting the fact But ϕ0 is invertible, whence52 that m is proper in A.  0

Remark. Proposition 7.7 shows that the quotient space G/H can be endowed with the structure of an algebraic variety (in fact, an affine algebraic variety). One easily sees that this is a quotient structure in the sense of 7.10. More generally, one can prove that for any k, if H is a k-subgroup of G, then the quotient of G by H “exists” and is defined over k, and is a quasi-projective variety. This, too, can be proved by using a suitable linear representation of G (cf. [5, §6]). 7.8. Proposition. Let G be an algebraic group, H be a closed subgroup, and k be a common field of definition for G and H. Then there exists a finite-dimensional vector space W that is defined over k, a morphism G → GL(W) that is defined over k, and a point w ∈ Wk such that H is the set of elements of G that stabilize the line [w] spanned by w. Proof. Let J ⊂ Ω[G] be the ideal of functions vanishing on H. It is defined over k and has a finite generating set. Thus, there exists (cf. §7A) a finite-dimensional subspace V of Ω[G] that is defined over k and invariant under left translations, such

that V ∩ J generates J. We then let W = d V, (d = dim V ∩ J), let [w] be the line corresponding to V ∩ J, and let ρ be the representation that is induced on W by the given representation of G on V.  7.9. Corollary. If X(H)k = {1}, then there exists a morphism G → GL(W) that is defined over k, and a point w ∈ Wk whose isotropy group is H. 51 im f

denotes the image of a function f , as was mentioned in footnote 12. fi ∈ Ω[G]H = I = ϕ0 (A), we have fi = ϕ0 (fi ), for some fi ∈ A.    0  mi . fi = ϕ (mi ) . fi = 1, so the injectivity of ϕ0 implies mi . fi = 1. ϕ0 52 Since

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Then

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Proof. Indeed, the hypothesis X(H)k = {1} shows that the representation of H on [w] is trivial.  7.10. Remarks. (1) Proposition 7.6 has a converse (due to Y. Matsushima [25]): “If G is reductive, H is a closed subgroup of G and G/H is an affine variety, then H is reductive.” For other proofs, see [6, Theorem 3.5] or [1]. In both cases, only characteristic zero is considered. However, it seems that by using Grothendieck cohomology, one can extend the proof of [6] to characteristic p > 0. (2) The proof of 7.7, suitably modified and extended, gives more generally: “Let X be an affine variety on which a reductive group H acts in such a way that every orbit is closed. Then, the quotient X/H ‘exists,’ and is an affine variety.” The expression “the quotient exists and is affine” means here that we can find an affine variety V, a morphism π : X → V, whose fibers are the orbits of H, such that if f : X → W is a morphism of X into an algebraic variety W, and f is constant on each orbit of H, then there exists a morphism53 g : V → W such that f = g ◦ π. In non-zero characteristic, this statement and the proof below remain valid if H is a torus, which is equivalent to saying that all the rational representations of H are completely reducible54 [36]. It will not be used in the sequel, and we will restrict ourselves to briefly indicating the proof, assuming some elementary notions of affine varieties. Let I = Ω[X]H . This is a finitely generated algebra (7.6). Let {f1 , . . . , fd } be a generating set for I. We therefore have I ∼ = Ω[T1 , . . . , Td ]/J, where J is the kernel of the homomorphism Ω[T1 , . . . , Td ] → Ω[X] defined by Ti → fi , (1 ≤ i ≤ d). Let V ⊂ Ωd be the affine variety whose ideal is J. Then π : x → f1 (x), . . . , fd (x) defines a morphism of X into V. We want to show that (V, π) satisfies the conditions imposed above. The fact that the invariant functions separate the disjoint closed sets that are invariant under H (7.6), together with the hypothesis, implies that the fibers of π are the orbits of H. Let us show that π is surjective. Let v ∈ V, and let A be the ideal of elements of Ω[V] ∼ = I that vanish at v. Let us show that B = A . Ω[X] is a proper ideal of Ω[X]. If this were not the case, we could find elements aj ∈ A, cj ∈ Ω[X] (1 ≤ i ≤ n), such that aj . cj = 1, whence:    aj . cj = 1, aj . cj = and 1 ∈ A, which is absurd. Thus B = Ω[X] and there exists x ∈ X annihilating B. Then, we have π(x) = v. The factorization condition formulated above can also be expressed as follows: let p, q ∈ Ω[X] and a ∈ X such that f = p/q is invariant under H, and q(a) = 0. Then there exist p , q  ∈ I such that f = p /q  and q  (a) = 0. [Here, p/q, p /q  are viewed as the elements of the total ring of fractions of Ω[X], on which H acts in an f

 

X →W note: I.e., the following diagram commutes: π ↓ g V 54 A representation is completely reducible if it is a direct sum of irreducible representations. 53 Translator’s

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obvious way. For f = p/q, there is an associated regular function on the open set U of points where q = 0, whose value at u is equal to p(u)/q(u).] Let E be a finite-dimensional subspace of Ω[X] that contains q and is invariant under H (7.2). We have a unique decomposition E = E + E where E = ker  ∩ E. Obviously, f . E = f . E + f . E , whence, by the uniqueness of this decomposition, f . E = (f . E) , so in particular: p = (f . q) = f . q  , which shows that f = p /q  . Therefore, our claim is established if q  (a) = 0. Given any s ∈ Ω[X] and h ∈ H, we can obviously write f = s . (p . h)/s . (q . h). We then also have: f = (s . (p . h)) /(s . (q . h)) , and to complete the proof, it suffices to show that we can choose s and h in such a   way that s . (q . h) (a) = 0. Let F be the subspace of Ω[X] generated by the translates q . h of q (h ∈ H) and let J be the ideal of Ω[X] generated by the functions fi − fi (a) and by F (1 ≤ i ≤ d). Let us show that 1 ∈ J. Assume that this is not the case. Hence, we can find b ∈ X, such that b is a zero of J. We therefore have fi (b) = fi (a), (1 ≤ i ≤ d), whereby r(b) = r(a) (r ∈ I), which implies the existence of h ∈ H, such that b = ha. From this we deduce: q(a) = (q . h−1 ) (h . a) = q . h−1 (b) = 0, which is absurd. Thus, 1 ∈ J. Therefore, we can find: ci , dj ∈ Ω[X], such that:

hj ∈ H

(1 ≤ i ≤ d; 1 ≤ j ≤ n),

    ci fi − fi (a) + dj . (q . hj ) = 1. i

j

By applying  to the two sides and evaluating at a, we obtain:   dj . (q . hj ) (a) = 0, j

and the claim follows.



C. Arithmetic groups. 7.11. In this section, V denotes a finite-dimensional vector space over C, endowed with a Q-structure. If G is an algebraic subgroup of GL(V) that is defined over Q, and L is a lattice of VQ , we denote by GL the subgroup of GQ that leaves L invariant, and call it the group of L-units of G: GL = {g ∈ GQ | g(L) = L}. Recall that two subgroups A, B of a group H are said to be commensurable if A ∩ B is of finite index in both A and B. Definition. Let G be a Q-subgroup of GL(V). A subgroup Γ of GQ is said to be arithmetic if there exists a lattice L of VQ such that Γ is commensurable with GL .

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It will follow from 7.13 that this property is independent of L and is invariant under Q-isomorphisms. If we identify V with Cn via a basis of L, then G can be identified with a Q-subgroup G of GL(n, C) and GL with GZ . Therefore, an equivalent definition is to say that Γ is arithmetic if there exists an embedding ρ : G → GL(n, C) that is defined over Q, and maps Γ onto a subgroup of ρ(G)Q that is commensurable with ρ(G)Z . In the case where55 ρ(Γ) ⊂ GL(n, Z), a subgroup Γ of Γ is a congruence subgroup (or principal congruence subgroup), if there exists an integer m > 0 such that: ρ(Γ ) = {x ∈ ρ(Γ) | x ≡ I mod m}. Such a subgroup is obviously normal and of finite index.56 7.12. Proposition. Let G be a subgroup of GL(n, C) that is defined over Q and let ρ be a representation of G that is defined over Q on a vector space V that is equipped with a lattice L of VQ . Then there exists a congruence subgroup of GZ that leaves L invariant. Proof. Choose a basis of L. Using this as the basis of V, define polynomials Pμ,ν by57 ρ(g)μ,ν = Pμ,ν (gij ). Since ρ is defined over Q, the coefficients of these polynomials are rational. So the same is true of the coefficients of the polynomials Qμ,ν defined by:58 ρ(g)μ,ν − δμ,ν = Qμ,ν (gij − δij ). But these have no constant term, since ρ(I) = I, and hence if m is denominator of all of their coefficients, we have Qμ,ν (gij − δij ) ∈ Z gij − δij ≡ 0 (mod m), i.e., g ≡ I (mod m). For such a choice of g entries ρ(g)μ,ν are integers, which guarantees the invariance of L under

a common as soon as the matrix ρ(g). 

55 The original French manuscript erroneously omits this condition. If the condition is not imposed, then m should be required to be relatively prime to the denominators of the matrix entries of the elements of ρ(Γ), in order for the congruence condition to define a subgroup. (However, most authors define congruence subgroups only for subgroups of GL(n, Z).) 56 There is a natural homomorphism φ : GL(n, Z) → GL(n, Z/mZ). Since GL(n, Z/mZ) m is finite, it is clear that ker φm is a normal subgroup of finite index in GL(n, Z). Since ρ(Γ) ⊂ GL(n, Z), this implies that Γ = ρ−1 (ker φm ) is a normal subgroup of finite index in Γ. 57 Each matrix entry ρ μ,ν of the representation ρ is a regular function on G. Assume, for simplicity, that det g = 1 for all g ∈ G. (This is true for the elements of the congruence subgroup of GZ corresponding to any m > 2, so this assumption causes no harm.) Then the regular function ρμ,ν (g) = ρ(g)μ,ν is a polynomial function of the matrix entries of g. We call this polynomial Pμ,ν . 58 The polynomials Q μ,ν are obtained from Pμ,ν by a change of variables. The point is that the homomorphism ρ must map the identity matrix to the identity matrix (ρ(I) = I), but equations of the form f (0) = 0 are better, so we change coordinates by translating by −I in both the domain and the range. δμ,ν is the Kronecker delta function, or, in other words, δμ,ν is the (μ, ν)-entry of the identity matrix I: δμ,ν is 1 if μ = ν, and it is 0 if μ = ν.

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7.13. Corollary. (1) If Γ is an arithmetic subgroup of G, then each lattice L of VQ is contained in a lattice that is invariant under Γ. (2) Let ϕ be an isomorphism that is defined over Q from G onto a Q-group G and let Γ, Γ be arithmetic subgroups of G and G . Then ϕ(Γ) is commensurable with Γ . (3) If ϕ is a morphism of G into G that is defined over Q and if Γ ⊂ GQ is arithmetic, then there exists an arithmetic subgroup Γ of G that contains ϕ(Γ). (4) If G = H . N is the semidirect product of a closed normal subgroup N and a closed subgroup H, and both are defined over Q, then HZ . NZ is an arithmetic subgroup of G. Proof. Let us prove (1). We can identify GL(V) with GL(n, C) in such a way that Γ is commensurable with GZ (7.11). In view of 7.12, there exists a finite-index subgroup Γ of Γ that stabilizes L . Hence, the set of lattices g(L ) that are images of L under elements of Γ is finite; the sum of these lattices is then a lattice that is invariant under Γ. (2) and (3) immediately follow from (1). Finally, let π be the projection of G onto H = G/N. For every g ∈ G, we obviously have π(g)−1. g ∈ N. By (3),59 there exists a finite-index subgroup Γ of GZ such that π(Γ) is contained  in HZ . Thus we have Γ ⊂ HZ . NZ . 7.14. Examples of arithmetic groups. 1) GL(n, Z) and SL(n, Z). 2) The group of units of a non-degenerate rational quadratic form (§5) 3) The group Sp(2n, Z) = {M ∈ GL(2n, Z) | t M . J . M = J}, where J denotes the matrix: ⎞ ⎛ 0 In ⎠. ⎝ −In 0 4) If k is a number field with basis {ω1 , . . . , ωd } over Q, this basis enables us to embed k∗ into GL(d, Q) through the regular representation. It then becomes the rational points over Q of a group G that is a d-dimensional torus defined over Q; the group G is the centralizer of k∗ in GL(d, C). If {ω1 , . . . , ωd } is a Z-basis of the ring of algebraic integers, then G ∩ M(d, Z) can be identified with the set of non-zero algebraic integers of k. The arithmetic group GZ is then the group of units of the number field k. 5) More generally, we can consider a finite-dimensional Q-algebra A, the group G of invertible elements of AC and the arithmetic group GL of units of a lattice L of AQ : GL = {g ∈ A | g . L = L}. Remark. The notion of arithmetic group adopted here is slightly more general than the traditional notion (independently of the fact that G is not necessarily a classical group). Indeed, classically, an arithmetically defined group is a subgroup of GZ that is determined by congruence conditions on the matrix entries; in particular, it always contains a congruence subgroup in the sense of 7.11. It may happen that an arithmetic group in the sense of 7.11 does not have this property, as has long 59 This follows more easily from the following direct consequence of 7.12: If ϕ is a morphism of G into G that is defined over Q and if Γ ⊂ GQ is arithmetic, then there exists an arithmetic subgroup Γ of G, such that ϕ(Γ) ⊂ Γ .

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been known for G = SL2 . However, this condition will not play a role here. In fact, we will even see that the principal theorems of reduction theory remain valid for every subgroup of GR (and not only of GQ ) that is commensurable with GZ . 7.15. In this book, we mainly consider reductive groups. It is usually not difficult to pass from the reductive case to the general case by using the following structure theorem. Theorem. Let k be a field of characteristic zero and G be a k-group. Then G is the semidirect product of a reductive k-subgroup H and a connected unipotent normal k-subgroup N. Every reductive k-subgroup of G is conjugate (by an element of Nk ) to a subgroup of H. The subgroup N is uniquely determined by the theorem: it is the unipotent radical of G, i.e., the maximal normal subgroup of G that consists entirely of unipotent matrices, and is sometimes denoted by Ru (G). It is a nilpotent group. Let us also note that, since N is connected, the group G is connected if and only if H is connected. In particular: G0 = H0. N. 7.16. Groups over a number field. Let k ⊂ C be a finite extension of Q of degree d, and let S be the set of monomorphisms of k into C. Thus, S contains d elements, and we have d = r1 + 2r2 , where r1 is the number of real places of k (i.e., of s for which s(k) ⊂ R) and r2 is the number of complex places (i.e., of pairs of elements of S formed by an s such that s(k) ⊂ R and the homomorphism s : x → s(x) ). Let Ok be the ring of integers60 of k. Let G be an algebraic subgroup of GL(n, C) that is defined over k. A subgroup of Gk is said to be arithmetic if it is commensurable with GOk . We now indicate, without proof, how this notion can be reduced to the seemingly more restrictive one in 7.11, by using the “restriction of scalars” functor Rk/Q . We refer to [43, Chap. I] for a description of this functor and will only indicate a construction of Rk/Q G that applies to our situation. Given s ∈ S, denote by sG the conjugate of G by s, or, in more technical terms, the group that is obtained from G by the change of base s : k → s(k); in less technical language, sG is the group whose ideal of definition in s(k)[X11 , . . . , Xnn ] is obtained by applying s to the coefficients of the elements of k[X11 , . . . , Xnn ] that vanish on G. Let us write the elements of M(n . d, C) as blocks of square matrices of order n. For g ∈ GL(n, k), denote by sg or s(g) the element of GL n, s(k) 61 that is s obtaineds by applying s to the entries of g. Now, let ι be the mapping 1 d g → (g), . . . , (g) , (si ∈ S) of GL(n, k) into GL(n  . d, C). Finally, let (αi ) be a Z-module basis of Ok , and let A be the matrix si (αj ) . In , which we know is invertible. There exists a subgroup G of GL(n . d, C) that defined over Q, such that: GQ = A . ι(Gk ) . A−1 ,

(1)

GZ = A . ι(GOk ) . A−1 .

The group G is isomorphic over C to the product of the sG (s ∈ S). Finally, let J be the set of archimedean places of k. We identify J with a subset of S consisting of the r1 real places and one representative from each pair consisting of a complex 60 The

original French manuscript uses a lowercase fraktur o for the ring of integers. s  1(g), . . . , sd(g) is a block-diagonal matrix with d blocks of order n.

61 Here,

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8. COMPACTNESS CRITERION

41

element of S and its complex conjugate. Let us denote by Gs (s ∈ J) the group sGR (resp. sGC ) if s is real (resp. complex). Then:  Gs . (2) GR ∼ = s∈J 

The group G is the group Rk/Q G that obtained from G by applying restriction of scalars from k to Q. If f : G → H is a k-morphism of k-groups, then there is a canonically defined Q-homomorphism Rk/Q f : Rk/Q G → Rk/Q H. Over C, this is the product of the conjugates sf : sG → s H of f . Suppose that k is an imaginary quadratic field. Then J consists of a single  complex place. We have GR ∼ = GC . We  pass from  GC to GR by associating to x y g = x + iy (x, y ∈ M(n, R)) the matrix . Let us note that, in this case, −y x GOk is discrete in G. In general, this is not the case, but GOk can be identified with a discrete subgroup of s∈J Gs by the “diagonal” map ι. Slightly more generally, if J is a subset of J such that Gs is compact for s ∈ J, s ∈ / J , then the projection  of ι(GOk ) into the product of the factors Gs (s ∈ J ) is discrete. Bibliographical note. For more details on the notions or results quoted here without proof, we refer the reader to62 [5, 13]. In fact, these monographs undertake the development of a theory that is valid for all characteristics, whereas, for the needs of this book, characteristic zero is quite sufficient. The reader who wishes to restrict to the latter and rely on the books by Chevalley [13] is warned that there is a slight difference between the notion of an algebraic group over a field k in [13] and that of an algebraic k-group in the sense of 7.1. The theorem of 7.15 was proved, first of all, for algebraic Lie algebras by Chevalley. The global version was established by Mostow [29] in the setting of [13], which implies that Gk is Zariski-dense in G. The passage from there to the general case was given in [7, §5.1]. The theorem of 7.10 (2) was proved by M. Nagata [32] and D. Mumford [31, §I.2, Theorem 1.1, p. 27]. 8. Compactness criterion We already noted (cf. §1) that the set R of lattices of Rn can be canonically identified with GL(n, R)/GL(n, Z), which allows it to be endowed with the natural topology of this quotient space. If G ⊂ GL(n, C) is an algebraic group, then GR /GZ can be embedded in an obvious way into R. 8.1. Proposition. Let G ⊂ GL(n, C) be an algebraic Q-group that either is reductive, or has no non-trivial rational character that is defined over Q. Then, GR /GZ is closed in R. Proof. The goal is to prove that GR . GL(n, Z) is closed in GL(n, R). In the two cases considered here, we see from 7.7, 7.9 that G has the following property: (P) There exists a representation π : GL(n, C) → GL(V) (on the right) that is defined over Q, and an element v ∈ VQ whose isotropy group is G. 62 Another good reference for information about algebraic groups over fields that are not required to be algebraically closed is [T. A. Springer, Linear algebraic groups, 2nd ed., Birkh¨ auser, 2009, MR2458469].

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By 7.13, there exists a lattice L of VQ that contains v and is invariant under GL(n, Z), which shows that v . GL(n, Z) is closed in VR . But GR . GL(n, Z) is the pre-image of v . GL(n, Z) in GL(n, R) under the map g → v . g from GL(n, R) to  VR . It follows that GR . GL(n, Z) is closed. 8.2. Proposition. Let G ⊂ GL(n, C) be a reductive algebraic Q-group, Γ be an arithmetic subgroup of G, and M be a subset of GR . The following are equivalent: (i) M is relatively compact modulo Γ. (ii) M is relatively compact modulo every arithmetic subgroup of G. (iii) |det g| is bounded above on M and there exists a constant c > 0 such that

g(x) ≥ c whenever g ∈ M and x ∈ Zn  {0}. (iv) |det g| is bounded above on M; furthermore, if (vj )j≥1 and (gj )j≥1 are sequences of elements of a lattice L of Qn and of M, respectively, such that gj . vj → 0 as j → ∞, then vj = 0 for all sufficiently large j. Proof. By 8.1, the image of GR /GZ in R is closed in R. The equivalence of (i) and (ii) follows from the commensurability of every pair of arithmetic subgroups. By assuming, without loss of generality, that Γ = GZ , we see that (i) is equivalent to saying that the image of M in R is relatively compact. The assertions (iii) and (iv) are merely translations of Mahler’s criterion (1.9), applied to M.  8.3. Lemma. Let G be an algebraic Q-group, Γ be an arithmetic subgroup, and π : G → GL(V) be a representation of G that is defined over Q. If GR /Γ is compact, then π(GR ) . v is closed in VR for every v ∈ VQ . Proof. By assumption,   we have GR = C . Γ, where C is compact, whence π(GR ) . v = π(C) . π(Γ) . v) . Therefore, it suffices to show that π(Γ) . v is closed; this follows from 7.13.63  Remark. This lemma is in fact a consequence of 7.13 and the following elementary remark. Let G be a locally compact group, and H be a subgroup such that G/H is compact. Let M be a locally compact space on which G acts and let m ∈ M be a point whose H-orbit is closed. Then, G . m is closed. If H is discrete, this shows in particular, by letting G act on itself by inner automorphisms, that the conjugacy class of each element of H is closed in G. 8.4. Theorem. Let G ⊂ GL(n, C) be a reductive Q-group and Γ be an arithmetic subgroup of G. The following are equivalent: (i) GR /Γ is compact; (ii) X(G0 )Q = {1}, and every element of GQ is semisimple. Proof. Let us first show that we can reduce this to the case where G is connected. The group (G0 )R is open and closed, normal, and of finite index in GR , so (G0 )R /(Γ ∩ G0 ) is open and closed in GR /Γ, and the latter is the union of a finite number of translates of (G0R )/(Γ ∩ G0 ); it follows that (i) for G is equivalent to (i) for G0 . On the other hand, in characteristic zero, an element of finite order is always semisimple, so in view of 7.3, if GQ contains an element u = 1 that is not semisimple, then every power um (m = 0) of u is not semisimple; since um ∈ G0 for suitable m, it follows that (ii) for G is equivalent to (ii) for G0 . 63 As

in the proof of Proposition 8.2.

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Henceforth, we assume G is connected. (i) ⇒ (ii). Let A : G → GL(1) be a character of G that is defined over Q and let v ∈ Q∗ . By 8.3, A(GR ) . v is closed in R; further, if A is non-trivial, this orbit contains at least the real numbers that are > 0, but does not contain the origin, so is not closed, whence A = 1. If we apply 8.3 to the representation of G obtained by letting it act on M(n, C) by inner automorphisms, we see that the conjugacy classes of elements of GQ are closed. However, let u ∈ GQ be a unipotent element = 1. By a theorem of JacobsonMorosow (see, e.g.,  [19,Lemma 7, p. 98]), we can find a morphism σ : SL2 → G 1 1 that maps u0 = onto u. The conjugacy class of u0 in SL(2, R) contains 0 1 the matrices:    −1    t 0 t 0 1 t2 , . = . u 0 0 t−1 0 1 0 t so it contains the identity element in its closure. The same is then also true for the conjugacy class of u in GR . It follows that GQ does not contain any nontrivial unipotent element, and so it must consist of semisimple elements (7.3). (ii) ⇒ (i). We first consider the case where G has trivial center {e}. Then the group G is semisimple and, moreover, the adjoint representation Ad of G on its Lie algebra g is faithful; since it is defined over Q, we can assume that Γ is the stabilizer of a lattice L of gQ . It is enough to show that the conditions of 8.2 (iv) are satisfied. The first is satisfied since G is semisimple, so |det g| = 1 for every g ∈ G. Write the characteristic polynomial of x ∈ g in the form:  Pi (x) . Tn−i , det(ad x − T . I) = (−T)n + coefficients where T is an indeterminate. The Pi (x) are polynomials with rational  with respect to a basis of g consisting of elements of L. Let P = i P2i . For every x ∈ L  {0}, we know that ad x is not nilpotent since gQ does not contain any non-zero nilpotent element,64 so P(x) = 0. But P has rational coefficients, and can be written in the form P (x) . q where P has integer coefficients, and q ∈ Q∗ . Thus, there exists a number c > 0 such that x ∈ L  {0} implies P(x) ≥ c. But if (gj ) and (vj ) are sequences of elements of GR and of L respectively, such that Ad gj (vj ) → 0, we have:   P(vj ) = P Ad gj (vj ) → 0, which implies that vj = 0 for j sufficiently large. Let us now move on to the general case. Let G∗ = G/Z(G) be the quotient of G by its center and (cf. 7.13) let Γ∗ be an arithmetic subgroup of G∗ that contains the image of Γ under the canonical projection π. Let us show that G∗ satisfies (ii). Since G∗0 = π(G0 ), it is clear that X(G∗0 )Q = {1}. Assume that G∗Q contains a non-semisimple element = e. Then, by 7.3, it contains a unipotent element = e, which must have infinite order, so G∗0 Q also contains a unipotent element x = e. The latter necessarily belongs to the derived group65 DG∗0 = H of G∗0 , so log x is a nilpotent element of hQ . Since π is an isogeny of DG0 onto DG∗0 , it follows that the Lie algebra of DG0 contains an element x = 0 that is rational over Q and 64 The exponential of any nilpotent element of the Lie algebra is unipotent, but every element of G is semisimple. 65 The derived group DG of a group G is the commutator subgroup [G, G].

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nilpotent. Then, exp x is a unipotent element = e of GQ , which contradicts the hypothesis on G. By what has already been proved, the quotient G∗R /Γ∗ is compact. Since the image of GR /Γ in G∗R /Γ∗ is closed (7.4), it suffices to show that the natural map GR /Γ → G∗R /Γ∗ is proper. This will follow from the following more general proposition.  8.5. Proposition. Let G, G be Q-groups, and π : G → G be a Q-morphism that is surjective. Assume that G is reductive, that X(G0 )Q = 1, and that ker(π) is commutative. Let Γ, Γ be arithmetic subgroups of G and G respectively, such that π(Γ) ⊂ Γ and let D = π−1 (Γ ) ∩ GR . Then, D/Γ is compact, and the map GR /Γ → GR /Γ induced by π is proper. Proof. The map GR /Γ → GR /Γ factors66 to GR /Γ → GR /D → GR /Γ . The first of these maps is a fibration with fiber D/Γ, and is proper if and only if D/Γ is compact. The second is an injection onto a closed subset, therefore, is proper.67 Since a composition of proper maps is proper, it suffices to show that D/Γ is compact. As any two arithmetic subgroups of a Q-group are commensurable, we immediately see that if our conclusion holds for one pair of arithmetic subgroups, it is also true for any other pair. Moreover, since G0 has finite index in G, the problem immediately reduces to the case where G is connected. Then,68 ker(π) is contained in the center C of G and the projection G → G/C factors through π. Let Γ∗ be an arithmetic subgroup of G∗ = G/C that contains the image of Γ (cf. 7.13). We have the maps: GR /Γ

α

GR /Γ

β

G∗R /Γ∗ .

To show that α is proper, it suffices to show that β ◦ α is proper, so we are reduced to the case where G = G/C. We need to prove that D/Γ is compact. To that end, let us consider an embedding i : G → GL(n, C) that is defined over Q. The center C of G defines a commuting family of semisimple endomorphisms of Cn . Hence, C is a semisimple C-module, and the centralizer A of C in M(n, C) is both a C-algebra and a semisimple ring, and contains G. For each g ∈ G, denote by β(g) the inner automorphism x → g . x . g −1 of A. The map β : g → β(g) is a morphism of G into GL(A) that is defined over Q, and its image can be identified with G = G/C. The intersection L = M(n, Z) ∩ A is an order of AQ , and in particular a lattice of AQ . We identify GL(A) with GL(m, C) (m = dim(A)) by using a basis of L, and we take Γ and Γ to be the groups GZ and GZ respectively. We indeed have β(Γ) ⊂ Γ ; moreover, D = {g ∈ GR | g . L . g −1 = L}. To show that D/Γ is compact, we apply Mahler’s criterion to the action of G on A by left translations. We use λ(g) to denote the left translation by g ∈ G on A. Let R be the space of lattices of AR . Since A contains the identity element, the GR /Γ → GR /Γ ↓ note: I.e., the following diagram commutes: −→ GR /D 67 In general, an injection onto a closed set does not have to be proper. However, the image of this map is a closed orbit of the GR -action, so it is homeomorphic to GR /D. Therefore it is clear that the map is a homeomorphism. 68 Since G is reductive. 66 Translator’s

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stabilizer of L in GR is precisely Γ. The group X(G)Q is trivial by assumption, so det λ(g) = 1 (g ∈ G). Also (by 8.1), GR /Γ is closed in R so, in view of 8.2, it suffices to verify that if (xj ) and (gj ) are sequences of elements of L and D respectively such that gj . xj → 0, then xj = 0 for j sufficiently large. For this, we may replace L by an arbitrary sublattice. But A is the direct sum of its minimal two-sided ideals that are defined over Q, and the direct sum of the L ∩ Ai = Li is a lattice of AQ that is contained in L. Therefore, we can restrict ourselves to the case where (xj ) is a sequence of elements of Li , for some fixed i. Endow AR with the norm:

a = trace(ta . a). We have: (1)

a . b ≤ a . b ,

a . b = b . a .

It suffices to show: (∗)

There is a constant c > 0 such that d . x ≤ c (d ∈ D, x ∈ Li ) implies x = 0.

By Hermite’s Theorem (1.8), there exists a constant c1 > 0 such that: min

x∈Li {0}

g . x ≤ c1 . |det g|1/n

(g ∈ GL(Ai,R ), n = dim Ai ).

Since det g = 1 if g ∈ G, this shows that, given d ∈ D, we can find an element zd ∈ Li  {0} such that zd . d−1 ≤ c1 . Choose a basis (yt ) of Li and let: c2 =

min

x∈Li {0}

x ,

c3 = max yt . t

We now want to show that (∗) holds for every c satisfying 0 < c < c2 /(c1 c3 ). Thus, we let d ∈ D and x ∈ Li be such that d . x ≤ c. Since the lattice Li is an order of Ai,Q , we have x . yt . zd ∈ Li (with zd as above), from which, by the definition of D, it follows that: d . x . yt . zd . d−1 ∈ L ∩ Ai = Li . In view of (1) and the properties of the ci , this implies:

d . x . yt . zd . d−1 ≤ d . x yt zd . d−1 ≤ c . c3 . c1 < c2 , d . x . yt . zd . d−1 = 0, x . yt . zd = 0. Since this holds for every element yt in the given basis of Li , it follows that x . Ai . zd . Ai = 0. But zd is non-zero and is rational over Q, and Ai is a minimal two-sided ideal over Q, so Ai . zd . Ai = Ai , and finally, x . Ai = 0, whence x = 0.  8.6. Examples. Here we present some classical cases of compactness that follow from 8.4; but they were well known before Theorem 8.4, and in fact, they inspired the statement of the general theorem. (1) Let F be a non-degenerate quadratic form on a vector space VQ over Q and let G = O(F) be the orthogonal group of F. Let Γ be an arithmetic subgroup of G, for instance, the group of units of a lattice L of VQ (cf. §5). Then, GR /Γ is compact if and only if “F does not represent zero rationally,” that is, F(x) = 0 (x ∈ VQ ) implies x = 0. To deduce this statement from 8.4, it suffices to show that the condition imposed on F is equivalent to the following: GQ consists of semisimple elements and X(G0 )Q = {1}.

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Let n = dim(V) ≥ 3. Then G is semisimple, so X(G0 ) = {1}. Let g ∈ GQ be unipotent. Then, there exists v1 ∈ VQ  {0} that is fixed by g. Let V1 be the subspace spanned by v1 . If v1 is not isotropic, then V is the direct sum of V1 and its orthogonal complement V1 , which is invariant under g. Thus, there exists v2 ∈ V1  {0} that is fixed by g. Arguing by induction, we then see that if F does not represent zero rationally, then g = 1, so (by 7.3) GQ consists of semisimple elements. Conversely, if V contains a non-zero isotropic vector, it is elementary to see that F is equivalent over Q to a form of the type x . y + F1 . Hence, the group G contains a Q-subgroup that is isomorphic over Q to the orthogonal group of the form xy + z 2 , and is therefore isomorphic to the quotient of SL2 by its center. It follows that GQ contains unipotent elements = 1. If n = 2, then G0 is a one-dimensional torus, so GQ consists of semisimple elements. We then easily see that the following conditions are equivalent: (i) F represents zero rationally; (ii) F is equivalent over Q to the form x . y; (iii) G0 is isomorphic over Q to the multiplicative group C∗ ; and we will see or recall69 in §10 that (1iii) is equivalent to X(G0 )Q = 1. (2) Let k be a number field of degree d over Q. Let G be the torus associated to k (cf. §7), G be the group of elements of G of norm 1, and Γ be the group of units of k. It is easy to see that X(G)Q = {1}. Since GQ can be identified with k∗ , it consists of semisimple elements, so by Theorem 8.4, GR /Γ is compact. On the other hand, it can be verified that GR = (R∗ )r1 ×(C∗ )r2 , where r1 (resp. r2 ) is the number of distinct embeddings of k into R (resp. the number of pairs of complex conjugate embeddings of k into C), so that GR is isomorphic, as a real Lie group, to the product Rr1 +r2 −1 × SO(2)r2 . The compactness of GR /Γ then means that, modulo torsion, Γ is a free abelian group on r1 + r2 − 1 generators, which is the essential part of Dirichlet’s Unit Theorem. (3) Let G be the group of invertible elements of AC , where A is a finitedimensional Q-algebra, let Γ be the group of units of a lattice of AQ and let G be the group of elements of reduced norm 1 (cf. §7). It can be verified that AQ is a division algebra if and only if GQ consists of semisimple elements and X(G)Q = {1}. Therefore, if AQ is a division algebra, then GR /Γ is compact (K¨ ate Hey’s Theorem). 8.7. Theorem. Let G be a Q-group and let Γ be an arithmetic subgroup of G. Then, the following are equivalent: (i) GR /Γ is compact; (ii) X(G0 )Q = {1}, and every unipotent element of GQ is in the unipotent radical (7.15) of G. Proof. (a) Assume G is unipotent. Then, (ii) is always satisfied (for the first part, this follows from 7.3), and we need to show that GR /Γ is compact. In view of 8.1 and 8.2, it suffices to show that if π : G → GL(V) is a Q-morphism, L is a lattice of VQ , and (gj ), (vj ) are sequences of elements of GR and L satisfying π(gj ) . vj → 0, then we have vj = 0 for j sufficiently large. To that end, we proceed by induction on dim(V). Since the group G is unipotent, each of its Q-representations admits a non-zero fixed vector that is rational 69 Cf.

10.6 (ii).

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over Q. In particular, there exists a non-zero linear  form λ on V that is defined  over Q and invariant under G. From λ π(gj ) . vj → 0, we conclude that, for j sufficiently large, we have vj ∈ ker(λ), so vj belongs to a proper Q-subspace of V that is invariant under G. We can then apply the induction hypothesis. (b) General case. We can write G = H . N where H is a reductive Q-group and N is a connected unipotent normal Q-subgroup (7.15). In view of 7.13 (4), we can take as arithmetic group a semidirect product Γ = Γ1 . Γ2 , where Γ1 (resp. Γ2 ) is an arithmetic subgroup of H (resp. N). We have established that NR /Γ2 is compact. It is then immediate that GR /Γ is compact if and only if HR /Γ1 is compact. On the other hand, X(N) = {1}, so X(G0 ) = X(H0 ) which, in view of 7.3, shows that (ii) for G is equivalent to (ii) for H. We are in a position to apply Theorem 8.4.  8.8. Remark. The proof of 8.4 is remarkably simple when G has trivial center. The argument for that case would easily carry over to the general case if any of the following four statements were true, for k = Q. Let k be a perfect field and G be a connected reductive k-group such that X(G)k = {1} and the k-rational elements of G are semisimple. Then: (a) There exists a faithful k-representation π : G → GL(V) such that the orbit π(G) . v of each element v of Vk does not have the origin in its closure (in the Zariski topology); (b) There exists a faithful k-representation π : G → GL(V) such that π(G) . v is closed, for every v ∈ Vk ; (c) For every k-representation π : G → GL(V), and every v ∈ Vk  {0}, the orbit π(G) . v does not accumulate at the origin; (d) For every k-representation π : G → GL(V) and every v ∈ Vk , the orbit π(G) . v is closed. I do not know if these statements are true, even when k = Q. Moreover, the question may also be posed over a non-perfect field, by replacing the hypothesis on G with: G is anisotropic over k (cf. §10). Examples show that (c) and (d) need not hold in this case. Let us show how (a) for k = Q would imply 8.4. Let Pi (1 ≤ i ≤ m) be homogeneous polynomials on V that have rational coefficients and are invariant  2 under G, and generate the ring J of invariants of G (7.6), and let P = i Pi . Since the invariant polynomials separate the disjoint invariant closed sets (7.6), the condition imposed on π implies that P(x) = 0 if x ∈ VQ  {0}. Then, given a lattice L of VQ , there exists a constant c > 0 such that P(x) ≥ c for x ∈ L  {0}. If(gj ) ∈ GR and (vj ) ∈ L (j = 1, 2, . . .) are such that π(gj ) . vj → 0, then we have P π(gj ) . vj = P(vj ) → 0, so vj = 0 for j sufficiently large, so Mahler’s criterion is satisfied. We conclude this section by proving, with the help of 8.5, a theorem on isogenies of Q-groups. 8.9. Theorem. Let G, G be Q-groups, r : G → G be an isogeny that is defined over Q and Γ be an arithmetic subgroup of G. Then r(Γ) is an arithmetic subgroup of G . Proof. It is clear that it is enough to consider the case where G is connected. We can find (7.13) an arithmetic subgroup Γ of G that contains r(Γ), and we need to show that r(Γ) has finite index in Γ .

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Assume  first that G is reductive and X(G)Q = {1}. By Proposition 8.5, the quotient r −1 (Γ ) ∩ GR /Γ is compact. Since the kernel of r is finite, the group r −1 (Γ ) is discrete, so the previous quotient is in fact finite, so r(Γ) has finite index in:   H = r r −1 (Γ ) ∩ GR . But since r(GR ) has finite index in GR (7.4), it is clear that H has finite index in Γ . Now let G be reductive. Put:   0  0  G1 = ker(χ) , G1 = ker(χ) . χ∈X(G)Q

χ∈X(G )Q

For every character χ ∈ X(G)Q , we have χ(GZ ) = ±1, so G1 ∩ Γ has finite index in Γ, and similarly, G1 ∩ Γ has finite index in Γ . On the other hand, it follows from elementary facts about characters (which will be recalled or proved in §10) that70 r(G1 ) = G1 and X(G1 )Q = {1}. We are therefore reduced to the case that has already been considered. In the general case, we have the decomposition G = H . N of 7.15, where H is reductive, and N is normal and unipotent, and we can assume that Γ = Γ1 . Γ2 is the semidirect product of an arithmetic subgroup Γ1 of H and an arithmetic subgroup Γ2 of N (7.13). The group G is the semidirect product of H = r(H) and N = r(N). By what has just been proved, r(Γ1 ) is an arithmetic subgroup of H . On the other hand, we have ker(r) ∩ N = {e}, since ker(r) is finite, so r is an isomorphism of N onto N , and, in view of 7.13, r(Γ2 ) is an arithmetic subgroup of N . Then (by 7.13),  r(Γ) = r(Γ1 ) . r(Γ2 ) is an arithmetic subgroup of G . 8.10. Corollary. Assume G is an almost direct product of normal Q-subgroups Gi (1 ≤ i ≤ m). Then Γ is commensurable with the group Γ that is generated by the intersections Γi = Γ ∩ Gi (1 ≤ i ≤ m), and each Γi is arithmetic in Gi (1 ≤ i ≤ m). Proof. It suffices to consider the case where G ⊂ GLn and Γ = GZ . Then, Γi = (Gi )Z , so Γi is arithmetic. The product map  Gi → G is a  ν of the inclusions Q-isogeny of G1 × . . . × Gm onto G, so Γ = ν (G1 )Z × . . . × (Gm )Z is arithmetic by 8.9, and is therefore commensurable with Γ.  8.11. Remark. Theorem 8.9 easily extends to the case of a surjective Q-morphism r [4, Theorem 6]. We restrict ourselves here to a sketch of the proof when G is connected and reductive. Let N be the identity component of the kernel of r. This is a normal Q-subgroup. Since G is reductive, there exists a connected normal Qsubgroup N of G such that G is the almost direct product of N and N . By 8.10, Γ is commensurable with (N ∩ Γ) . (N ∩ Γ) and N ∩ Γ is arithmetic in N . The group r(Γ) is commensurable with r(N ∩Γ). Since the restriction of r to N is a Q-isogeny of N onto G , it follows that r(N ∩ Γ) is arithmetic (8.9), so r(Γ) is arithmetic. Bibliographical note. Two precursors of the compactness criterion were already mentioned in 8.6: one is Dirichlet’s Unit Theorem and its generalization to division algebras; the other is about the group of units of quadratic or hermitian forms that do not represent zero rationally, examples of which can already be found in FrickeKlein [16], and also in the works of Siegel (for instance, [39]) and his students. A 70 See

10.7 (b).

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9. FUNDAMENTAL SETS (FIRST TYPE)

49

general theorem on algebras with involution, which includes these classical cases, can be found in [42]. The statement of 8.4 was conjectured by R. Godement. It was proved in [6], and also in [30]. It is the latter proof that has been presented here. Both of these papers also establish the generalization stated in 8.7. Theorem 8.9 is proved, by a different method, in [6, §6.11]. 9. Fundamental sets (first type) 9.1. Definition. A subgroup G of GL(n, R) is said to be self-adjoint with respect to a positive-definite symmetric bilinear form Q on Rn if, for every g ∈ G, the adjoint71 of g with respect to Q is also in G. In particular, if Q is the Euclidean inner product,72 we will simply say self-adjoint. The adjoint of an element a ∈ GL(n, R) with respect to the usual Euclidean inner product is the transpose ta of a. 9.2. Proposition. Let G be a semisimple algebraic subgroup of GL(n, R) that is defined over R. There exists an element u ∈ GL(n, R) such that u . GR . u−1 is self-adjoint. In other words, GR is self-adjoint with respect to some positive-definite form. The proof will be preceded by two lemmas. 9.3. Lemma. Let H be a self-adjoint subgroup of GL(n, R) and M be an algebraic subgroup of GL(n, R) that is defined over R, such that H has finite index in MR . Then, H = K . P, where K = O(n) ∩ H and P = S ∩ H, and where S denotes73 the space of positive-definite symmetric matrices. We have P = exp p, where p is the intersection of the Lie algebra h of H with the space s of real symmetric matrices, and K is a maximal compact subgroup of H. Proof. Recall that: • gl(n, R) decomposes into the direct sum74 of s and the Lie algebra of O(n), • GL(n, R) = O(n) . S, and each g ∈ GL(n, R) can be written uniquely as the product of an element of O(n) and of an element of S, depending continuously on g, and • the exponential map defines a homeomorphism of s onto S. On the other hand, every relatively compact subgroup of S consists only of {e}. Hence, Lemma 9.375 amounts to the following statement: if g = k . s is an element of H (k ∈ O(n), s ∈ S), then s ∈ H and s = exp X, for some X ∈ p. But tg = s . k−1 , and s2 = tg . g is then an element of H by assumption. More generally, for every n ∈ N, s2n ∈ H. We can write s = exp X, where X ∈ s. Let us prove, for t ∈ R, that exp tX is in M. Since M is defined by polynomial equations, and since s may adjoint g ∗ of g is defined by: Q(g ∗. x, y) = Q(x, g . y) for all x, y. Q(x, y) = x1 y1 + . . . + xn yn . The original French manuscript calls this the unit form (or identity form). 73 This is the same set that was denoted H in §2. 74 This is a direct sum of vector spaces, not of Lie algebras. 75 Other than the statement at the end about K. 71 The 72 I.e.,

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II. ALGEBRAIC GROUPS

be assumed to be diagonal, we see that we are reduced to proving the following statement (which is immediate): if P is a polynomial in r variables, the relation: P(enx1 , enx2 , . . . , enxr ) = 0

(n ∈ Z),

P(etx1 , etx2 , . . . , etxr ) = 0

(t ∈ R).

implies: Thus, exp tX belongs to M, so also to MR for every t ∈ R. Since H is of finite index in MR , exp tX ∈ H, whence X ∈ h. Finally, let L be a compact subgroup of GR . Then any element x ∈ L ∩ P is semisimple, with eigenvalues that are positive real numbers of modulus one, so x = e; therefore L ∩ P = {e}. It follows that K is maximal compact.  9.4. Lemma. There exists u ∈ GL(n, R) such that u . G0R . u−1 is self-adjoint. Proof. Let g = k + p be a Cartan decomposition of the Lie algebra g of G. The compact form k + ip generates a compact subgroup of GL(n, C), which must be conjugate to a unitary group.76 Thus, we see that after conjugation, we may assume that: k ⊂ o(n), The group

G0R

and p ⊂ s.

is generated by exp k and exp p, so it is self-adjoint.



9.5. Proof of 9.2. By 9.3 and 9.4, we may assume, after replacing GR by a conjugate, that G0R = K . P where K is the group generated by k = o(n)∩g, and P = exp p, (p = g ∩ s). Let us show that K is its own normalizer in G0R . It is enough to show that P ∩ N(K) = {e}. Let p ∈ P ∩ N(K) and k ∈ K; then: k . p = p . k = k. k−1. p . k

(k ∈ K),

whence k = k and p = k−1. p . k . Thus, p centralizes K. The unique element X ∈ p such that p = exp X then belongs to the centralizer of k. But we know that k is its own normalizer in g whence X = 0 and p = e. Denote by K the normalizer of K in GR . We then have K ∩ G0R = K. Since all Cartan decompositions of g are conjugate77 under inner automorphisms of G0R , we have78 GR = K. G0R = K. P. It follows that: K /K = K /(K ∩ G0R ) = GR /G0R is finite (so K is compact). Let Q be the average of the images of the Euclidean inner product under K /K. This is a positive-definite symmetric bilinear form that is invariant under K . On the other hand, K normalizes p (because it is the orthogonal complement of k with respect to the Killing form), so it also normalizes P. 76 This conjugation is in GL(n, C), not GL(n, R). It implies there is a positive-definite selfadjoint matrix J ∈ Mn (C), such that (k + ip) . J + J .t(k − ip) = 0 for all k ∈ k and p ∈ p. Letting J0 = Re J, this implies k . J0 = −J0 .tk (by letting p = 0) and p . J0 = J0 .tp (by letting k = 0). This means that the matrix k is skew-adjoint, and the matrix p is self-adjoint, with respect to the positive-definite symmetric bilinear form on Rn that corresponds to J0 . 77 Equivalently, all maximal compact subgroups of G0 are conjugate under G0 . R R 78 This uses the Frattini argument: If H ⊂ G ⊂ G, and every G-conjugate of H is conjugate 0 to H via an element of G0 , then G = G0 . NG (H). (So also G = NG (H) . G0 , since G = G−1 .) Proof: Given g ∈ G, there is some g0 ∈ G0 , such that g0 g H = H, so g0 g ∈ NG (H). Therefore g ∈ g0−1. NG (H) ⊂ G0 . NG (H).

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9. FUNDAMENTAL SETS (FIRST TYPE)

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Hence, the elements of P are self-adjoint79 with respect to Q. Since GR = K. P, Proposition 9.2 is proved.  9.6. Definition. Let G be an algebraic group that is defined over Q, and Γ be an arithmetic subgroup of G. We say that Ω ⊂ GR is a fundamental set for Γ if it satisfies the following conditions: (F0 ) K . Ω = Ω, for a suitable maximal compact subgroup K of GR ; (F1 ) Ω . Γ = GR ; (F2 ) For b ∈ GQ , the set {γ ∈ Γ | Ω . b ∩ Ω . γ = ∅} is finite (“Siegel property”). For example, Siegel domains that are sufficiently large are fundamental sets for Γ = GL(n, Z) in GL(n, R) (cf. §§1 and 4). 9.7. Remarks. (1) Property (F2 ) is equivalent to the following seemingly stronger property: (F2 )

If C is a finite subset of GQ , then {γ ∈ Γ | Ω . C ∩ Ω . C . γ = ∅} is finite.   −1  −1 Indeed, let Γ = Γ ∩ ⊂ C−1. Γ and, in c∈C c . Γ. c . This group satisfies Γ . C view of 7.13, is arithmetic, therefore of finite index in Γ. Let D be a finite subset of Γ such that Γ = D . Γ . If γ = d . γ (d ∈ D, γ ∈ Γ ) and c1 , c2 ∈ C are such that Ω . c1 ∩ Ω . c2 . d . γ = ∅, then we have: −1 Ω . γ−1 1 ∩ Ω . c2 . d . c1 = ∅

(γ1 = c1 . γ. c−1 1 ∈ Γ).

If Ω satisfies (F2 ), then there are only a finite number of possibilities for γ1 , whereby (F2 ) follows. (2) One of the reasons that we impose (F2 ) for every b ∈ GQ , and not just for b = e, is so that the existence of a fundamental set for one arithmetic subgroup of G implies its existence for every arithmetic subgroup of G. To see this, it suffices to show that if Γ ⊂ Γ are two arithmetic subgroups, then Γ has a fundamental set if and only if Γ has one. Let Ω be a fundamental set for Γ. Then, a fortiori, it satisfies (F0 ) and (F1 ) for Γ . There exists a finite subset C of GQ such that Γ = C . Γ. Now, if γ = c . γ (c ∈ C, γ ∈ Γ) is such that Ω . b ∩ Ω . γ = ∅, then Ω . b ∩ Ω . C . γ = ∅, so γ ranges over a finite set (cf. (1)). Conversely, let Ω be a fundamental set for Γ , and let Ω = Ω. C. This satisfies (F0 ) and (F1 ) for Γ. In view of (1), it satisfies (F2 ) for Γ , so a fortiori for Γ. 9.8. Theorem. Let G be a semisimple algebraic Q-group in GL(n, C), Γ be an arithmetic subgroup of G and S be a standard Siegel set in GL(n, R) that is a fundamental set for GL(n, Z). Let u ∈ GL(n, R) be such that u . GR . u−1 is selfadjoint (9.2). Then, there exists a finite subset B ⊂ GL(n, Z) such that   u−1. S . b ∩ GR Ω= b∈B

is a fundamental set for Γ in GR . 79 For k ∈ K and p ∈ P, we have kp ∈ P, so k . x, k . p . y = k . x, kp . k . y = kp . k . x, k . y = k . p . x, k . y. This means that p is self-adjoint with respect to the image of the Euclidean inner product under any element of K. Therefore, it is also self-adjoint with respect to the average Q.

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II. ALGEBRAIC GROUPS

Proof. We may assume (cf. the preceding remark) that Γ = GZ . We can find a representation (on the right) of GL(n, C) on a vector space V, a lattice L of VQ that is invariant under GL(n, Z), and a point v ∈ L such that v . GL(n, R) is closed, the representation is defined over Q, and the isotropy group of v in GL(n, R) is GR (cf. 7.7 and 7.13 (1)). Then, the isotropy group of v  = v . u−1 is u . GR . u−1 . By the finiteness lemma (6.2), the set (v . S)∩L is finite. (The diagonalizability condition (iii) of the lemma is satisfied because the representation is defined over Q.)80 A fortiori, the intersection   −1 (v . S)∩ v . GL(n, Z) is also a finite set. Let v . b−1 1 , . . . , v . bm be the points of this set (bi ∈ GL(n, Z)). Put H = u . GR . This is obviously the set of all g ∈ GL(n, R), such that v . g = v. Each h ∈ H can be written h = s . b where s ∈ S and b ∈ GL(n,  Z). From v . s . b = v, we get v . s = v . b−1 . Thus, this is an element of (v . S) ∩ v . GL(n, Z) and there exists i such that: v . s = v . b−1 i . −1 Then, v . b−1 i . b = v, so bi . b is in GR ∩ GL(n, Z), so b ∈ bi . GZ . Finally:    S . bi . GZ , GR = u−1. H ⊂ u−1. S . bi . GZ . H⊂



i

i

−1

Put Ω = i (u . S . bi ∩ GR ). We will prove that Ω is a fundamental set for GZ in GR . We already saw that Ω satisfies (F1 ). Let us prove (F0 ). By Lemma 9.3, u . GR . u−1 has a maximal compact subgroup  K that is contained in O(n). In particular, K. S = S. Then, K1 = u−1. K . u is a maximal compact subgroup of GR that satisfies (F0 ). It remains to prove (F2 ). If Ω . b ∩ Ω . γ = ∅, we can find indices i and j, such that: u−1. S . bi . b ∩ u−1. S . bj . γ = ∅, or, equivalently: S ∩ S . bj . γ . b−1. b−1 = ∅. i A straightforward variant81 of Siegel’s theorem (4.6) shows that the set of γ satisfying this relation is finite, and the result follows.  9.9. Remark. Theorem 9.8 actually remains valid for every reductive Q-group. (Indeed, the theorem and its proof generalize to without any modification.) By using 7.15, we can pass without difficulty from this case to the existence of fundamental sets for an arbitrary Q-group. Indeed, if G = H . N (where H is reductive, N is normal and unipotent, and both are defined over Q), then we can take for Γ the semidirect product Γ = Γ1 . Γ2 of an arithmetic subgroup Γ1 of H by an arithmetic subgroup Γ2 of N. Let Ω1 be a fundamental set for Γ1 in HR , K be a maximal compact subgroup of HR such that K . Ω1 = Ω1 , and (by 8.7) Ω2 be a compact subset of NR such that NR = Ω2 . Γ2 . Let us show that Ω = Ω1 . Ω2 is a fundamental set for Γ. The group 80 And

by 10.3. statement of Siegel’s theorem requires the matrix entries to be integers, but the same proof works whenever there is an upper bound on the denominators of the matrix entries. 81 The

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9. FUNDAMENTAL SETS (FIRST TYPE)

53

NR is homeomorphic to a Euclidean space, so K is also maximal compact in GR , whence (F0 ) follows. We have: GR = HR . NR = Ω1 . Γ1 . NR = Ω1 . NR . Γ1 = Ω1 . Ω2 . Γ2 . Γ1 = Ω . Γ, whence (F1 ). Let b ∈ GQ and let γ ∈ Γ such that Ω . b ∩ Ω . γ = ∅. We can write uniquely: b = h.n

(h ∈ HQ , n ∈ NQ ),

γ = γ1 . γ2 ,

(γ1 ∈ Γ1 , γ2 ∈ Γ2 ).

We then have: Ω1 . h . (h−1. Ω2 . h) . n ∩ Ω1 . γ1 . (γ−1 1 . Ω2 . γ1 ) . γ2 = ∅,

(1) whence: (2)

Ω1 . h ∩ Ω1 . γ1 = ∅,

(3)

h−1. Ω2 . h . n ∩ (γ−1 1 . Ω2 . γ1 ) . γ2 = ∅.

Condition (F2 ) for H, combined with (2), shows that there are only finitely many possible values for γ1 . The relation (3) then implies that γ2 belongs to a compact set that depends only on b and Ω, whence our claim. We will see later that Ω is of finite invariant volume when G is semisimple or, more generally, when X(G0 )Q = {1}. 9.10. Let X = K\GR . The group Γ acts properly on X by right translations. A subset Ω of X is said to be a fundamental set for Γ if it satisfies the condition: (F1 )

Ω . Γ = X,

and the condition (F2 ). It is clear that the natural projection of the fundamental set Ω for Γ in GR of 9.8 satisfies these conditions. We know that the space X is connected and simply connected (in fact, it is homeomorphic to a Euclidean space). It follows, by a well-known topological argument, that the image Γ of Γ in the group of homeomorphisms of X is finitely presented.82 We also see, as in 4.8, that the finite subgroups of Γ constitute a finite number of conjugacy classes. It is easy to deduce from this that these properties are true for every arithmetic group [3]. 9.11. For completeness, we will describe one more generalization of the finiteness theorem of 6.4. In addition to the result mentioned in 7.10 (1), the proof makes use of (the case k = 2 of) the following generalization of 9.2: “Let G1 ⊃ G2 ⊃ . . . ⊃ Gk be reductive R-subgroups of GL(n, C). Then there exists u ∈ GL(n, R) such that the groups u . (Gi )R . u−1 are selfadjoint (1 ≤ i ≤ k).” Theorem. Let G be a reductive Q-group, and Γ be an arithmetic subgroup of G. Let π : G → GL(V) be a Q-morphism and let L be a lattice of VQ that is invariant under Γ. Let v ∈ VQ be a point whose orbit X = G . v under G is closed. Then, L ∩ X consists of a finite number of orbits of Γ. Proof. We assume that G is embedded in GL(n, C) in such a way that Γ is contained in GL(n, Z) (7.13). 82 For

example, see Theorem 4.2 on page 195 of the book [PR] listed on page vii.

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II. ALGEBRAIC GROUPS

Let H be the isotropy group of v. It is defined over Q. Since G/H can be identified with an affine variety, H is reductive (7.10 (1)). Thus, there exists a Q-morphism: σ : GL(n, C) → GL(W), and a point w ∈ WQ such that the isotropy group of w in GL(n, C) is H, and83 w . GL(n, C) is closed (7.7). Let M be a lattice of WQ that is invariant under GL(n, Z). Let us show that84 M ∩ w . G consists of a finite number of orbits of Γ. We can assume that Γ = GZ . Let u ∈ GL(n, R) such that u . GR . u−1 and u . HR . u−1 are self-adjoint. We can find a Siegel set S of GL(n, R) and a finite subset C of GL(n, Z) such that GR ⊂ Ω . Γ, with Ω = u−1. S . C. Thus, it suffices to show that M ∩ w . Ω is finite and for this, it is enough to show that M ∩ w . u−1. S is finite. But let w = w . u−1 . Its isotropy group u . HR . u−1 in GL(n, R) is self-adjoint, so we can apply 6.2. To finish the proof of the theorem, it suffices to show the existence of an equivariant Q-isomorphism ϕ : X → Y (Y = w . G) such that the image of L ∩ X is contained in a lattice M that is invariant under GL(n, Z). But this is immediate. Indeed, X and Y are two realizations of the quotient G/H (cf. the remark in 7.7). The maps g → g . v and g → g . w induce isomorphisms of G/H onto X and Y respectively, whence the existence of an equivariant Q-isomorphism ϕ of X onto Y. Let us take coordinates in V and W that come from bases of L and M respectively. Then the coordinates of ϕ(x) (x ∈ X) are polynomials in the coordinates of x, with rational coefficients. Hence, there exists an integer q such that q . ϕ(x) ∈ M if x ∈ L, whence ϕ(X ∩ L) ⊂ (1/q) . M.  9.12. Applications. Let k be a number field, Ok be the ring of integers of k, and F be a homogeneous form of degree m ≥ 2 on Cn with coefficients in Ok , and with non-zero discriminant. Then its orbit X under the action of SL(n, C) is closed (for m = 2, this is clear (6.3); for m ≥ 3, see 6.5). Therefore, by using 9.11 and the passage from k to Q given by restriction of scalars (7.16), it can be shown that the set of forms in X with coefficients in Ok is the union of a finite number of orbits of SL(n, Ok ). For k = Q(i), Ok = Z[i], this is the result of Jordan mentioned in 6.5. For the proof in general, we generalize 9.11 by replacing Q with k, replacing L with an Ok -lattice, and taking Γ to be an arithmetic group in the sense of 7.16. The above argument remains valid if L is free. Otherwise, we use the fact that L has finite index in a free Ok -lattice L , and note that L is invariant under a finite-index subgroup of Γ (which follows from 7.13 by restriction of scalars). Bibliographical note. Proposition 9.2 and its generalization mentioned in 9.11 are due to G. D. Mostow [28]. A proof also appears in [6, Theorem 1.9]. The construction of fundamental sets given in 9.8 is taken from [6]. It depends crucially on the finiteness lemma of §6, and constitutes a generalization of the process of Hermite described in §5. The theorem in 9.11 is proved in [6, Theorem 6.9].

83 We

assume G acts on the right on W. the proof considers only the subset w . GR of w . G. This is sufficient, because (w . G)R is the union of finitely many orbits of GR (see 7.4). 84 Actually,

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10.1090/ulect/073/03

CHAPTER III

Fundamental Sets with Cusps 10. Algebraic tori 85

10.1. Let k be a field, T be a torus that is defined over k, X(T) = Mor(T, GL1 ) be its group of characters, and Y(T) be the group Mor(GL1 , T) of its one-parameter subgroups. The subgroup of X(T) (resp. Y(T)) consisting of its characters (resp. its one-parameter subgroups) that are defined over k is denoted by X(T)k (resp. Y(T)k ). Definition. We say that T is k-split, or splits over k, if T is isomorphic over k to a direct product Gm × . . . × Gm of copies of the multiplicative group. The algebraic closure k of any field of definition k obviously splits any given torus, but in fact an appropriate finite Galois extension will suffice; this is clear at least when k is perfect. 10.2. If k is a splitting field of the torus T, then there exists an isomorphism that is defined over k from T onto the group of diagonal matrices: x = diag(x1 , . . . , xd ). After this identification, each character is of the form: md 1 x → xm 1 · . . . · xd ,

and each one-parameter subgroup is of the form: (t ∈ Ω∗ ),

t → diag(tm1 , . . . , tmd )

with mi ∈ Z. The abelian groups X(T) and Y(T) associated to a torus T are therefore isomorphic to Zd , where d denotes the dimension of the torus. On the other hand, if a ∈ X(T) and b ∈ Y(T), then the composition a ◦ b is a character of GL1 . Thus, there exists an integer n ∈ Z, such that (a ◦ b)(t) = tn . The formula a, b = n defines a Z-bilinear form on X(T) × Y(T) that actually establishes a duality between X(T) and Y(T). The Galois group M of a Galois extension k of a field of definition k of T acts in a natural way on X(T) and Y(T). The image sa = s(a) of a ∈ X(T) under s ∈ M satisfies:   s s (x ∈ Tk ), a( x) = s a(x) and similarly for the action of M on Y(T). The groups X(T)k and Y(T)k then appear as the sets of fixed points of M; they are direct factors.86 We can extend 85 For our purposes, we may assume k is a subfield of C, so char k = 0, although more general fields will sometimes be discussed. 86 They are direct factors when X(T) and Y(T) are considered as abelian groups, but sometimes they are not direct factors in the category of M-modules, because there might not be a complementary subgroup that is invariant under M.

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III. FUNDAMENTAL SETS WITH CUSPS

these actions to the vector spaces: X(T)Q = X(T) ⊗ Q,

Y(T)Q = Y(T) ⊗ Q.

We therefore obtain two representations of M, which are contragredient to each other, in view of the identity: sa, sb = a, b

(a ∈ X(T), b ∈ Y(T)),

which immediately follows from the definitions. It follows that X(T)k ⊗ Q and Y(T)k ⊗ Q have the same dimension,87 or in other words: (1)

rank X(T)k = rank Y(T)k .

Since X(T) is finitely generated,88 the kernel of these representations has finite index in M, thus, the representations of M on X(T)Q and Y(T)Q are completely reducible.89 Let us note that there is a one-to-one correspondence between the k-subtori of T and the M-invariant subspaces of Y(T)Q . It is defined by associating to a k-subtorus S, the subspace Y(S)Q of Y(T)Q , and to a subspace V of Y(T)Q , the subtorus generated by the images ofb ∈ V ∩ Y(T). If the subspaces Vi (i ∈ I), are associated to the k-subtori Si , then i∈I Vi corresponds to the identity component  of i∈I Si and the sum of the Vi to the torus generated by the Si . 10.3. Proposition. For a torus T that is defined over k, the following properties are equivalent: (i) T is k-split. (ii) Y = Yk . (iii) X = Xk . (iv) For every representation ρ that is defined over k, ρ(T) is a k-split torus. Proof. Almost all the implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (i) are trivial. Let us simply prove that (iv) follows from (iii). If K/k is a Galois extension that splits T, then the space V of the representation can be written over K as a direct sum of the subspaces: Vχ = {v ∈ V | ρ(t) . v = χ(t) . v, ∀t ∈ T}. If s is an element of the Galois group M = Gal(K/k), it is clear that we have the equality: (χ ∈ X(T)).

s(Vχ ) = Vs(χ)

The hypothesis X = Xk means that for every χ ∈ X(T), we have χ = s(χ), (s ∈ M). The spaces Vχ are therefore invariant under M, which proves that they are defined over k.  10.4. Corollary. (i) A quotient of a k-split torus is k-split. (ii) Assume the sequence of tori: 0

T

T

T

0

87 Each

of these is the space of fixed points of the corresponding representation. each χ ∈ X(T) is defined over some finite extension of k. 89 Maschke’s Theorem states that every characteristic-zero finite-dimensional representation of a finite group is completely reducible. 88 Also,

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10. ALGEBRAIC TORI

57

is exact and defined over k. Then the torus T is k-split if and only if T and T are k-split. Proof. The first statement simply applies the implication (i) ⇒ (iv) of the proposition. The exactness of the sequence of vector spaces: 0

X(T )Q

X(T)Q

X(T )Q

0

follows easily from that of the sequence of tori.90 In view of the complete reducibility of the action of the Galois group of a Galois extension that splits the three tori, the sequence of spaces of fixed points: 0

X(T )Q k

X(T)Q k

X(T )Q k

0

is also exact. We deduce from this the equality: rank X(T)k = rank X(T )k + rank X(T )k . This implies the desired conclusion, thanks to 10.3 (i ⇔ iii) and the absence of  torsion in the quotients of the form X(T)/X(T)k . 10.5. Definition. A reductive algebraic group (7.5) G is said to be anisotropic over k if it is defined over k, and contains no nontrivial k-split torus. When there is no ambiguity about k, we will also say anisotropic for anisotropic over k. If char(k) = 0, this condition is equivalent to: Gk consists of semisimple elements and X(G0 )k = {0}. 10.6. Proposition. (i) If k is a common field of definition for a torus T and a subtorus S, then there exists a subtorus S of T that is defined over k, such that T = S . S and S ∩ S is finite. (ii) A k-torus T can be written over k as an almost direct product of two subtori: T = Td . Tc where Td is k-split91 and Tc is anisotropic over k. This decomposition is unique, and is preserved by all k-morphisms. Td is the unique maximal k-split subtorus, and Tc is the unique maximal k-anisotropic subtorus of T. The torus Tc is the identity component of the intersection of the kernels of the characters of T that are defined over k. Proof. (i) If K/k is a finite Galois extension that splits T, then its Galois group M acts completely reducibly on Y(T) ⊗ Q and leaves the subspace Y(S) ⊗ Q invariant; we can then find a complementary subspace that is invariant under M and therefore defines a suitable subtorus S . (ii) Let M be as in the proof of (i). The subspace V = Y(T)k ⊗ Q of fixed points of M in Y(T)Q admits a unique complementary subspace W that is invariant under M. The torus T is the almost direct product of the k-subtori associated to V and W (10.2) and the first (resp. second) is split (resp. anisotropic) over k in view of 10.3 (resp. 10.2 (1)), which establishes the first statement of (ii). Let S be a k-subtorus of T. If it is split over k, then 10.4 (i) implies that it is contained in Td ; if it is anisotropic over k, then it corresponds to an M-invariant subspace U of Y(T)Q that does not contain the trivial representation of M, and is therefore it is immediate that the sequence 0 → X(T ) → X(T) → X(T ) → 0 of abelian groups is exact, which is a stronger statement. 91 The split part of T is denoted T because the French word for “split” is “d´ eploy´ e.” (Some d authors writing in English denote it Ts .) The anisotropic part is often denoted Ta , but Tc is used here because, as will be explained in 10.8, its group of real points is compact (when k ⊂ R). 90 Indeed,

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58

III. FUNDAMENTAL SETS WITH CUSPS

contained in W, so S ⊂ Tc . Also, it is clear that the image under a k-morphism of a k-anisotropic torus is k-anisotropic; that of a k-split torus is k-split in view of 10.4, whence the decomposition is preserved by k-morphisms. The last claim of (ii) is straightforward.  10.7. Proposition. (a) Assume that G is a connected k-group and that G is a connected normal k-subgroup of G. Then, the image of the restriction homomorphism i∗ : X(G)k → X(G )k has finite index. Furthermore, if G is reductive, and G = Z(G)0 , then i∗ is injective. (b) A connected k-group is the almost direct product of a k-split torus S and a subgroup G1 such that X(G1 )k = {0}. This subgroup is the identity component of the intersection of the characters of G that are defined over k. This decomposition is unique, and is preserved by all surjective92 k-morphisms. Proof. (a) Assume first that G is reductive, and G = Z(G)0 . We know that G is the almost direct product of G and its derived group H. Every character of G is obviously trivial on H, so i∗ is injective. On the other hand, if a ∈ X(G )k , there exists a power am of a that is trivial on the finite group H ∩ G , therefore am is in the image of i∗ , so coker(i∗ ) is finite in this case. Let G be reductive. Then G is also reductive. Put: S = Z(G)0 ,

and

S = ZG (G )0 .

We have a commutative diagram, where the arrows denote restriction homomorphisms: X(G)k

μ

X(G )k β

α

X(S)k

ν

X(S )k

We already saw that α, β are injective, with finite cokernel. On the other hand, 10.6 immediately implies that the cokernel of ν is finite. The same must also be true for the cokernel of μ. In the general case, let U be the unipotent radical of G (cf. 7.15). The quotient G/U is reductive (for example, this follows from 7.15, but it is actually much more elementary). Then, G ∩ U is the unipotent radical of G . We have the following commutative diagram: μ

X(G)k

β

α

X(G/U)k

X(G )k

ν

  X G /(G ∩ U) k

where μ, ν are induced by the inclusion maps, and α, β by the canonical projections. The maps α and β are isomorphisms since X(U) = X(G ∩ U) = {1}. We are then reduced to the reductive case. (b) In view of 10.6, the torus Z(G)0 can be written uniquely as the almost direct product S . Sc of a k-split torus S and a k-anisotropic torus Sc . Then, put 92 The

original French manuscript erroneously omits the word “surjective.”

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10. ALGEBRAIC TORI

59

 0 G1 = Sc . [G, G]. It is clear that G1 ⊂ ker(a) . The reverse inclusion a∈X(G)k immediately follows from (a). The last claim is then clear.  10.8. Application to the real case. Let T be a torus that is defined over R, and d be the dimension of T. The Galois group M = Z/2Z of C/R acts on Y(T)Q , which decomposes into a sum of d invariant lines. Therefore, the torus is the almost direct product of d one-dimensional tori. It remains to determine the structure of T when d = 1. In that case, it is not difficult to see that T admits a completely reducible faithful representation93 on a vector space V of dimension ≤ 2. We must distinguish two cases: (i) M acts trivially on X(T), so (10.3), T is isomorphic over R to GL1 , and TR = R ∗ . (ii) M does not act trivially on X(T). Then it maps each character to its inverse, so V is the direct sum of two invariant lines Va , Va−1 , where a is a generator of X(T). The group T ⊂ GL(V) is therefore isomorphic over C to the group of matrices   z 0 (z ∈ C∗ ), 0 z −1 i.e., to the special orthogonal group of the form x . y. It can then be seen that T is isomorphic over R to the special orthogonal group SO(2, C), so T = SO(2) is compact. Returning to the case where d is arbitrary, we then see that:  a T0R = SO(2) × (R∗+ )b , (a = dim Tc , b = dim Td ). In particular, the following conditions are equivalent: (i) T is anisotropic over R; (ii) TR is compact; (iii) the non-trivial element of M = Gal(C/R) maps a ∈ X(T) to a−1 ∈ X(T). If these conditions are satisfied, then TR is a torus in the usual sense,94 and is connected. Therefore, in general, we have, in view of 10.6,95 TR = (Tc )R . (Td )R . In particular, TR /(TR )0 is a group of type (2, 2, . . . , 2). 10.9. To conclude, let us mention that if G is a connected k-group, and T is a ksplit k-subtorus of G, then the map Gk → (G/T)k is surjective. This follows from the vanishing of the group H1 (k, T) in the sense of Galois cohomology (see, e.g., [37, Chap. X, §1], or, for a (non cohomological) proof of a more general result96 [35, Theorem 10], [5, Corollary 15.7, p. 206]). Bibliographical note. For more details on the content of this section, one may consult [8, §1], which also gives additional references, or [5, §8, pp. 111–127]. The groups X(T) and Y(T) were introduced in [14, Exp. 9] (reprinted in [15]). For the results in 10.2–10.6, we followed the exposition of [8, §1]. precisely, a representation that is defined over R. TR ∼ = SO(2)a is an a-dimensional torus in the terminology of elementary topology. 95 Although it is true that T = (T ) . (T ) , this is not immediate from 10.6, because, c R R d R in general, an almost-direct product of R-groups can have real points that are not a product of real points of the factors. Instead, it can easily be verified from the fact that every R-torus is isomorphic to a direct product of copies of three basic examples: GL1 , SO(2, C), and RC/R GL1 , whose real points are R∗ , SO(2), and C∗ . 96 Cf. 11.10. 93 More 94 I.e.,

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60

III. FUNDAMENTAL SETS WITH CUSPS

11. Parabolic subgroups. Bruhat decomposition A. Generalities. 11.1. Definition. A parabolic subgroup of a connected algebraic group G is a closed subgroup H, such that the quotient G/H is a complete variety. It is known that if B is a Borel subgroup of G (that is, a maximal connected solvable subgroup), then G/B is a projective variety. It follows easily that the parabolic subgroups of G are the closed subgroups that contain a Borel subgroup.97 Let us first give an existence theorem for parabolic subgroups. 11.2. Theorem. Let G be a connected reductive group that is defined over k. The following two conditions are equivalent: (i) G contains a proper parabolic subgroup that is defined over k; (ii) G contains a non-central k-split torus; and are equivalent to the following whenever G is semisimple and k has characteristic zero: (iii) Gk contains a non-trivial unipotent element. Henceforth, we consider a group G that is defined over k and a maximal k-split torus S. Thanks to the conjugation theorem asserting that “the maximal k-split tori are conjugate via Gk ,” one can see that the definitions that we will now present do not have any essential dependence on the choice of S. 11.3. Definition. (1) The dimension of S is called the k-rank of G; we denote it by rankk (G). (2) The finite group N(S)/Z(S) is “the” Weyl group of G over k; we denote it by kW(G) or kW. (3) The k-roots of G are the non-trivial characters of S in the adjoint representation; they form a finite set denoted by kΦ(G) or kΦ. Thus, α ∈ kΦ means that there exists a non-zero element X of g, such that for all s ∈ S, we have: Ad(s) . X = s . X . s−1 = α(s) . X. The finite group kW has a natural action on X(S) and Y(S); for instance, if χ ∈ X(S) and if xw ∈ N(S) represents w ∈ kW, we have: (w . χ)(s) = χ(x−1 w . s . xw ). Since this operation leaves kΦ invariant, the group kW acts on the set of k-roots. We can now state a theorem that describes the structure of minimal parabolic subgroups and also states the Bruhat decomposition. 11.4. Theorem. Let G be a connected reductive group that is defined over k. (i) The minimal parabolic k-subgroups are all conjugate to each other under Gk . Each of them can be written as the semidirect product: P = Z(S) . U, of its unipotent radical U (which is normal) by the centralizer Z(S) of a maximal k-split torus S. (This centralizer is reductive and connected.) Furthermore, if we 97 Thus,

this may be taken to be the definition of being parabolic.

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11. PARABOLIC SUBGROUPS. BRUHAT DECOMPOSITION

let M =

61



0 ker χ , then M is the largest connected anisotropic subgroup χ∈X(Z(S))k

of Z(S), M ∩ S is finite and Z(S) = M . S, which yields:98 P = M . S . U. (ii) We have both the “Bruhat decomposition” of Gk : Gk = Uk . N(S)k . Uk , and the equality: N(S) = N(S)k . Z(S), which makes it possible to write Gk as the finite union of mutually disjoint “cells”:  Gk = U k . x w . Pk , w∈kW

where {xw } is a system of representatives of kW in N(S)k . (iii) As a group acting on Y(S)Q , kW is generated by the symmetries with respect to hyperplanes defined by the k-roots.99 The set of k-roots kΦ is a “root system” which, when G is semisimple, spans X(S)Q . To make this last assertion precise, we must define the abstract notion of “root system.” 11.5. Definition. Given a vector space V over Q that is endowed with a positivedefinite100 inner product ( | ), a finite subset R of V is called a root system if it satisfies the following axioms: (i)

0∈ / R and

(ii)

α, β ∈ R

⇒

(iii)

α, β ∈ R

⇒

α ∈ R ⇒ −α ∈ R. (α, β) ∈ Z. (β, β) sβ (α) = α − nα,β β ∈ R.

nα,β = 2

These axioms imply that if α, β are two roots, then ! α = c β ⇒ c ∈ ± 12 , ±1, ±2 . Given a linear ordering101 on kΦ, there is a unique set k Δ of rankk (G) positive roots, such that every k-root is a linear combination with integer coefficients, all of the same sign,102 of elements of k Δ: the elements of k Δ are the simple roots associated to the chosen ordering. In the situation of Theorem 11.4, V is the vector space X(S)Q , which we endow with a positive-definite inner product that is invariant under the finite group kW. 98 When k = R, this is known as the Langlands decomposition of P, and is traditionally written P = M A N (using the letters A and N in the place of S and U, respectively). 99 Each k-root α is a linear functional on Y(S)Q . The kernel of this linear functional is the hyperplane defined by α. 100 The inner product is required to be positive definite on V ⊗ R, not only on V. 101 Choose a linear functional f on X(S)Q , such that the restriction of f to Φ is injective, k and define α < β if f (α) < f (β). 102 Except that some of the coefficients may be zero.

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62

III. FUNDAMENTAL SETS WITH CUSPS

11.6. Remarks on 11.4. (1) The anisotropic part M of Z(S) is normalized by N(S): we can immediately see this103 from the equality N(S) = N(S)k . Z(S). (2) The Weyl chambers of G are the connected components in Y(S)R of the complement of the union of the hyperplanes defined by the roots. The group kW permutes them simply transitively. Each chamber C defines an ordering of kΦ: the root α is positive if it takes positive values on all of C. (3) We can order kΦ in such a way that the Lie algebra of U is: " u= gα , where gα = {X ∈ g | ∀s ∈ S, Ad(s) . X = α(s) . X}. α>0

We will then say that this ordering of kΦ is associated to U. In this way, there is a one-to-one correspondence between: (i) the sets of positive elements, or the sets of simple roots, for the different orderings of kΦ; (ii) the Weyl chambers in Y(S)R ; (iii) the minimal parabolic k-subgroups that contain Z(S); (iv) the maximal unipotent k-subgroups that are normalized by S. When we have a decomposition P = M . S . U, it is to be understood that kΦ is equipped with the ordering that is associated to U; the group U therefore determines a particular set of simple roots k Δ. (4) Each cell Gw,k = Uk . xw . Pk of the Bruhat decomposition of Gk gives rise to a unique decomposition of each of the elements that it contains. For this, it suffices to write: Gw,k = (Uw )k . xw . Pk , where we put: Uw = xw . U−. x−1 z ∩ U, where U− denotes the unipotent subgroup that corresponds to the ordering of kΦ that is opposite104 to the one defined by U. Furthermore, let: Uw = xw . U . x−1 w ∩ U. The group U is equal to Uw . Uw and, in fact, the product map: Uw × Uw → U is an isomorphism of algebraic varieties. As for the Lie algebra of Uw , it is equal to the sum of the gα , where α runs through the roots α > 0 such that w−1 (α) < 0. Finally, we should mention that the open cell Gw,k corresponds to the element w of the Weyl group that maps U to U− , i.e., the element that maps the given ordering of kΦ to its opposite. 11.7. To classify all of the parabolic subgroups of G that are defined over k, we will start with a fixed minimal parabolic k-subgroup P = Z(S) . U, and construct 103 M is obviously normal in Z(S) (e.g., because it contains the commutator subgroup). Any automorphism of Z(S) that is defined over k (e.g., conjugation by an element of N(S)k ) must preserve M, so N(S)k also normalizes M. 104 The opposite ordering 0 for some α ∈ / θ. We have gβ ⊂ z(Sθ ) iff mα = 0 for all α ∈ / θ. 106 (i) implies that all minimal parabolic k-subgroups of G are conjugate to each other via G k (if G is connected and reductive). By the Frattini argument (see footnote 78), we conclude that if G is any reductive k-group (which might not be connected), and P is a minimal parabolic k-subgroup of G0 , then Gk = (G0 )k . NG (P)k . This observation remains true even without the assumption that G is reductive. This is seen by using the semidirect product decomposition G0 = L . U, where L is connected and reductive (see 7.15), and noting that U is contained in every parabolic subgroup of G0 (because it is contained in every Borel subgroup). 105 Write

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64

III. FUNDAMENTAL SETS WITH CUSPS

Assume that k has characteristic zero.107 Then the notions of maximal k-split torus and of minimal parabolic k-subgroup are preserved by every surjective kmorphism f : G → G . If f is an isogeny, this easily follows108 from 10.4 and 10.6. Consequently, if G = G1 . G2 is the almost direct product of the connected ksubgroups G1 and G2 , and P (resp. S) is a minimal parabolic k-subgroup (resp. a maximal k-split torus) of G, then P = (P ∩ G1 ) . (P ∩ G2 )

(resp. S = (S ∩ G1 )0. (S ∩ G2 )0 ),

and P ∩ Gi (resp. (S ∩ Gi )0 ) is a minimal parabolic k-subgroup (resp. a maximal k-split torus) of Gi . Now, if N is the identity component of the kernel of f , then, since G is reductive and connected, we can find a connected k-subgroup N of G such that G is the almost direct product of N and N , and f : N → G is an isogeny. We may then use the preceding remarks. (If char(k) = 0, this remains true if we assume that f is separable and that the Lie algebras of N and N are transverse.) 11.9. Let P and P be two parabolic subgroups that are conjugate to each other. An element x ∈ G such that x . P . x−1 = P canonically defines an isomorphism a → xa from X(P) onto X(P ), defined by xa(g) = a(x−1. g . x), (g ∈ P). Since P is equal to its normalizer, every element y ∈ G such that y . P . y −1 = P is of the form x . p (p ∈ P). But every group acts trivially by inner automorphisms on its group of characters, so xa = ya, (a ∈ X(P)). It follows that given two conjugate parabolic ∼ subgroups, there is a canonical isomorphism X(P) → X(P ). If moreover, P and P are defined over k, then we may assume x ∈ Gk , whence we also have a canonical isomorphism of X(P)k onto X(P )k . We will often identify X(P) with X(P ) (and X(P)k with X(P )k ), through this isomorphism. 11.10. Let G ⊂ GL(n, Ω) be a k-group. It is said to be triangularizable over k if there exists an element x ∈ GL(n, k) such that x . G . x−1 is contained in the group of upper triangular matrices. G must therefore be solvable. If G is connected, and k is perfect, this condition is equivalent to the existence of a composition series: G = G0 ⊃ G1 ⊃ . . . ⊃ Gs = {e}, consisting of connected k-subgroups such that the quotient Gi /Gi+1 is isomorphic over k to either the additive group of Ω, or to GL1 (1 ≤ i ≤ s − 1). A k-group satisfying this condition is a k-split solvable k-group. The result of Rosenlicht [35] that was cited in 10.9 implies the more general fact that if G is a k-split solvable k-subgroup of a connected k-group H, then the fibration of H by G is locally trivial over k. In particular, the map Hk → (H/G)k is surjective, so (H/G)k = Hk /Gk . Assume that G is connected and reductive, and that k has characteristic zero. Then every connected k-subgroup of G that is triangularizable over k (resp. is unipotent) is conjugate over k to a subgroup of the subgroup S . U (resp. U), of 11.4. To conclude, we will prove some technical lemmas, which will be used in the following sections. We keep the preceding notation. 107 We also retain the assumption of Theorem 11.8 that G is a connected reductive k-group. However, the reductivity assumption could easily be eliminated by using 7.15. 108 The cited results apply to tori. For a minimal parabolic subgroup P, note that the kernel Z of any isogeny is central, and is therefore contained in N(P) = P. So (G/Z)/(P/Z) ∼ = G/P is projective, which means that P/Z is parabolic. If P/Z properly contained a parabolic k-subgroup of G/Z, then its inverse image would be properly contained in P.

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11. PARABOLIC SUBGROUPS. BRUHAT DECOMPOSITION

65

11.11. Lemma. Let α ∈ k Δ and w ∈ kW be such that for every β ∈ have:109  w(β) = nβγ (w) . γ, with nβα (w) ≥ 0.

k Δ,

we

γ∈k Δ

Then, every representative xw of w belongs to the maximal proper parabolic subgroup: Pk Δ{α} . Proof. Since the k-roots that appear in P = Pk Δ{α} are the sums of the  form γ∈k Δ q(γ) . γ, where we impose the constraint q(α) ≥ 0, the assumption means that each simple root is mapped by w to a root of P ; then the same holds for every  positive root; since xw normalizes Z(S), it follows that xw . P . x−1 w ⊂ P . Due to the  conjugacy of all minimal parabolic subgroups of P (11.4), there exists y ∈ P , such that: −1 y . xw . P . x−1 = P. w .y

But P is equal to its normalizer (11.8), so y . xw ∈ P, so xw ∈ P .



11.12. Lemma. Consider either two subsets of Sk , or two algebraic k-subgroups of S. Denote them by A and A , and assume they are conjugate by g ∈ Gk . Then there exists h ∈ N(S)k , such that: h . a . h−1 = g . a . g −1

(a ∈ A).

Proof. Indeed, S = g . S . g −1 is a maximal k-split torus that is contained in Z(A ), and so is S. Therefore, there exists z ∈ Z(A )k , such that: z . S. z −1 = S. The element h = z . g belongs to N(S)k and since g . a . g −1 ∈ A , we have: z . g . a . g −1. z −1 = g . a . g −1

(a ∈ A). 

11.13. Lemma. Let N be a unipotent group that is defined over a field k of characteristic 0. Consider a sequence of ideals {ni }0≤i≤q+1 of its Lie algebra n, such that: [n, ni ] ⊂ ni+1 ,

{0} = nq+1 ⊂ . . . ⊂ ni+1 ⊂ ni ⊂ . . . ⊂ n0 = n.

Let n = a ⊕ b be a decomposition of n into a direct sum of two subspaces such that: ni = (ni ∩ a) ⊕ (ni ∩ b)

(i = 1, . . . , q).

Then, the map: ϕ : (a, b) → ea . eb is an isomorphism of the variety a × b onto N. 109 The original French manuscript writes m(γ, β) here, instead of n (w), but that notation βγ is never used later (see, for example, 15.1 (1)).

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III. FUNDAMENTAL SETS WITH CUSPS

Proof. By the assumption, nq is central: we may consider the projection π of N onto the quotient group N = N/Nq ; we put a = dπ(a) and b = dπ(b) and we choose a complementary subspace a (resp. b ) of nq ∩ a (resp. nq ∩ b) in a (resp. b). The proof proceeds by induction on the integer q; if g ∈ N: 







π(g) = ea . eb = π(ea . eb ), where a , b (hence also a , b ) are regular functions of π(g), and, therefore, of g, whence: 



g = ea . eb . z,

z ∈ Nq , a ∈ a , b ∈ b .

where

But also:110 z = eu. ev ,

where

u ∈ nq ∩ a and

v ∈ nq ∩ b,

with u, v depending regularly on z, and therefore on g, and: 



g = ea . eb . eu . ev = ea



+u



. eb

+v

,

since nq is central.



B. Examples. 11.14. G = GLn or SLn . We take our maximal k-split torus to be the torus D of diagonal matrices; this is a Cartan subgroup,111 because Z(D) = D. Since a matrix in D can be written as diag{λi }, we may choose as positive roots the maps λi /λj with i < j, which gives as simple roots the αi = λi /λi+1 . The minimal parabolic subgroup corresponding to this ordering is the solvable group of upper triangular matrices, which is equal to D . N, where N is the unipotent subgroup consisting of the upper triangular matrices with 1 on the diagonal. Thus, G/P is the usual flag variety. More generally, we can associate to an increasing sequence of integers: 1 ≤ d1 < d2 < . . . < dp ≤ n − 1, the canonical flag of signature (d1 , d2 , . . . , dp ): (e1 , . . . , ed1 ) ⊂ (e1 , . . . , ed2 ) ⊂ . . . ⊂ (e1 , . . . , edp ). Their stabilizers are precisely the parabolic subgroups that contain P. Thus, in particular, we have: ⎧⎛ ⎧⎛ ⎧⎛ ⎞⎫ ⎞⎫ ⎞⎫ 0 ⎬ ∗ ⎬ ∗ ⎬ ⎨ 1 ⎨ λ1 ⎨ λ1 .. .. .. ⎠ N= ⎝ ⎠ ⎠ P= ⎝ D= ⎝ . . . ⎭ ⎭ ⎩ ⎩ ⎭ ⎩ 0 λn 0 λn 0 1 110 This is obvious because n is abelian (so exp : n → N is an isomorphism of algebraic q q q groups), but the original French manuscript says it is by induction. 111 A Cartan subgroup is a maximal torus.

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11. PARABOLIC SUBGROUPS. BRUHAT DECOMPOSITION

Pαj

⎧⎛ λ1 ⎪ ⎪ ⎪⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎪ ⎨⎜ ⎜ = ⎜ ⎜ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎪ ⎝ ⎪ ⎪ ⎩

..

∗

. λj−1 B λj+2

0

..

. λn

⎞⎫ ⎪ ⎪ ⎪ ⎟⎪ ⎪ ⎟⎪ ⎪ ⎟⎪ ⎪ ⎟⎪ ⎟⎬ ⎟ , ⎟⎪ ⎟⎪ ⎪ ⎟⎪ ⎪ ⎟⎪ ⎪ ⎠⎪ ⎪ ⎪ ⎭

67

(B ∈ GL(2, R)),

Pj = PΔ{αj } = {(aik ) ∈ G | aik = 0, (k ≤ j < i)}. 11.15. G is split over k. This means that S is a Cartan subgroup: Z(S) = S. Then P is solvable. For every extension k of the base field k, we have the isomorphism: k Φ

∼

→ kΦ.

The groups GLn , SLn are split over the prime field. Another important special case is that of the symplectic group:   0 In t . Sp(2n, Ω) = {X ∈ GL(2n, Ω) | X . J . X = J}, where J = −In 0 The diagonal matrices in Sp(2n, Ω) are the matrices of the form: ⎞ ⎛ λ1 ⎟ ⎜ .. ⎟ ⎜ . 0 ⎟ ⎜ ⎟ ⎜ λ n ⎟. ⎜ −1 ⎟ ⎜ λ 1 ⎟ ⎜ ⎟ ⎜ . . ⎠ ⎝ . 0 −1 λn They form a k-split torus of dimension n, which is a Cartan subgroup. If we identify Sp(2n, Ω) with the group of automorphisms G of Ω2n that preserve the bilinear form with matrix:112 ⎞ ⎛ 1    0 Jn ⎟ ⎜ . , where Jn = ⎝ ⎠, .. −Jn 0 1 then we can take for S (resp. P) the intersection of G with the group of diagonal matrices (resp. upper triangular matrices). We then immediately see that the roots ±1 (1 ≤ i ≤ j ≤ n), each of G (in multiplicative notation) are the characters λ±1 i . λj 113 with multiplicity one. 11.16. G = O(F), the orthogonal group of a non-degenerate quadratic form on kn . If p is the index of F, that is, the dimension of the maximal isotropic subspaces,    0 Jn 0 In is transformed to by an appropriate permutation of the basis  −In 0 −Jn 0 vectors, so the two representations of Sp(2n, Ω) are conjugate to each other. In the second representation, the diagonal matrices are diag(λ1 , . . . , λn , λ−1 n , . . . , λ1 ). This has the advantage that P is upper triangular, whereas the minimal parabolics are a bit more complicated in the first representation. 113 Except that λ . λ−1 = λ−1. λ = 1 is not a root. i i i i 

112

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III. FUNDAMENTAL SETS WITH CUSPS

then the Witt decomposition [11, §4, no 2] enables us form: ⎛ ⎞ ⎛ 0 0 J ⎜ . F = ⎝0 F0 0⎠ , where J = ⎝ .. J 0 0 1

to write the matrix F in the ⎞ 1 ⎟ ⎠

is of order p and where F0 does not represent 0 over k: F(x) = 2(x1 xn + . . . + xp xn−p+1 ) + F0 (xp+1 , . . . , xn−p ). We can then take our maximal k-split torus to be the subgroup: ⎧⎛ ⎞⎫ λ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ ⎪ ⎟⎪ . ⎪ . ⎪ ⎜ ⎪ ⎟⎪ . 0 ⎪ ⎪ ⎪ ⎜ ⎪ ⎟ ⎪ ⎪ ⎪ ⎪ ⎟ λ ⎨⎜ p ⎜ ⎟⎬ ⎟ I S= ⎜ n−2p ⎜ ⎟⎪ . ⎪ −1 ⎟⎪ ⎪⎜ λ ⎪ p ⎜ ⎪ ⎟⎪ ⎪ ⎪ ⎪ ⎟⎪ ⎪⎜ . ⎪ ⎪ . ⎪ ⎪ ⎝ ⎠ . 0 ⎪ ⎪ ⎪ ⎪ ⎩ −1 ⎭ λ1 The centralizer of S is Z(S) = S × O(F0 ). If Q is the canonical isotropic flag: (e1 ) ⊂ (e1 , e2 ) ⊂ . . . ⊂ (e1 , . . . , ep ), its stabilizer is the minimal ⎧⎛ ⎨ P1 P2 P = ⎝ 0 P4 ⎩ 0 0

parabolic subgroup: ⎞⎫ P3 ⎬ P5 ⎠ , ⎭ P6

where P1 and P6 are upper triangular, P4 belongs to the orthogonal group O(F0 ), P6 is determined by P1 and P3 , and P5 is determined by P1 , P2 , and P4 . Furthermore, the unipotent radical U of P (such that P = Z(S) . U) is the set of matrices: ⎧⎛ ⎞⎫ ⎨ P1 P2 P3 ⎬ I P5 ⎠ , U= ⎝0 ⎩ ⎭ 0 0 P6 where, this time, P1 and P6 are unipotent (and upper triangular): we can read the relations between P1 , . . . , P6 in the Lie algebra of U: ⎧⎛ ⎞⎫ ⎨ A1 A2 A3 ⎬ 0 A5 ⎠ . u= ⎝0 ⎩ ⎭ 0 0 A6 A1 and A6 are upper triangular with all diagonal entries equal to zero. Since a matrix X of u is constrained by: t

X . F + F . X = 0,

we have the relations: t

A1 . J + J . A6 = 0,

t

A2 . J + F0 . A5 = 0, t

A3 . J + J . A3 = 0,

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11. PARABOLIC SUBGROUPS. BRUHAT DECOMPOSITION

69

which in particular require A3 to be the antisymmetric with respect to the nonprincipal diagonal. The positive k-roots may be obtained by looking at how S acts on u by the adjoint representation: • in A1 , we find λi /λj with 1 ≤ i < j ≤ p and multiplicity one; • in A6 , we find again114 the same roots with the same multiplicity; • in A3 , we have the roots λi . λj with 1 ≤ i < j ≤ p and multiplicity one; • finally, if F0 = 0, then we obtain eigenspaces in A2 and A5 that all have dimension n − 2p, and whose eigenvalues are the roots λi with 1 ≤ i ≤ p. We see that the simple k-roots are: αi = λi . λ−1 i+1

(1 ≤ i ≤ p − 1),

plus an additional root αp that is equal to λp−1 λp in the case n = 2p (F0 = 0), and is equal to λp in the case n = 2p (F0 = 0). We highlight two interesting special cases: (1) n = 2p or 2p + 1: Z(S) = S, so S is a Cartan subgroup; this is the split case; (2) n = 2p + 2: Z(S) is a torus: this is a Cartan subgroup, but different from S; this is the case of a group that is quasi-split over k, which means that P is solvable (as it is in the split case). C. Study of the real case. We begin by comparing the Iwasawa, Bruhat, and Cartan decompositions of the group of real points of a reductive R-group. The main case of interest is, of course, that of semisimple groups. However, for later use, we consider the more general reductive case. 11.17. Cartan decomposition. Let G be a reductive R-group, let Z and G , respectively, be the identity component of the center and the derived group of G0 . Also let K be a maximal compact subgroup of GR . Then K ∩ Z and K ∩ G are maximal compact subgroups of ZR and GR respectively. Let p be the direct sum of the Lie algebra of the group of real points of the maximal R-split torus Zd of Z, and the orthogonal complement of g ∩ k in gR with respect to the Killing form. We then have: gR = k ⊕ p, and the map (k, p) → k . exp p (k ∈ K, p ∈ p) is a real analytic diffeomorphism from K × p onto GR = K . ep . These are the Cartan decompositions of gR and GR respectively. The associated Cartan involution of gR is the involutive automorphism: θ : k + p → k − p

(k ∈ k, p ∈ p),

and in GR , it is the unique involutive automorphism whose set of fixed points is K and whose restriction to ZR is the Cartan involution of ZR in the sense of 10.8. 11.18. Iwasawa decomposition. We start with a Cartan decomposition of GR ; we choose a maximal subalgebra a of p (the subalgebra is necessarily commutative115 ) word “again” may be misleading. The root λi /λj has multiplicity one, because its root space projects injectively into both the A1 -space and the A6 -space. 115 To say that a is a subalgebra means [a, a] ⊂ a, so [a, a] ⊂ p. However, for all p , p ∈ p, 1 2   we have θ [p1 , p2 ] = [θ(p1 ), θ(p2 )] = [−p1 , −p2 ] = [p1 , p2 ], so [p, p] ⊂ k. Therefore, if a is a subalgebra of p, then [a, a] = {0}, which means that a is commutative. 114 The

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70

III. FUNDAMENTAL SETS WITH CUSPS

and a linear ordering of the set of its roots. Let:  n= gα , where gα = {x ∈ gR | [a, x] = α(a) . x, (a ∈ a)}. α>0

The Iwasawa decomposition: gR = k ⊕ a ⊕ n of gR corresponds to the Iwasawa decomposition: GR = K . A . N

(A = exp a, N = exp n)

of GR . The product map is a real analytic diffeomorphism of K × A × N onto GR , and likewise: (k ∈ K, a ∈ a, n ∈ n)

(k, a, n) → k . ea . en

is a real analytic diffeomorphism of K × a × n onto GR . The group N is normalized by A. Note that N is unipotent, that A is commutative, and that N and A are homeomorphic to Euclidean spaces. The Iwasawa and Cartan decompositions therefore provide two ways of representing GR as a topological Cartesian product of a Euclidean space and a maximal compact subgroup. In particular, we see that K intersects each connected component of GR . Every maximal compact subgroup of GR is conjugated by an inner automorphism to K, and its intersection with A . N consists only of {e}. It follows that the preceding assertions remain valid if we replace K by any maximal compact subgroup of GR . The corresponding decompositions of gR and GR will also be called Iwasawa decompositions. 11.19. Let GR = K . A . N be an Iwasawa decomposition of GR . Then, there exists a unique minimal parabolic R-subgroup P of G and a maximal R-split torus S of P such that we have: P = M . S . U = Z(S) . U, (notation of 11.4) with: A = (SR )0 ,

N = UR ,

MR ⊂ K.

Since PR ⊃ A . N, the homogeneous space:  GR /PR = (Uw )R . xw , w∈RW

(cf. 11.6) is a quotient of K, so the elements of a system of representatives of the relative Weyl group RW can be chosen to be in116 K, and in fact even in K0 , as we will see that GR /PR is connected. Let P = Pθ = M. S. U be a parabolic R-subgroup of G0 that contains P, where S = Sθ and M is the maximal R-anisotropic subgroup of the centralizer Z(S )0 of S in G0 . 116 We may assume that S and G are self-adjoint, and that K = G ∩ SO(n) (cf. 11.25). R R Then N(S)R is also self-adjoint,  so K0 ∩ N(S)R is a maximal compact subgroup of N(S)R (by 9.3). Therefore N(S)R = K ∩ N(S) . N(S)R . However, the automorphism group of any torus is discrete,   so we have N(S)0R ⊂ Z(S)R . We conclude that N(S)R = K ∩ N(S) . Z(S)R , which means that a representative of each element of the Weyl group can be chosen to be in K.

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11. PARABOLIC SUBGROUPS. BRUHAT DECOMPOSITION

71

The groups SR and MR are stable under the Cartan involution117 associated to K, so K ∩ Z(S )R and K ∩ MR are maximal compact subgroups of Z(S )R and MR respectively, and K ∩ S is the set of elements of order two in SR . We have: (1)

GR = K . PR = K . PR0 = K . MR0 . A. N ,

(N = UR , A = SR0 ).

The product map defines a homeomorphism of (K . MR0 ) × A × N onto GR and a proper map of K × MR0 onto K . MR0 . In fact, K . MR0 is the quotient of K × MR0 by the equivalence relation: (x, y) ≈ (x . k, k−1. y)

(x ∈ K, y ∈ MR0 , k ∈ K ∩ MR0 ).

Thus, we can write each element g ∈ GR in the form: (2)

g = k .m.a.n

(k ∈ K, m ∈ MR0 , a ∈ A , n ∈ N ).

The elements a, n and the product k . m are uniquely determined by g and depend on it real analytically, whereas k and m are determined only up to multiplying by an element of the compact group K ∩ MR0 . We may also consider the coarser decomposition: (k ∈ K, p ∈ P0R ),

g = k .p

where k and p are determined up to multiplying by an element of K ∩ P0R . The latter is a maximal compact subgroup of P0R . It follows that if K is a maximal compact subgroup of GR , then: G = K. PR0 , and K ∩ PR0 is a maximal compact subgroup of PR0 . Indeed, there exists g ∈ GR such that K = g . K . g −1 . Since K acts transitively on GR /PR , the isotropy groups (in K) of the points of GR /PR are conjugate in K. In particular, the subgroup K ∩ PR0 = g(K ∩ g −1. PR0 . g) . g −1 is isomorphic to K ∩ PR0 . Therefore, it must be a maximal compact subgroup of PR0 . 11.20. Proposition. Let G be a connected algebraic R-group and P be a parabolic R-subgroup of G. Then GR /PR is connected. Proof. Since P contains the unipotent radical of G, we may assume, by passing to the quotient, that G is reductive. Let us first assume that P is minimal among the parabolic R-subgroups. By the Bruhat decomposition, we have:  GR /PR = (Uw )R . xw . w∈RW

Moreover, one of these “cells” is open (see 11.6 (4)), hence dense. Since it is connected, it follows that GR /PR is connected, and this proves our claim for minimal P. In the general case, let P be a minimal parabolic R-subgroup of G that is contained in P. The canonical map GR /PR → GR /PR is surjective and continuous,  hence GR /PR is also connected. 117 This compatibility with the Cartan involution assumes that a ⊂ p. So we are not considering an arbitrary Iwasawa decomposition in the sense of the final paragraph of 11.8.

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72

III. FUNDAMENTAL SETS WITH CUSPS

In view of 11.8, we have (G/P)R ∼ = GR /PR , whence we also have: 11.21. Corollary. The map (GR )0 → (G/P)R is surjective. From this it follows118 that we can find representatives of the Weyl group of G in (GR )0 , by lifting them from GR /PR ; thus,119 we can find them in K0 . 11.22. Proposition. If G is a connected reductive group that is anisotropic over R, then the group GR is compact and connected in the usual topology. Proof. G is isogenous to the product of its derived group G , which is semisimple, and its connected center T, which is an R-anisotropic torus (10.7). The group TR is connected and compact (10.8) and G0R = GR0 . T0R has finite index in GR (7.4). Therefore, for the first assertion, we are reduced to the case where G is connected and semisimple. We may then write (11.17, 11.18): GR = K . A . N = K . ep . But N consists of unipotent elements, hence consists only of {e}, whence we also have A = {e} and GR = K is compact. By a theorem of Chevalley [13, vol. III, Proposition 2 on p. 230], every compact subgroup L of GL(n, R) is “algebraic,” in the sense of [13, vol. II]; this means that L is the group of all the real points of some R-subgroup of GL(n, C), which we may assume to be the smallest algebraic subgroup LC of GL(n, C) that contains L. We have l = gl(n, R) ∩ lC and dimR l = dimC lC . Since an algebraic group over C is connected in the usual topology if and only if it is connected in the Zariski topology, it follows that L and LC are either both connected, or both disconnected. If GR were not connected, then (G0R )C would be strictly contained in G, and of same dimension as G, which is absurd since G is connected.  11.23. Proposition. Let G be a connected group that is defined over R and let S be a maximal R-split torus of G. The canonical map: π0 (SR ) → π0 (GR ), is surjective, where π0 denotes the group of connected components in the usual topology. Proof. G is the semidirect product of a reductive group H and the unipotent radical N. Since NR is homeomorphic to a Euclidean space, applying the exact sequence of homotopy groups to GR /NR ∼ = HR gives the isomorphism: ∼

π0 (HR ) ← π0 (GR ). 118 A representative n of any element of the Weyl group may be chosen in N(A) (see 11.4 (ii)). R Then n P is a parabolic R-subgroup that is obviously conjugate to P. We conclude from 11.21 that 0 g . n P = P. After multiplying g on the left by an element of N, there exists g ∈ (GR ) , such that we may assume g ∈ N(A). So g . n ∈ N(A) ∩ P = Z(A). Thus, g −1 represents the same element of the Weyl group as n. 119 This argument is similar to footnote 116, but replaces N(S) and K with N(S) ∩ G0 and R R R K ∩ G0R = K0 .

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11. PARABOLIC SUBGROUPS. BRUHAT DECOMPOSITION

73

We can therefore assume that G is reductive, and choose a minimal parabolic R-subgroup that contains Z(S). Since the group Z(S) is isomorphic to the quotient of P by its unipotent radical, we obtain as above, an isomorphism:   ∼ π0 Z(S)R ← π0 (PR ). It only remains to prove the surjectivity of the maps:   π0 (PR ) → π0 (GR ), and π0 (SR ) → π0 Z(S)R , which, in view of the exact sequence of homotopy groups, amounts to proving that GR /PR and Z(S)R /SR are connected. For GR /PR , see 11.20. Finally, since S   is R-split, the group Z(S)R /SR is isomorphic to the group Z(S)/S R (10.9); but the latter is connected (by 11.22), because Z(S)/S is reductive, connected120 and R-anisotropic.  11.24. Definition. Let G ⊂ GLn be a semisimple algebraic group that is defined over R. We say that an Iwasawa decomposition K . A . N of GR is in good position with respect to an Iwasawa decomposition K0 . A0 . N0 of GL(n, R), if the following conditions are satisfied: (i) K ⊂ K0 , A ⊂ A0 and N ⊂ N0 ; (ii) if a character of A0 is positive for an ordering that is associated to N0 , then its restriction to A is a character of A that is positive for an ordering that is associated to N. Note that the inclusion N ⊂ N0 is an immediate consequence of (ii). 11.25. Proposition. Let G ⊂ GLn be a semisimple algebraic group that is defined over R. Given an Iwasawa decomposition K . A . N of GR , conjugating by an appropriate element of GL(n, R) can make GR self-adjoint, and put K . A . N in “good position” with respect to the canonical Iwasawa decomposition of GL(n, R). Proof. Denote by: k ⊕ a ⊕ n = k ⊕ p, the Iwasawa-Cartan decomposition of gR associated to K . A . N. From the proof of Proposition 9.2 (see 9.5), a preliminary conjugation by an element of GL(n, R) allows us to embed K in K0 = O(n) and ep in the space S of positive-definite symmetric matrices. These inclusions will not be altered by any of the conjugations that follow, because they will be carried out by elements of O(n), and GR will therefore remain self-adjoint. Since any two maximal abelian subalgebras of the space121 S of real symmetric matrices are conjugate under O(n), there exists a ∈ O(n), such that aA ⊂ A0 . Now, the linear ordering of X(aA) that is associated to a N can be extended to a linear ordering of X(A0 ); this determines a Weyl chamber in Y(A0 )R and, to recover the usual Weyl chamber (which is associated to N0 ), it suffices to conjugate by some element b of N(A0 ) ∩ O(n) (cf. 11.6 (2), (3)). The group baA is still contained in  A0 and the condition (11.24(ii)) of Definition 11.24 holds for baA and ba N. 120 We

know from 11.4 (i) that Z(S) is reductive and connected. is the same set that was denoted s in 9.3.

121 This

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III. FUNDAMENTAL SETS WITH CUSPS

Remark. Obviously, we could state the same proposition for any Iwasawa decomposition K0 . A0 . N0 of GL(n, R) by replacing “self-adjoint” with “invariant under the Cartan involution that is compatible with K0 . A0 . N0 .” Bibliographical note. For the proofs of the results stated in (A), we refer the reader to [8] where one will also find other references. The results on the Cartan and Iwasawa decompositions recalled in 11.17, 11.18, are classical for real semisimple groups. The analogues for reductive groups are established in [13] or [6, §1]. Proposition 11.23 is due to Matsumoto [24]. Our proof is taken from [8, Th´eor`eme 14.4]. 12. Siegel sets In this section, F is a subfield of R, G is a reductive F-group, P is a minimal parabolic F-subgroup of G0 , S is a maximal F-split torus of P, U is the unipotent radical of P; M is the maximal F-anisotropic F-subgroup of the centralizer Z(S)0 of S in G0 , and FΔ is the set of simple F-roots of G with respect to S, for the ordering that is associated to U. 12.1. We will write FA for122 S0R . For each t > 0, we put: FAt

= {a ∈ FA | aα ≤ t, ∀α ∈ FΔ},

Pt = {p ∈ PR | |pα | ≤ t, ∀α ∈ FΔ}, P0t = Pt ∩ P0 . We have P0R = M0R . FA . UR , so: P0t = Pt ∩ P0R = M0R . FAt . UR . 12.2. Lemma. Let ω be a relatively compact subset of MR . UR and let t > 0. Then the union of the sets a . ω . a−1 (a ∈ FAt ) is relatively compact. Proof. The set ω is contained in the product of a compact subset of MR and a compact subset of UR . Since MR is centralized by FA, it suffices to consider the case where ω ⊂ UR . The exponential map is a homeomorphism of uR onto UR , and it commutes123 with PR , where PR acts on uR by the adjoint representation, and acts on UR by inner automorphisms. Thus, it remains to show that if ω is compact in uR , then the union of the subsets Ad a(ω) (a ∈ FAt ) is relatively compact. But uR admits a basis that consists of eigenvectors of SR corresponding to positive F-roots. Since these are linear combinations with positive coefficients of simple F-roots, there exists a constant t > 0 such that aα ≤ t for every positive root α and every a ∈ FAt . The restrictions of the operators Ad(a) (a ∈ FAt ) to uR therefore form a bounded set of operators, whereby the lemma follows.  12.3. Definition. Let K be a maximal compact subgroup of GR . A Siegel set of GR (with respect to K, P, S) is the product set: S = St,ω = K . FAt . ω,

(1)

where ω is a compact neighborhood of e in M0R . UR . 122 S0 R

means (SR )0 . We have S0 = S (because tori are connected), so there is never any reason to write (S0 )R . 123 I.e., exp(px) = p exp(x) for p ∈ P and x ∈ u . R R

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12. SIEGEL SETS

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Similarly, we define an open Siegel set by taking ω to be a relatively compact open neighborhood of e in M0R . UR , and by replacing ≤ with < in the definition of FAt . We say that S is normal if FA is invariant under the Cartan involution associated to K. Note that the notion of Siegel set depends on the chosen field of definition; we should actually speak of a Siegel set over F, a precision that we will omit when it does not cause confusion. Let F be a subfield of R that contains F. Then each Siegel set of GR over F is contained in a Siegel set over F . Indeed, we can start by taking a minimal parabolic F -subgroup P = M. S. U of G0 , such that we have: S ⊃ S,

U ⊃ U,

M ⊂ M,

and assume that we have fixed compatible orderings of X(S ) and X(S). We have P = FPθ , where θ is a subset of FΔ (notation of 11.7). On the other hand, the restrictions of the simple F -roots to S are simple F-roots (or zero).124 From this, we easily see that S is contained in125 a Siegel set126 over F . In particular, S is a Siegel set over F if rankF (G) = rankF (G). In the case where G is anisotropic over F, a Siegel set is simply a compact (or relatively compact) neighborhood of the identity element that is invariant under left-translation by elements of K. At the other extreme, if G is an R-split torus, then S = GR . Assume that G is the almost direct product of two F-subgroups G1 , G2 , let Si be a Siegel set of Gi (i = 1, 2), and let K be a maximal compact subgroup of GR that contains the maximal compact subgroup Ki of (Gi )R that arises in the definition of Si (i = 1, 2). Then, K . S1 . S2 is a Siegel set of GR . This follows from the definition and the remarks in 11.8. It is not always sufficient to consider S1 . S2 because (G1 )R . (G2 )R may be a proper subgroup of GR (though it always has finite index). As a result, the product K1 . K2 of maximal compact subgroups of (G1 )R and (G2 )R is an open subgroup – which may be proper – of a maximal compact subgroup K of GR . In that case, K . S1 . S2 is the union of the translates of S1 . S2 by a system of representatives of K/(K1 . K2 ). By using 11.8 and what precedes it, we see that if f : G → G is a surjective F-morphism, and S is a Siegel domain of G, with respect to K, P, S, then f (S) is contained in a Siegel domain, with respect to f (P), f (S), and a maximal compact subgroup that contains f (K). If F = R, or, more generally, if S is a maximal R-split torus, and if S is normal, then MR ⊂ K and we may choose ω to be a compact subset of UR . In particular, we recover the Siegel sets of GL(n, R) that were introduced in §1, which will be called standard Siegel sets of GL(n, R). 124 See

[8, Proposition 6.8, p. 107]. S = K . FAt . M0 . U0 , where M0 and U0 are compact subsets of MR and UR , respectively. Choose compact K1 ⊂ K ∩ MR , A1 ⊂ S ∩ MR , ω1 ⊂ (M. U ) ∩ MR , such that M0 ⊂ K1 . A1 . ω1 . Then 125 Write

S ⊂ K . FAt . (K1 . A1 . ω1 ) . U0 = K . (FAt . A1 ) . (ω1 . U0 ) is contained in a Siegel set over F (since FAt ⊂ FAt ). 126 The original French manuscript erroneously states that S is contained in a set of the form K . L . ω, where L = {a ∈ FAu | aα ≥ u > 0, ∀α ∈ θ}, for suitable constants u > u > 0, and where ω is a compact subset of MR . UR . The factor FAt should be inserted between K and L.

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12.4. It is clear that K . S = S. On the other hand, if η is a compact subset of P0R , then there exists t > 0 and a compact subset ω of M0R . UR such that: (1)

St,ω . η ⊂ St ,ω .

It suffices to prove this when η = α . β . γ, where α, β, γ are compact subsets of FA, M0R and U0R respectively. But we have:127 FAt . ω . α . β . γ

⊂ FAt . α . (α−1. ω . α) . β . γ.

Thus, it suffices to take t > 0 such that FAt . α ⊂ FAt and put: ω = α−1. ω . α . β . γ. Along the same lines, let us note that if η is a compact subset of GR , then there exists t > 0 such that: (2)

η . K . Pt ⊂ K . Pt ,

η . K . P0t ⊂ K . P0t .

Since GR = K . P0R , we have η . K ⊂ K . α . β . γ, with α, β, γ compact subsets of M0R ,  FA and UR respectively. Then, it suffices to choose t such that β . FAt ⊂ FAt . 12.5. Lemma. Assume that X(G0 )F = {1}. Then every Siegel set has finite Haar measure. Proof. The proof is in principle similar to the one given in §1 for the Siegel sets of SL(n, R). The map (k, p) → k . p is a proper map of K × P0R into GR . The inverse image of a Haar measure of GR is invariant under left translations by K and right translations by P0R . Therefore, it is the product of a Haar measure dk on K with a right invariant Haar measure dp on P0R . We have:



dg ≤ dk . dp, S

K

FAt . α . β

where α (resp. β) is a compact neighborhood of e in M0R (resp. UR ). The group P0R is the semidirect product of Z(S)0R and the normal subgroup UR , both of which are unimodular. It follows that dp = ds . du . ρ(z), where ds (resp. du) is a Haar measure on Z(S)0R (resp. UR ) and where: ρ(z) = |det(Ad z)uR |

(z ∈ Z(S)0R ).

On the other hand, Z(S)0R = M0R . FA and since M is anisotropic over F, the character ρ is equal to one on M. Hence, we can write: ds = dm . da . ρ(a), where dm and da are Haar measures on MR and FA respectively, whence:





dg = c dm du ρ(a) da. (1) S

α

β

FAt

The first two integrals on the right are finite and non-zero. The character ρ is the sum of all the positive roots (each  counted with its multiplicity), and can therefore be written in the form ρ = α∈FΔ cα . α, with cα ∈ N∗ . The exponential map is an isomorphism of Fa onto FA that carries a Lebesgue measure to a Haar measure. Since X(G0 )F = {1}, the set of α form a basis of X(S). Hence, their differentials 127 α−1. ω . α

should be read as {a−1. w . a | a ∈ α, w ∈ ω}.

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12. SIEGEL SETS

77

form a system of coordinates on sR . Thus, the rightmost integral in (1) is of the form:  log t  ecα . α dα, c. α∈FΔ

−∞

and is finite since cα > 0 (α ∈ FΔ).

 128

Siegel domain over F of G, and let 12.6. Proposition. Let S be a normal c ∈ GR be such that S . c ∩ S is not compact. Then c belongs to a proper parabolic subgroup that contains P. More precisely, we have c ∈ Pθ where θ is the complement in FΔ of the set ψ of simple k-roots β for which we can find a sequence of elements β xj ∈ S such that129 xj → 0 and xj . c ∈ S. Proof. (i) Assume, for the moment, that c ∈ GF . For η ∈ FΔ, we have: PFΔψ ∩ PFΔη = PFΔ(ψ∪{η}) . Thus, it suffices to prove that if130 xαj → 0 (xj , xj . c ∈ S; j = 1, 2, . . .), then c ∈ Pθ (θ = FΔ  {α}). Let: xj = kj . mj . sj . nj ,

xj . c = kj . mj . sj . nj ,

be decompositions of xj and xj . c, with respect to K, M, S, U and: (u, v ∈ UF ; q ∈ N(S)F ),

c = u.q .v

be the Bruhat decomposition of c. We already noted (11.19) that:   N(S)R = K ∩ N(S) . Z(S)R . Thus, we can write q = w . z (w ∈ K ∩ N(S)R , z ∈ Z(S)R ), and we have: kj . mj . sj . nj . u . w . z . v = kj . mj . sj . nj . Also: kj . mj . sj . nj . u . w . z . v = kj . mj . sj(nj . u) . mj . sj . w . z . v −1

−1

= kj . w . dj . w mj . w sj . z . v

(dj = w

−1

. mj . s j

(nj . u)).

The elements mj and nj remain in compact subsets since xj ∈ S, so in view of 12.2, the element dj remains bounded as j → ∞. The same therefore holds for its components in K, M, S, U. Since w−1. mj . w ∈ M (11.6), we immediately see that: −1

sj = sdj . w sj . sz , where sz and sdj are the components in S of z and dj . β Let β ∈ FΔ. Since sdj is bounded, and sj ≺ 1 (because xj . c ∈ S), it follows that: −1

( w sj ) β = sj

w(β)

≺1

(j = 1, 2, . . . ; β ∈ FΔ).

128 The original French manuscript omits the word “normal,” both here and in results in §15 that (as is well known) implicitly rely on this assumption, such as by appealing to 11.19 or 14.4. 129 xβ is an abbreviation for (s )β . xj j 130 α and β have been interchanged from their use in the original French manuscript, in order to be consistent with the notation of 11.11, which is applied later in the proof.

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III. FUNDAMENTAL SETS WITH CUSPS

 As in 11.11, we may write w(β) = γ nβγ (w) . γ. We then have:  nβγ (w) . γ w(β) = sj ≺ 1. sj γ∈FΔ

Suppose nβα (w) < 0. Then w(β) < 0, so nβγ (w) ≤ 0 for every γ, so each factor on the right is 1. Since the product is ≺ 1, this implies sαj  1, contradicting the assumption. Consequently, we have nβα (w) ≥ 0 for every β ∈ FΔ, so 11.11 shows that c ∈ FPθ . (ii) Let T be a maximal R-split torus that contains S, with tR orthogonal to k. We endow X(T) and X(S) with compatible orderings, where the second is associated with P. Let P be a minimal parabolic R-subgroup corresponding to the positive R-roots. Then S is contained in a Siegel domain S over R, relative to K, P , T (12.3). Let η be the set of elements of R Δ whose restriction to S is an element of ψ. Part (i) of the proof, applied to k = R and S shows that c belongs to the parabolic R-group R Pζ , with ζ = R Δ  η. But it follows from the definitions that  R Pζ = FPθ . 12.7. In order to obtain the reduction theorems when G is not connected, we will need a slight strengthening of 12.4. We keep the preceding notation and we denote by H the subgroup of GR consisting of the connected components of GR that contain an element n of the normalizer of FA. Obviously, int(n) maps the positive roots to the positive roots for another ordering so in view of 11.6, 11.21, the component n . G0R also contains an element n that normalizes FA and permutes the positive roots.131 In this case, n also normalizes UR , so it normalizes PR = NG0R (UR ), as well. That being said, let us show that 12.4 remains valid132 if S is normal, and η is replaced by a compact subset of the normalizer Q of PR in H. The group L = NH (FA) ∩ Q meets every connected component of PR since PR = Z(FA)R . UR , and meets every connected component of H by the argument in the first paragraph. Since L ∩ UR = {e} and L . UR ⊃ PR , it immediately follows that Q is the semidirect product of L and UR . On the other hand, L is stable133 under the Cartan involution associated to K, so (9.3) the group L∩K is a maximal compact subgroup of L, so134 L = (L ∩ K) . Z(FA)0R . In view of 12.4, we are reduced to the case where η ⊂ L ∩ K. Let g ∈ L. Then int(g) leaves S and U invariant, so it permutes the simple F-roots, whence g . FAt . g −1 = FAt . On the other hand, g normalizes M0R . U0R because this group is the identity component of the intersection of the kernels of the α ∈ FΔ, viewed as characters of PR . Thus, if η ⊂ L ∩ K, we have:135 K . FAt . ω . η ⊂ K . η . FAt . (η−1. ω . η) = K . FAt . ω

(ω = η−1. ω . η),

whence the claim. 131 By

an argument similar to part of footnote 118. S . η is contained in a Siegel set. 133 If g ∈ L, then g permutes the roots of A (because g normalizes A). Moreover, g preserves F F the set of positive roots (because g normalizes P), so g also preserves the complement, which is the set of negative roots. The Cartan involution θ inverts FA, and therefore sends negative roots to positive roots, so we conclude that θ(g) preserves the set of positive roots. This means that θ(g) normalizes UR , and therefore also PR , so we see that θ(g) ∈ L. 134 We have L = (L∩K) . L0 ⊂ (L∩K) . N ( A)0 ⊂ (L∩K) . Z( A)0 (since N( A)0 ⊂ Z( A)0 , H F F F F R R R R as in footnote 116). The reverse inclusion is clear. 135 η−1. ω . η should be interpreted to be −1. ω . g. g∈η g 132 I.e.,

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13. FUNDAMENTAL SETS (SECOND TYPE)

79

13. Fundamental sets (second type) The goal of this section is to prove Theorem 13.1, which provides fundamental sets that are often more useful than those of §9. However, we will restrict ourselves here to establishing that they satisfy the conditions (F0 ) and (F1 ) of 9.6, postponing to §15 the study of the Siegel property. 13.1. Theorem. Assume G is a reductive Q-group. Let P be a minimal parabolic Q-subgroup of G, S be a maximal Q-split torus of P, K be a maximal compact subgroup of GR , and Γ be an arithmetic subgroup of G. Then there exists a Siegel set S (with respect to K, P, S) and a finite subset C of GQ such that: (1)

GR = S . C . Γ. Before proceeding with the proof, let us give a corollary.

13.2. Corollary. If X(G0 )Q = {1}, then GR /Γ has finite invariant measure. Proof. This follows from the theorem and the fact that S has finite Haar measure (12.5).  13.3. We first show that if 13.1 holds for one choice of K, then it also holds for any other maximal compact subgroup K . By 12.4, there exists t > 0 such that: K . Pt ⊂ K. Pt .   c Let Γ = Γ ∩ c∈C Γ . This is an arithmetic group (7.13) that clearly satisfies: (1)

(2)

Γ. C ⊂ C . Γ.

By the compactness criterion (8.7), there exist compact subsets α ⊂ M0R , β ⊂ UR such that: (3)

M0R = α . (Γ ∩ M),

UR = β . (Γ ∩ U).

Using the fact that K. P0t = K . M0R . QAt . UR , we see that: K. Pt . C ⊂ K. QAt . α . (M ∩ Γ ) . UR . C = K. QAt . α . UR . (M ∩ Γ ) . C ⊂ K. QAt . α . β . Γ. C, whence, in view of (1), (2): (4)

K . Pt . C ⊂ K. QAt . ω. C . Γ

(ω = α . β).

13.4. Since K meets every connected component of GR , it obviously suffices to prove the theorem when G is connected. Assume that G is embedded in GLn . We saw (9.8, 9.9) that given u ∈ GL(n, R) such that u . GR . u−1 is self-adjoint, there exists a standard Siegel domain S of GL(n, R) and a finite subset B of GL(n, Q) such that GR = (u−1. S . B ∩ GR ) . Γ. On the other hand, we know that we can find u such that u . GR . u−1 is selfadjoint and even in “good position” in GL(n, R) (cf. 11.25). Theorem 13.1 is therefore a consequence of the facts that we just recalled, and the following. 13.5. Theorem. We keep the assumptions and notation of 13.1. Assume that G is connected and that SR is invariant under the Cartan involution associated to K. Let u be an element of GL(n, R) such that u . GR . u−1 is in good position, let

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80

III. FUNDAMENTAL SETS WITH CUSPS

b ∈ GL(n, Q) and let S0 be a standard Siegel set of GL(n, R). Then there exists a Siegel set S of GR , with respect to K, P, S, and a finite subset C of GQ such that: (1)

u−1. S0 . b ∩ GR ⊂ S . C . Γ.

The proof is in principle very much like that in [6, §7], which establishes the finiteness of the volume by proving an inclusion similar to the above, but where S is a Siegel domain over R of GR and C is a finite subset of GR . 13.6. Notation. K0 . A0 . N0 is the standard Iwasawa decomposition of GL(n, R). We write QA for S0R . Let GR = K . A . N be an Iwasawa decomposition of GR such that A ⊃ QA, and such that N and U are associated to the positive roots for compatible orderings of X(AC ) and X(S). In particular, N ⊃ UR . Since all Iwasawa decompositions of GR are conjugate, we may assume,136 after replacing u with u . v (v ∈ GR ), that: N = N0 ∩ uG,   and that there is compatibility between the ordering of X (A0 )C and the ordering of X(uAC ) that is obtained from the given ordering of X(AC ) by the isomorphism induced by u.

(1)

u

K = K0 ∩ uG,

A = A0 ∩ uG,

u

u

13.7. (a) Let us first note that, in view of 13.3 (4), it suffices to establish the existence of t > 0 and a finite subset C of GQ such that: (1)

u−1. S0 . b ∩ GR ⊂ K . Pt . C.

(b) Let q ∈ GL(n, Q). Let us show that if the assertion (1) holds for q G, then it also holds for G. To that end, we apply (1), by replacing u, G, b, with u . q −1 , q G and b . q −1 . This proves the existence of t > 0 and of a finite subset C of q GQ such that: (2)

q . u−1. S0 . b . q −1 ∩ q G ⊂ q K . q Pt . C .

It then suffices to conjugate the two sides of this relation by q −1 to obtain (1). (c) By replacing G with q G (q ∈ GL(n, Q)), we may assume that S ⊂ D (D being the group of diagonal matrices of GLn ), that UR ⊂ N0 , that137 u ∈ Z(S), and that the orderings of X(D) and X(S) are compatible. By applying the theorem on conjugation of triangularizable subgroups over Q to GLn (11.10), we can first reduce to the case where: (3)

S ⊂ D,

UR ⊂ N.

In view of 13.6, we have: (QA) ⊂ uA ⊂ A0 ,

u

u

UR ⊂ u N ⊂ N0 .

Thus, u S and S are two subtori of D, that are conjugate via an element of GL(n, R). Since D is a maximal Q-split138 torus of GLn , there exists (11.12) an element v ∈ N(D)Q , such that: v . x . v −1 = u . x . u−1

(x ∈ QA).

136 12.4 implies we may multiply u−1 on the left by any element of G , so we may multiply u R on the right by an element of GR . 137 The original French manuscript does not explicitly state that u ∈ Z(S), but this observation will be helpful in the proof of (e). 138 This implies that every subtorus of D is defined over Q.

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13. FUNDAMENTAL SETS (SECOND TYPE)

81

Thus, we have u = z . v (z ∈ Z(u S)Q ). Since the orderings of X(D) and X(uAC ) are compatible, and also those on X(uAC ) and X(u S) = X(v S), the same holds for the orderings of X(S) and X(v S); in particular, v UR ⊂ N. The group v G satisfies the conditions stated in (c),139 hence our claim follows from (b). (d) The Lie algebra n0 of N0 is the direct sum of the subalgebra:  n1 = n0 ∩ z(S) = (gα )R , α>0, α(S)=1

and the subspace: 

n+ =

(gα )R ,

α>0, α(S) =1

which is clearly an ideal. It follows (11.13) that the product map is a homeomorphism of N1 × N+ onto N0 (N1 = exp n1 , N+ = exp n+ ). The group Z(S) leaves gα and gα ∩ u invariant. Since it is reductive and defined over Q, there exists a Z(S)-invariant Q-subspace nα,2 that is complementary to gα ∩ u in gα . Let n2 be the sum of the nα,2 (α > 0, α(S) = 1) and then let N2 = exp(n2 )R . For every β > 0 that is non-trivial on S, the sum of the subspaces (gα )R (α ≥ β, α(S) = {1}) is an ideal of n+ . Thus, Lemma 11.13 shows that the product map is a homeomorphism of N2 . UR onto N+ , which results in a topological decomposition N0 = N1 . N2 . UR . (This is in fact biregular over Q, but we will not need this fact.140 ) We will now prove that it suffices to prove (1) when: (4)

b = q −1. v

(q ∈ N(D)Q , v ∈ N1 . N2 ).

By using the Bruhat decomposition of GL(n, Q) (cf. §3), we can write: (5)

b = n. t . w . n

(n, n ∈ (N0 )Q , t ∈ DQ , w ∈ N(D)Q ∩ O(n)).

On the other hand, n = v . v  (v ∈ N1 . N2 , v  ∈ UR ). There exists a standard Siegel domain S of GL(n, R) such that S0 . n. t ⊂ S , whence: u−1. S0 . b ∩ G ⊂ u−1. S. w . v . v  ∩ G ⊂ (u−1. S. w . v ∩ G) . v  , hence,141 our claim follows. (e) Since S ⊂ D, the group Z(S) is self-adjoint in GL(n, R), so the components k, a, n of u in the standard Iwasawa decomposition of GL(n, R) belong to Z(S). Let us put: −1

−1

K = u K0 = n −1

. a−1

K0 ,

−1

A = n

. a−1

−1

A0 = n A0 .

−1

Obviously, N0 = n . a N0 , so K. A. N0 is the Iwasawa decomposition of GL(n, R) that is conjugate to K0 . A0 . N0 by n−1. a−1 . By writing S0 = K0 . A0,t . ω (ω a u · v −1 = z ∈ Z(u S), we replace S with u S = v S, and u with u · v −1 . this fact will eliminate a technical issue later in the proof (see footnote 141). 141 v  ∈ G (because the decomposition N = N . N . U is biregular over Q), so v  can be 0 1 2 Q R incorporated into the finite set C. 139 Since

140 Actually,

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82

III. FUNDAMENTAL SETS WITH CUSPS

compact subset of N0 ), we have: u−1. S0 . q −1. v = u−1. K0 . A0,t . ω . q −1. v = n−1. a−1. K0 . A0,t . ω . q −1. v ⊂ K . n−1. A0,t . ω . q −1. v = K. At . ω. (n−1. q −1. n) . n−1. v, where t is such that a−1. A0,t ⊂ A0,t and where ω = n−1. ω . n. Now we have n−1. q −1. n ∈ N(A ) and n−1. v ∈ N1 . N2 , and we also know K. At . ω is a Siegel set with respect to K , A , N0 , so we are reduced to proving the following lemma: 13.8. Lemma. We keep the notation of 13.5. Let K. A. N0 be an Iwasawa decomposition of GL(n, R) such that142 K ⊃ K, A ⊃ QA, N0 ⊃ UR , and such that N0 and UR are associated with the positive roots of orderings that are compatible on X(AC ) and X(S). Let S be a Siegel set of GL(n, R) with respect to K , A , N0 , and let q ∈ N(A )R , v ∈ N1 . N2 (notation of 13.7). For each w ∈ QW(G, S), let xw be a representative of w in N(S)Q . Then, there exists t > 0 such that:  S. q −1. v ∩ G ⊂ K . Pt . x w . w∈QW(G) 

Each coset of N(A ) modulo Z(A ) meets K (11.19). Therefore, there exists z ∈ Z(A ) such that z . q −1 ∈ N(A ) ∩ K . Since S. z is contained in a Siegel domain with respect to K , A , N0 , we see that we may assume: q ∈ N(A ) ∩ K , and we will do so. Let x ∈ S and let x = k. a. n be its Iwasawa decomposition, with respect to K , A , N0 . Assume moreover that x ∈ G . v −1. q. We can write: x . q −1. v = k. a. n . q −1. v = k . m . s . u,

(1) where:

k ∈ K,

m ∈ M0R ,

s ∈ QA,

u ∈ UR ,

(see 11.19). As has been noted, the product k . m and the elements s and u are uniquely determined by the left-hand side, and depend continuously on x. The elements k and m are determined only up to multiplying by an arbitrary element of the compact group K ∩ M0R . But this lack of uniqueness is not an issue for us here, because we are only interested in knowing whether the elements range over bounded subsets of GR . (i) We show that the set of elements msu (notation of (1)) is bounded, where x ranges over S ∩ G . v −1. q. We have: 

x . q −1. v = k. q −1. qa n . qa . v1 . v2

(2)

(v1 ∈ N1 , v2 ∈ N2 ).



Let c = qa n. We can write it in the form: (3)

n

−1

c = kc . mc . ac . nc K and A of 13.7 (e), we have K = S = S (because n ∈ Z(S)). 142 For

(kc ∈ K , mc ∈ M0R , ac ∈ QA, nc ∈ UR ). u−1K

0

⊃

u−1K

0

∩ G = K and A =

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n−1A

0

⊃

13. FUNDAMENTAL SETS (SECOND TYPE)

83

Since c remains bounded as x runs through S (by 12.2), we know that each factor of the right-hand side is also bounded. Substituting in (2), we obtain: (k = k. q −1. kc ∈ K ), x . q −1. v = k. mc . ac . qa . (qa )−1. nc . qa . v1 . v2     x . q −1. v = k. mc . ac . qa . v1 . v1−1. (qa )−1. nc . qa . v1 . v2 .

(4)

Since nc ∈ UR ⊂ N2 . UR = N+ and since both qa ∈ A and v1 ∈ N1 normalize N+ , the second product in parentheses in the right-hand side is in N+ , while the first belongs to K. Z(S)R . Since v2 ∈ N+ , we obtain, by comparing (1) and (4): (5)

m . s ∈ K. mc . ac . qa . v1 ⊂ K. Z(S)R ,

u . v2−1 = v1−1. (qa )−1. nc . qa . v1 ∈ N+ .

The element m . s ∈ Z(S) leaves UR and N2 invariant (cf. 13.7 (d)), so msu is the component in UR of ms(u . v2−1 ). Therefore, it suffices to see that the latter element is bounded. In view of (5), this reduces to proving that: z = int(mc . ac . qa . v1 ) . int(v1−1. (qa )−1 ) (uc ) = int(mc . ac ) (uc ) is bounded, but this follows from the fact pointed out earlier (after (3)), that mc , ac , uc are bounded. (ii) Proof of Lemma 13.8. Let: C = {a ∈ QA | aα ≤ 1,

(α ∈ QΔ)}.

For w ∈ QW(G, S), let: Mw = {x ∈ S ∩ G . v −1. q | xw . s . x−1 w ∈ C}, where s is defined by (1). Since C is a fundamental domain for QW in QA, the set S. q −1. v ∩ G is the union of the Mw . q −1. v (w ∈ QW), and it suffices to show that there exists t > 0 such that: (6)

Mw . q −1. v ⊂ K . Pt . xw .

We can find (by 11.19) an element yw ∈ K such that yw . x−1 w ∈ Z(S). Then, there exists t > 0 such that: Pt . yw ⊂ Pt . xw , and we are reduced to proving that we have: (7)

Mw . q −1. v ⊂ K . Pt . yw ,

for suitable t > 0. We have (see (1)): x . q −1. v = k . m . s . u = k . msu . m . s, −1 −1 −1 −1 −1 = k . yw . yw . (msu) . yw . (yw . m . yw ) . (yw . s . yw ). z = x . q −1. v . yw −1 . We have a decomposition: Put d = yw . (msu) . yw

d = kd . md . sd . ud

(kd ∈ K, md ∈ M0R , sd ∈ QA, ud ∈ UR ),

whence: −1 −1 ) . (yw . s . yw ) z = k. md . sd . ud . (yw . m . yw

−1 (k = k . yw . kd ∈ K).

The element yw normalizes S, so also M (cf. 11.6). Since M normalizes UR , we see that we can write: −1 −1 ) . (sd . yw . s . yw ) . (u ), z = (k. md . yw . m . yw

where u ∈ UR is a conjugate of ud in PR , and where the three factors defined by the parentheses are the components of z in K . M0R , QA and UR respectively. Thus

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III. FUNDAMENTAL SETS WITH CUSPS

−1 the component sz in QA of z is equal to sd . (yw . s . yw ). By (i), sd runs through a  −1 −1 ∈ C. bounded subset when x ∈ S ∩ G . v . q. Suppose x ∈ Mw . Then yw . s . yw Therefore, there exists t > 0 such that: −1 sz = sd . yw . s . yw ∈ QAt ,



which establishes (7).

Bibliographical note. In the case of a linear group over a division algebra over Q, Theorem 13.1 is equivalent to a classical reduction theorem, which can be found in [42] for instance. The general case was announced in [2], and the proof given above is the one to which [2] alludes. A very different proof, based on the consideration of the adelic group of G, was later obtained by Godement-Weil [17]. We will give an analogue that uses only GR in §16. 14. Fundamental representations. Associated functions 14.1. Let G be a connected reductive R-group, P be a parabolic R-subgroup of G, and χ ∈ X(P). We will say that a continuous function Φ : GR → R≥0 is of type (P, χ) if it satisfies: Φ(g . p) = Φ(g) . |pχ |

(g ∈ GR , p ∈ PR ).

We first make some simple remarks on functions of this type. 14.2. (a) There always exists a function of type (P, χ) that is > 0, and invariant on the left under a given maximal compact subgroup K. Indeed, we have the decomposition GR = K . P0R , and it is clear that |pχ | = 1 if p belongs to a compact subgroup of PR , in particular, if p ∈ K ∩ P. By putting Φ(k . p) = |pχ |, we therefore obtain a function on GR that satisfies our conditions. (b) Let Φ, Φ be two functions of type (P, χ), and assume Φ > 0. Then Φ ≺ Φ on GR . To see this, note that the function Φ/Φ admits an upper bound c > 0 on K. Since GR = K . P0R and we have: (Φ/Φ )(g . p) = (Φ/Φ )(g)

(g ∈ GR , p ∈ PR ),



it follows that Φ(g) ≤ c . Φ (g), (g ∈ GR ). In particular, we see that if Φ, Φ are both strictly positive, then Φ  Φ . (c) Let C be a compact subset of GR and let Φ > 0 be of type (P, χ). Then: Φ(c . g)  Φ(g)

(g ∈ GR ; c ∈ C).

Indeed, we have, by putting g = k . p (k ∈ K, p ∈ PR ): Φ(c . g)/Φ(g) = Φ(c . k)/Φ(k), and it suffices to note that since K is compact, the right-hand side is  1 when c ∈ C, k ∈ K. 14.3. Let π : G → GL(V) be a finite-dimensional representation of G. Assume that V contains a P-invariant line D, and let χ be the character of the associated one-dimensional representation of P. We endow V with a Hilbertian norm , and let e0 be a vector that spans D. Then, the function: Φ(g) = π(g) . e0 is obviously > 0, and of type (P, χ); if π(h) (h ∈ G) is unitary, then Φ is invariant under left-translation by h.

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14. FUNDAMENTAL REPRESENTATIONS. ASSOCIATED FUNCTIONS

85

The most important properties of the functions of type (P, χ) will be established by considering functions that are constructed in this way. 14.4. Let k be a subfield of R, and assume P is a minimal parabolic k-subgroup. Let S be a maximal k-split torus of P and k Δ be the set of simple roots of G relative to S, for the ordering associated to P. We assume X(S) ⊗ Q is endowed with a euclidean inner product ( , ) that is invariant under the relative Weyl group kW(G). An element χ ∈ X(P)k is said to be dominant if (χ, α) ≥ 0 (α ∈ k Δ). For each α ∈ k Δ, we pick an element Λα ∈ X(P)k that is trivial on Z(G)0 , and satisfies: (Λα , β) = dα . δαβ

(1)

(dα > 0, α, β ∈ k Δ),

and the Λα are called the fundamental dominant k-weights of DG. Thus, an element χ ∈ X(P)k is dominant if and only if some positive multiple of χ is the sum of a linear combination with positive143 integer coefficients of the Λα , and a character that is trivial on P ∩ DG. Given a dominant weight χ that is trivial on Z(G)0 ∩ DG, there exists an irreducible representation π : G → GL(V) that is defined over k, and a P-invariant line Dπ ⊂ V that is defined over k, on which P acts via χ. For μ ∈ X(S), let: Vμ = {v ∈ V | π(s) . v = sμ . v, (s ∈ S)}. This is a k-subspace of V. We say that μ is a k-weight of π if Vμ = {0}. It is known that V is the direct sum of the Vμ , that Vχ is one-dimensional, and that every k-weight of π is of the form:  cα (μ) . α (cα (μ) ∈ N). (2) μ=χ− α∈k Δ

As a matter of fact, this is proved in [8, §12] when G is semisimple. But the extension to the case considered here is immediate; indeed, let χ1 , χ2 be the restrictions of χ to Z(G)0 and DG∩P, and let σ be the representation of DG of dominant weight χ2 that is given by [8, loc. cit.]. Since χ is assumed to be trivial on Z(G)0 ∩ DG, it is then clear that χ1 ⊗ σ is a representation of G with the required properties. Let us also note that every χ ∈ X(P)k admits a positive multiple that is trivial on Z(G)0 ∩ DG, since this group is finite. After choosing a maximal compact subgroup K whose Lie algebra is orthogonal144 to that of SR , we endow VR with a euclidean inner product for which ρ(g) is unitary (resp. self-adjoint) whenever g ∈ K (resp. g ∈ SR ); this is always possible. In particular, the subspaces Vμ are mutually orthogonal. The function145 Φπ : g → π(g) . e0 (e0 : unit vector spanning Dπ ) is then > 0, of type (P, χ), and invariant on the left under K. We will denote by Φα the function that is obtained in this way for χ = Λα . Given a dominant χ ∈ X(P)k we obtain a function that is > 0 and of type (P, χ) by putting:  d /m Φαα , (3) Φ = |χ1 |1/m . α∈k Δ

where the dα are the positive integers such that m χ = χ1 + m ∈ N and a character χ1 that is trivial on P ∩ DG.

 α

dα Λα for suitable

always in this text, “positive” means ≥ 0. sets that are normal (12.3) satisfy this assumption that the Lie algebra of K is orthogonal to the Lie algebra of SR . 145 Φ will often be referred to as a standard function of type (P, χ). π 143 As

144 Siegel

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III. FUNDAMENTAL SETS WITH CUSPS

In view of 14.2 (b) every function of type (P, χ) (resp. and > 0) is essentially bounded by Φ (resp. is comparable to Φ). Using the decomposition P = M . S . U of P, we can write g ∈ GR in the form: (kg ∈ K, mg ∈ MR , sg ∈ S0R , ug ∈ UR ).

g = kg . mg . sg . ug χ

We then have π(g) . e0 = π(kg ) . sg . e0 , whence: Φπ (g) = sχg .

(4)

14.5. Example. Let G = SLn . Let P (resp. S) be the group of upper triangular (resp. diagonal) matrices of SLn and let K = SO(n). The fundamental

representations of G are the exterior powers i (1 ≤ i ≤ n − 1) of the identity representation.146 In the notation of §1, the simple roots are the quotients aii /ai+1,i+1 (i = 1, . . . , n − 1), the dual basis Λα consists of: Λi = a11 · . . . · aii

(i = 1, . . . , n − 1);

the irreducible representation of dominant weight Λi is the i-th exterior power. In

i n R , the P-invariant line is spanned by e1 ∧ . . . ∧ ei . In this case, the functions Φα are none other than the functions Φi of §1. 14.6. Proposition. We keep the preceding notation. Let χ ∈ X(P)k be dominant and let Φ be a function that is > 0 and of type (P, χ). Let S be a Siegel domain of GR with respect to K, P, S. Then: Φ(x . y) Φ(x) . Φ(y)

(x ∈ S; y ∈ GR ).

Proof. It suffices to prove this for the function Φ = Φπ of 14.4. We can write:  fμ (g), (fμ (g) ∈ Vμ ; g ∈ GR ), (1) π(g) . e0 = μ

where μ runs through the k-weights of π. Therefore, since the spaces Vμ are pairwise orthogonal:  Φ(g)2 =

fμ (g) 2 . μ

Using the decompositions P = M . S . U and GR = K . P0R (cf. 11.19), we get: (kx ∈ K, mx ∈ M0R , sx ∈ S0R , ux ∈ UR ),

x = kx . mx . sx . ux whence:

π(x . y) . e0 = π(sx ) . π(mx . ux . y) . e0 . We have: π(sx ) . π(mx . ux . y) . e0 =



sμx . fμ (mx . ux . y),

μ

hence Φ(x . y)2 =



(sx )2μ fμ (mx . ux . y) 2 ;

μ χ

but (by 14.4 (4)), we have Φ(x) = sx , so in view of (14.4 (2)), we have:   sx−2cα (μ) . α . fμ (mx . ux . y) 2 . (2) Φ(x . y)2 = Φ(x)2 μ 146 The

α

identity representation is the homomorphism g → g from G to GLn .

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14. FUNDAMENTAL REPRESENTATIONS. ASSOCIATED FUNCTIONS

87

Since x is in a Siegel domain, we have sαx ≤ t. The cα (μ) being ≥ 0, it follows that the product arising in front of fμ (mx . ux . y) 2 in (2) is 1 when x ∈ S, whence:   (x ∈ S, y ∈ GR ) Φ(x . y)2 Φ(x)2 .

fμ (mx . ux . y) 2 μ 2

= Φ(x) . Φ(mx . ux . y)2 . The elements mx and ux range over compact sets since x ∈ S, therefore (by 14.2 (c)), Φ(mx . ux . y)  Φ(y), and the proposition follows.  14.7. Notation. In the sequel, we will use the Iwasawa and Bruhat decompositions simultaneously. We will write the decomposition of x ∈ GR according to K, M, S, and U: (1)

(kx ∈ K, mx ∈ M0R , sx ∈ S0R , nx ∈ UR ).

x = kx . mx . sx . nx

On the other hand, we will use the same letter to denote both an element w ∈ kW(G) and a representative of w in N(S)k , chosen once and for all. Let W be the set of these representatives. Let U− be the maximal unipotent k-subgroup corresponding to the negative k-roots and let Uw be the largest subgroup V of U such that w−1. V . w ⊂ U− . We know (§11) that given g ∈ Gk , there exists a unique wg ∈ W such that we have: (2)

(ug ∈ (Uwg )k ; zg ∈ Z(S)k ; vg ∈ Uk ),

g = ug . wg . zg . vg

where the factors on the right are all uniquely determined by g. 14.8. Proposition. We keep the notation of 14.6. Let Φ be a function that is > 0 and of type (P, χ), where χ is dominant. Then: χ (i) Φ(sg )  Φ(g)  sg , (g ∈ GR ), (ii) Φ(u) 1, (u ∈ U− R ), (iii) A subset C of U− R is bounded if and only if ψ  1 on C for every function ψ > 0 of type (P, η), and every dominant η ∈ X(P)k . Proof. (i) In view of 14.2 (b), it suffices to prove (i) for the function Φπ ; in that case we have equality (see 14.4 (4)). (ii) It suffices to consider the function Φπ of 14.4. But we know that U− acts as the identity modulo the sum of the spaces Vμ (μ = χ), so:  π(u) . e0 = e0 + fμ (u), (u ∈ U− ), μ =χ

2

Φ(u) = 1 +



fμ (u) 2 ≥ 1.

μ =χ −

(iii) Since U ⊂ DG, we can limit ourselves to the case where G is semisimple. The necessity of the condition is obvious. Let η be a dominant weight that is not orthogonal to any simple k-root. For the converse, it suffices to show that if Φ > 0 is of type (P, η), and Φ  1 on C, then C is bounded. We may assume that Φ is associated to an irreducible representation (π, V) of dominant weight η. The assumption on η implies that P is the entire stabilizer of the line Dπ that is spanned by e0 [8, §12]. It follows that the map ϕ : u → π(u) . e0 is an injective map of U− into V. By a theorem of Rosenlicht [36, Theorem 2], ϕ(U− ) is closed. It follows that ϕ induces a homeomorphism of U− onto ϕ(U− ). Since the hypothesis implies that the image ϕ(C) of C is bounded, it follows that C is bounded. 

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III. FUNDAMENTAL SETS WITH CUSPS

14.9. Corollary. Let B be a subset of GR . Then the set of components sb (b ∈ B) is bounded if and only if Φ  1 on B for every function Φ > 0 that is of type (P, χ) and every dominant weight χ. Proof. The necessity of the condition follows from 14.8 (i). Conversely, if this χ condition is met, then we see from 14.8 that sb  1 (b ∈ B) for every dominant χ. Since the set of these characters contains a coordinate system on S0R , the converse follows.  14.10. Corollary. In the notation of 14.7, we have, for every dominant weight χ and every function Φ > 0 that is of type (P, χ): Φ(g)  Φ(sg ) Φ(zg )

(g ∈ Gk ).

Proof. We can limit ourselves to considering a function Φ = Φπ . We have:147 Φ(g) = Φ(wg . wg−1. ug . wg . zg . vg )  Φ(wg−1. ug . wg . zg ) = Φ(wg−1. ug . wg ) . Φ(zg ). Since wg−1. ug . wg ∈ U− , the corollary follows from 14.8 (i), (ii).



15. The Siegel property We will now extend the results of §4 (for GLn ) to the case of a general reductive group, by using similar proofs. 15.1. In this section, k is a field of characteristic zero, G is a reductive k-group, P is a minimal parabolic k-subgroup of G0 , and S is a maximal k-split torus of P. We fix an ordering of X(S) that is associated to P, and we denote by D a basis of X(S) ⊗ Q that is contained in X(S) and that consists of the set k Δ of simple k-roots of G, together with some characters that are trivial on S ∩ DG. For every w ∈ kW(G, S) and α ∈ k Δ, we put:  (1) w(α) = nαβ (w) . β. β∈k Δ

The coefficients nαβ (w) are therefore integers, all with the same sign. The following theorem is due to Harish-Chandra (unpublished) when k = R. The proof in the more general case considered here is essentially the same. 15.2. Theorem. Assume that k ⊂ R and that G is connected. Let S be a normal Siegel domain of GR with respect to K, P, S. Let C = C−1 be a symmetric148 χ subset of Gk such that, for each dominant weight χ, we have |zc | 1 on C. Then, CS = {c ∈ C | S . c ∩ S = ∅} is relatively compact in GR . Let us first note that the condition imposed on C may also be expressed as Φ(zc ) 1 on C, for every dominant weight χ and every function that is > 0 and of type (P, χ).

148 A

χ

χ

the final equality, note that π(zg ) . e0 = zg · e0 (so |zg | = Φ(zg )) because zg ∈ P. subset C of a group is symmetric if C = C−1 .

147 For

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15. THE SIEGEL PROPERTY

89

Proof. (i) We show that uc , sc and szc remain bounded as c runs through CS . Let c ∈ CS . Then, we can find x ∈ S such that x . c ∈ S. In view of 14.6, we have, for every dominant weight χ and every function Φ > 0 that is of type (P, χ): Φ(x) = Φ(x . c . c−1 )

Φ(x . c) . Φ(c−1 )

(c ∈ CS ; x ∈ Sc−1 ∩ S)

Φ(x) . Φ(c) . Φ(c−1 ). This implies: (1)

Φ(c) . Φ(c−1 ) ≺ 1

(c ∈ CS ).

Since C = C−1 and Φ(c) Φ(zc ) (cf. 14.10), the hypothesis and (1) imply that Φ  1 on CS , whence also Φ(sc )  1  Φ(zc ) on CS ; this shows that sc and szc are bounded. But we have (proof of 14.10): Φ(c)  Φ(wc−1. uc . wc ) . Φ(zc ), so Φ(wc−1. uc . wc )  1. Proposition 14.8 then implies that wc−1. uc . wc is bounded. This is therefore also the case for uc . (ii) To prove the theorem, we proceed by induction on the k-rank of G. If it is zero, then S is compact, so the theorem is obvious. Thus, we assume that rankk (G) > 0 and that the theorem holds for G if rankk (G ) < rankk (G). We first consider the case where X(G)k = {1}. For each w ∈ kW, put: CS,w = CS ∩ (Pk . w . Pk ), where once again, we use w to denote a representative of w in N(S)k that has been chosen once and for all. We have to prove that CS,w is bounded and to that end, let us distinguish two cases: (a) w does not belong to any proper parabolic subgroup that contains P. Let: (1)

M = S ∩ S . C−1 S,w ,

M = S ∩ S . CS,w .

We have CS,w ⊂ M−1. M . Therefore, it suffices to show that M and M are bounded. Let x ∈ M and y = x . c ∈ M . By using the Iwasawa decompositions of x, y and the Bruhat decomposition of c, this equality may be written: kx . mx . sx . nx . uc . w . zc . vc = ky . my . sy . ny . We also want to find the Iwasawa decomposition of the left-hand side so as to be able to compare the components of the two sides. We have: y = kx . mx . sx(nx . uc ) . mx . sx . w . zc . vc −1

−1

= kx . mx . sx(nx . uc ) . w . w mx . w sx . zc . vc . Put: d = mx . sx(nx . uc ) . w = kd . md . sd . nd ,

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III. FUNDAMENTAL SETS WITH CUSPS −1

Since w ∈ N(S), we have w mx ∈ M and and centralize each other, this implies149

w−1

sx ∈ S. Since M and S normalize U

−1

−1

y = (kx . kd ) · (md . w mx . mzc ) . (sd . w sx . szc ) . (nd . vc )

(nd ∈ UR ),

so −1

sy = sd . w sx . szc . We see from 12.2 that150 d is bounded, so sd is bounded. From (i), we know that szc is also bounded. Thus, we have, for every α ∈ k Δ: sαy  (w−1. sx . w)α = sxw(α) .

(2)

In the notation of 15.1, we have:  nαβ (w) . β sx . sxw(α) = β

Fix β ∈ k Δ. In view of the assumption made on w, and 11.11, there exists α ∈ k Δ such that nαβ (w) < 0. We then have nγβ (w) ≤ 0 for every γ ∈ k Δ, so each factor β on the right is 1 (since x ∈ S). But sy ≺ 1 since y ∈ S; thus, we must have: n (w) . β

sx γβ

1

(γ ∈ k Δ). β sx

In particular, since nαβ (w) = 0, this yields  1, and since this holds for every β ∈ k Δ, this shows that sx is bounded. But mx and ux are automatically bounded, so M is bounded. Taking (2) into account, we see that sy is bounded, so y is also bounded. (b) The element w belongs to a proper parabolic subgroup P that contains P. We then have CS,w ⊂ P and hence it suffices to show that C = CS ∩P is bounded. Let x, y ∈ S be such that x . c = y (c ∈ C ). By multiplying both sides by kx−1 , we see that we can assume x ∈ P ⊂ P , so also y ∈ P . It follows that: C −1 = C = {c ∈ C ∩ P | (S ∩ P ) . c ∩ (S ∩ P ) = ∅}.

(3)

It suffices to consider the case where P is maximal proper. Then, in view of 11.7, 11.8, there exists β ∈ k Δ such that P = Pθ (θ = k Δ  {β}), and we have the canonical decompositions: P = Z(Sθ ) . Vθ ,

(4)

Z(Sθ ) = Lθ . Mθ . Sθ ,

where rankk (Lθ ) = rankk (G) − 1, rankk (Mθ ) = 0, and Sθ is one-dimensional. The group Lθ is semisimple, and θ can be identified with the set of simple k-roots of Lθ for an ordering that is associated to P ∩ Lθ . We may assume151 w ∈ Lθ . The unipotent radical U of P is the semidirect product over k of Lθ ∩ U by Vθ . The first equality of (4) also represents a semidirect product over k. In the equality: c = uc . w . zc . vc = hc . bc

(hc ∈ Z(Sθ )k , bc ∈ Vθ,k ),

149 A

mistake in the original French manuscript has been corrected: it defines d to be and states that kd . md . sd is the Iwasawa decomposition of w (so kd normalizes M), which cannot be expected. 150 Since x ∈ S, we know m and n are bounded. From (i), we know u is also bounded. x x c 151 Write w = v . z (z ∈ Z(S ), v ∈ V ). Then v S = v . z S = w S ⊂ w S = S, so [v, S ] ⊂ S. θ θ θ θ θ θ However, Vθ is normal in Pθ , so we also have [v, Sθ ] ⊂ Vθ . Since S ∩ Vθ = {e}, this implies v ∈ Z(Sθ ). But also v ∈ Vθ , so v = e. This means w ∈ Z(Sθ ). Since Mθ and Sθ centralize Sθ , we may assume those components of w are trivial, which means w ∈ Lθ . sx(n

x . uc ) . w,

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15. THE SIEGEL PROPERTY

91

we see by using the Bruhat decomposition of hc in Z(Sθ )k , that we have: hc = uc . w . zc . bc

(5)

vc = bc . bc

(uc , bc ∈ Lθ ∩ Uk ; zc ∈ Z(S)k ).

By enlarging S, we may assume that S = S∩P is the product of its projections152 onto Z(Sθ ) and Vθ , where the second projection is compact, hence:  −1   (6) hc ∈ S ∩ Z(Sθ ) . S ∩ Z(Sθ ) (c ∈ C ). We will first show that hc is bounded. Let π1 , π2 , π3 be the canonical projections of Z(Sθ ) onto its quotients Q1 , Q2 , Q3 by Mθ . Sθ , Lθ . Mθ and Lθ . Sθ respectively. The restriction to Z(Sθ )R of the map: π = π1 × π2 × π3 : Z(Sθ ) → Q = Q1 × Q2 × Q3 has finite kernel, and its image is closed because it contains the connected component of e in Q, for the usual topology. Hence, this is a proper map, and it suffices to show that πi (hc ) is bounded (i = 1, 2, 3). Let i = 2. By (5), we have π2 (hc ) = π2 (zc ) = π2 (szc ), whence the claim follows in this case, since szc is bounded, according to (i). For153 i = 3, we have π3 (hc ) = π3 (zc ). Noting that the projection of S to Q3 is bounded (since Q3 is anisotropic), it is immediate that π3 (zc ) is bounded.  Let T = π1 S ∩ Z(Sθ ) . We immediately verify from the definitions that T is a normal Siegel set of (Q1 )R (with respect to the projections K1 , P1 , S1 , of K ∩ Lθ , P ∩ Lθ and S ∩ Lθ ). By (6), we have: π1 (hc ) ∈ T−1. T.

(7)

Since π1 is a k-morphism, we have π1 (hc ) ∈ (Q1 )k , and the equality C = C −1 of (3) immediately implies that the set of all hc is also symmetric. π1 is a k  isogeny of Lθ onto Q1 . The homomorphism π∗1 : X(P1 ) → X Z(Sθ ) induced by π1 transforms dominant weights to dominant weights. If we apply π1 to the relation hc = uc . w . zc . bc , we see that we obtain the Bruhat decomposition of π1 (hc ) with respect to P1 , S1 . In particular, π1 (zc ) is the component of π1 (hc ) relative to Z(S1 ). It follows that the set of all π1 (hc ) satisfies all our assumptions. Since rankk (Q1 ) = rankk (Lθ ) < rankk (G), we can apply the induction hypothesis, so π1 (hc ) is bounded. It still remains to show that bc is bounded. But the equality x . c = y can be written:154     n x , n y ∈ U ∩ Lθ ;     kx . mx . sx . nx . nx . hc . bc = ky . my . sy . ny . ny , nx , ny ∈ Vθ whence:  ny = h−1 c . nx . hc . bc .

  S = K . FAt . ω. We may assume ω = Z(Sθ ) ∩ ω . (Vθ ∩ ω). Also, we know from 11.9 that K ∩ Z(Sθ ) is a maximal compact subgroup of Z(Sθ )R , and therefore also of PR . So K ∩ P ⊂ Z(Sθ ). Hence   S ∩ P = (K . FAt . ω) ∩ P = (K ∩ P ) . FAt . ω ⊂ S ∩ Z(Sθ ) . (Vθ ∩ ω). 153 The original French manuscript mistakenly uses the same argument for i = 3 as for i = 2, by asserting that (i) establishes the boundedness of zc , rather than only of szc . −1 154 Or, since x, y ∈ P : h . b y y = hx . bx . hc . bc = (hx . hc ) . (hc . bx . hc . bc ), so by =  ) and b (= n ) are bounded, . b . h . b (since h ∈ Z(S ) normalizes V ). Note that b (= n h−1 x c c c x y c θ θ x y since x, y ∈ S. 152 Write

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We just showed that hc is bounded. As for nx , ny , they are also bounded since x, y ∈ S, so bc is bounded. (iii) This completes the proof when X(G)k = {1}. In the general case, G can be written as the almost direct product G = G1 . Z where Z is a k-split torus and X(G1 )k = {1}. Let π : G → G = G/Z be the canonical projection. Then π defines a k-isogeny of G1 onto G , so X(G )k = {1}. On the other hand, π(P) and π(S) are a minimal parabolic k-subgroup and a maximal k-split torus of G respectively, and it immediately follows from the definitions that π(S) = S is a normal Siegel set of GR . What has already been proved therefore shows that π(CS ) is relatively compact. Since Z is a k-split torus, the homomorphism π maps Gk onto Gk , for each extension k of k, from which we conclude the existence of a compact subset Q ⊂ GR such that CS ⊂ Q . ZR . Let us write c ∈ CS in the form c = qc . rc (qc ∈ Q, rc ∈ ZR ) where qc and rc are uniquely determined up to multiplying by χ an element of the compact set Q−1. Q ∩ ZR . In view of (14.10), we have sc 1, χ whence rc 1, for every dominant weight χ. But the set of restrictions to Z of the dominant weights of G contains a finite index subgroup of X(Z), so rc is bounded, and this completes the proof.  15.3. Corollary. Assume k = Q. Let E be a symmetric subset of GQ that consists of elements with bounded denominators. Then ES = E ∩ S−1. S is finite. (In this statement, we have identified G with a Q-subgroup of GLn . Let us note that for a symmetric subset M of GQ , the condition of having “bounded denominators” is independent of the chosen matrix representation, because (see 7.1, 7.2) if ϕ : G → GLm is a Q-morphism, then the matrix entries of ϕ(g) are polynomials, with rational coefficients, in the entries gij of g and in det g −1 .) Proof. Since the matrix entries of the elements of E are rational numbers with bounded denominators, we have |det x| 1 on E, and since E is symmetric, we also have |det x| ≺ 1 so: (1)

|det x|  1

(x ∈ E).

Given w ∈ QW, put Gw = Uw . w . P and Ew = E ∩ Gw . We have: w−1. Gw ⊂ U−. P = U−. Z(S) . U. Denote by zy the component in Z(S) of an element y of U−. Z(S) . U. We then have:  zw −1. x = zx

(x ∈ Gw ).

On the other hand, w−1. Ew consists of elements of GQ with bounded denominators, whose determinant is, in view of (1), comparable to 1 in absolute value. Thus, we are reduced to proving that if F ⊂ U− Q . PQ consists of elements with bounded denominators that satisfy (2)

|det x|  1

(x ∈ F),

then: (3)

χ

|zx | 1

(x ∈ F;

χ a dominant weight).

It suffices to consider a positive integer multiple of each dominant weight. As in (14.4), given χ that is trivial on Z(G)0 ∩D(G), let π : G → GL(V) be an irreducible representation that is defined over Q, and let Dπ be a P-invariant line, on which P

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acts via χ. There exists a basis (ei ) of VQ that consists of eigenvectors of S, whose first vector spans Dπ . If x = u . zx . v (u ∈ U− , v ∈ U), then:    χ χ π(x) . e1 = π(u) . zx . e1 = zx . e1 + ci (u) . ei , i≥2

hence: (4)

χ

zx = π(x)11 .

But π(x)11 is a regular function on G that is defined over Q. Hence, it is a polynomial, with rational coefficients, in the entries xij of x and in (det x)−1 . After multiplying by a suitable power of det x, and taking (2) into account, we see that it suffices to prove that if R is a polynomial with rational coefficients in the entries of x, which does not vanish on F, then |R(x)| 1 on F. But if m is a multiple of the denominators of the coefficients of R and the elements of F, then we obviously have m |R(x)| ≥ 1 (x ∈ F), and the claim follows.  15.4. Theorem. Let k = Q and let S be a normal Siegel set of GR (with respect to K, P, S). Let A be a finite subset of GQ and Γ be an arithmetic subgroup of G. Then S . A has the Siegel property for Γ. Proof. We must show that, given q ∈ GQ , M = {γ ∈ Γ | S . A . q ∩ S . A . γ = ∅} is finite. We may assume155 that Γ ⊂ GZ ; then: E = A . Γ. q −1. A−1 ∪ A . q . Γ. A−1 is a symmetric subset of GQ , and the denominators of its elements are bounded, so we are reduced to 15.3 if156 G is connected. Let us now assume that G is not connected. By enlarging A, we may limit ourselves to the case where Γ ⊂ G0 . As the minimal parabolic Q-subgroups of G0 are all conjugate over Q, we have GQ = NG (P)Q . (G0 )Q (see 11.8). Thus, we can find finite subsets A , B of NG (P)Q and A , B of (G0 )Q such that A ⊂ A. A , A . q ⊂ B. B . On the other hand (12.7), there exists a normal Siegel set S that contains157 S . (A ∪ B ). Let C = A ∪ B . It suffices to show the finiteness of: M = Γ ∩ (S. C)−1. S. C. Since GR = K . (G0 )R , K . S = S , and Γ, C ⊂ G0 , we also have: M = Γ ∩ (S. C)−1. (S. C)

(S = S ∩ G0 ),

hence, we are reduced to the connected case that was already considered.



By combining 12.5, 13.1 and 15.4, we obtain the following theorem, which is the principal result in the reduction theory of arithmetic groups. 155 This

uses 9.7 (2). implicitly assumes the hypothesis of 15.2 that G is connected, but there is no such restriction here in 15.4. 157 Any a ∈ A ∪ B is in N (P) , so a S is a maximal Q-split torus of P. Hence, a S is G Q conjugate to S via an element of UQ . Since UR is connected, this implies that G0R . a contains an element of N(S). Therefore, a is in the subgroup H of 12.7. It also normalizes PR . Hence, A ∪ B is a (compact) subset of the subgroup Q of 12.7. 156 15.3

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15.5. Theorem. Let k = Q and let Γ be an arithmetic subgroup of G. There exists a normal Siegel set S over Q of G and a finite subset C of GQ such that Ω = S . C is a fundamental set for Γ in GR . This set is compact if rankQ (G) = 0 and has finite Haar measure if X(G0 )Q = {1}. For every finite subset A of GQ , the set Γ ∩ (ΩA)−1. (ΩA) is finite. We conclude this section by giving an interpretation of C. 15.6. Proposition. Let k = Q. Let Γ be an arithmetic subgroup of G, and P be a parabolic Q-subgroup.158 Then GQ is the union of a finite number of double cosets PQ . c . Γ. Let C be a finite subset of GQ . We have PQ . C . Γ = GQ if and only if there exists a normal Siegel set S, with respect to K, P, S, such that S . C . Γ = GR . Proof. It suffices to prove the first statement for P = P. Taking 15.5 into account, we then see that it is enough to establish the second statement. First, suppose that we have S . C . Γ = GR . Let g ∈ GQ . Then S . g meets only a finite number of translates S . c . γ (c ∈ C, γ ∈ Γ), by 15.4. Since S . C . Γ = GR , there exists a finite number of elements ci ∈ C, γi ∈ Γ (1 ≤ i ≤ m) such that:  S . ci . γi . S.g ⊂ i

Since C is finite, there exists a sequence of elements xj ∈ S, and an index i such that:159 (sxj )α → 0 (j → ∞),

xj . g ∈ S . ci . γi

(α ∈ QΔ).

−1

It then follows from 12.6 that ci . γi . g ∈ PQ , hence g ∈ PQ . C . Γ. Conversely, suppose that GQ = PQ . C . Γ. By 15.5, there exists a finite subset C of GQ and a normal Siegel set S such that S . C. Γ = GR . There exists a finite subset F of PQ such that C ⊂ F . C . Γ, whence GR = S . F . C . Γ. It then suffices to note (12.4) that S . F is contained in a normal Siegel set relative to K, P, S.  15.7. Corollary. Let H be a Q-group,160 and let P be the normalizer of a minimal parabolic Q-subgroup P of H0 . Let Γ be an arithmetic subgroup of H. There exists a finite subset C of H0Q that is contained in the intersection of the kernels of the elements of X(H0 )Q , such that HQ = Γ. C . PQ . Proof. It immediately follows from (11.8) that HQ = H0Q . PQ . Thus, we may assume that H is connected. Let U be the unipotent radical of H. Then H = L . U is the semidirect product of a reductive Q-group L by U (7.15) and X(H)Q can be identified with X(L)Q , by restriction. Let M be the identity component of the kernel of the elements of X(L)Q and let S be the largest central Q-split torus of L. Then L = M . S, and M ∩ S is finite (10.7). Obviously S . U ⊂ P, and P ∩ M (resp. P ∩ L) is a minimal parabolic Q-subgroup of M (resp. L). The spaces M/(P ∩ M), L/(P ∩ L) and H/P are Q-isomorphic. From (11.8), we see that: MQ /(P ∩ M)Q = HQ /PQ . G is not connected, P may be a parabolic subgroup of G0 , or of G. is assumed here that S is not compact. If that is not the case, then rankQ G = 0, so P = G, so it is obvious that PQ . C . Γ = GQ . 160 The point of this corollary is that H is not assumed to be reductive (or connected). 158 When 159 It

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Therefore, if C is a finite subset of MQ such that: (M ∩ Γ) . C . (P ∩ M)Q = MQ , we also have Γ. C . PQ = HQ .



Finally, to provide a convenient reference, let us mention yet another simple consequence of the Siegel property. 15.8. Proposition. Let k = Q and let S be a normal Siegel set of GR . Let A be a finite subset of GQ . Then, for every g ∈ GR , the set S ∩ g . Γ. A is finite. Proof. Let V = {γ ∈ Γ | S . A−1. γ ∩ S . A−1 = ∅}. The set V is finite in view of (15.4), and it is immediate that if x = g . σ . b (σ ∈ Γ, b ∈ A) belongs to S ∩ g . Γ. A, then each of the other elements of this set is of the form g . σ . v . a (v ∈ V, a ∈ A).  Appendix: Commensurator and the Siegel property. In addition to GR and Γ, the condition (F2 ) involves the group GQ , which is not directly associated to GR and Γ. Here, we will show that in fact, one may replace GQ with a group that is closely related to it, but may be larger, and is defined from GR and Γ. This leads to a stronger formulation of (F2 ), which may be applied to any group and its subgroups (15.13), and is also satisfied by arithmetic groups (15.15). 15.9. Definition. Let H be a group and L be a subgroup. The commensurator of L in H, which we denote by161 CommH (L), is the set of all h ∈ H, such that h L is commensurable with L. Since the notion of “being commensurable” is an equivalence relation on the collection of all subgroups of H, it is immediate that CommH (L) is indeed a group, which depends only on the commensurability class of L. 15.10. Lemma. Let H, H be groups, π : H → H be a surjective homomorphism with finite kernel N, and L be a subgroup of H. Then    CommH (L) = π−1 CommH π(L) .    Proof. It is clear  that CommH (L) ⊂ π−1 CommH π(L)  . Conversely, let   −1  π(L) . Since N is finite, the subgroups π π(L) = L . N and x ∈ π−1 Comm H  π−1 π(x L) = x L . N are commensurable. But these groups are commensurable with  L and x L respectively, so x ∈ CommH (L). 15.11. Lemma. Let G be connected and almost simple over Q, and let L be a subgroup of GQ . (i) If L is infinite, and CommG (L) is Zariski-dense, then L is Zariski-dense. (ii) If G is simple over Q, and L is Zariski-dense, then CommG (L) ⊂ GQ . (A connected Q-group is simple (resp. almost simple) over Q is it does not have any proper normal Q-subgroup that is = {e} (resp. that has dimension > 0).) 161 The

original French manuscript denotes the commensurator of L by C(L).

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Proof. Given a subgroup M of G, we denote by A (M) the smallest algebraic subgroup of G that contains M. (This is also the closure of M in the Zariski topology.) If M ⊂ GQ , then A (M) is defined over Q. If M has finite index in M, it is immediate that A (M ) has finite index in A (M), so A (M )0 = A (M)0 . It follows that if M is commensurable with M, then A (M)0 = A (M )0 . (i) Since L is infinite, A (L)0 has dimension > 0. In view of the preceding paragraph, this is a group that is defined over Q, and is normalized by CommG (L). But since CommG (L) is Zariski dense in G, this implies that A (L)0 is normal in G. We therefore have A (L)0 = G, from which (i) follows. (ii) We identify G with a subgroup of SLn . Then C[G] is the algebra generated by 1 and the matrix entries gij of elements of G. We let g act on C[G] by f → gf , where: f (x) = f (g −1. x . g)

(g, x ∈ G).

g

We obtain a rational representation σ of G on the vector subspace of C[G] that is spanned by the gij , and this representation is defined over Q. Since G is simple over Q, and in particular, has center consisting only of {e}, σ is a Q-isomorphism of G onto σ(G). Therefore, g ∈ G belongs to GQ if and only if the map f → gf sends Q[G] into Q[G]. Let g ∈ CommG (L) and f ∈ Q[G]. We can write: g

f = f0 + c1 f1 + . . . + cm fm

(fi ∈ Q[G]; 0 ≤ i ≤ m),

with (1, c1 , . . . , cm ) linearly independent over Q. Let x ∈ L be such that we have g −1. x . g ∈ L. Then: f0 (x) + c1 f1 (x) + . . . + cm fm (x) = f (g −1. x . g) ∈ Q, hence, fi (x) = 0, (1 ≤ i ≤ m). Since L ∩ g −1. L . g has finite index in L, it is also  Zariski-dense in G, so fi = 0 (1 ≤ i ≤ m), and gf = f0 ∈ Q[G]. Remarks. (i) holds if Q is replaced with an arbitrary field k. The previous argument shows that A (L)0 is normal in G. In view of its definition, it is k-closed. But in a connected reductive k-group, each connected normal subgroup is defined over a separable extension of k, therefore162 A (L)0 is defined over k, so once again, we have A (L)0 = G. (ii) and its proof also remain valid for an arbitrary field if the hypothesis “G is simple over k” is replaced with “G is simple over k and isomorphic to its adjoint group.” 15.12. Proposition. Assume G is connected and semisimple. Let N be the largest normal Q-subgroup of G whose set of real points is compact, π : G → G = G/N be the canonical projection, and Γ be an arithmetic subgroup of G. (i) If N is finite, then Γ is Zariski-dense in G. (ii) We have CommG (Γ) = π−1 (GQ ). Proof. (i) We obviously may assume G = {e}. Then G = N, so GR is not compact. Since GR /Γ has finite invariant volume (13.2), Γ is infinite. On the other hand, in view of (7.13 (2)), CommG (Γ) ⊃ GQ . But, by a theorem of Rosenlicht (cf. [5, Corollary 18.3]), GQ is dense in G. We then apply (15.11 (i)) to each Q-simple factor of G (because of 8.10). 162 Every k-closed subset of affine space is defined over a purely inseparable extension of k. So the subgroup A (L)0 is defined over both a purely inseparable extension and a separable extension. It therefore must be defined over k.

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(ii) In view of (15.10) and the fact that π(Γ) is arithmetic (8.11), it suffices163 to consider the case where N = {e}, and to show that, in this case, CommG (Γ) = GQ . The group G is the direct product of Q-simple groups Gi . Further, in view of (8.10), Γ is commensurable with the product of the groups Γ ∩ Gi , and Γ ∩ Gi is arithmetic in Gi . Thus, we are reduced to the case where G is simple over Q, and GR is not compact. Then, Γ is Zariski-dense by (i), so CommG (Γ) ⊂ GQ by (15.11 (ii)). Since the reverse inclusion follows from (7.13 (2)), this concludes the proof.  15.13. Given a group H, a subgroup L and a subset Ω of H, we consider the following conditions: (F− 2 ) For every c ∈ L, the set {x ∈ L | Ω . c ∩ Ω . x = ∅} is finite; (F+ 2 ) For every c ∈ CommH (L), the set {x ∈ L | Ω . c ∩ Ω . x = ∅} is finite. It is immediate164 that if Ω satisfies (F+ 2 ) for L, then it also satisfies this condition for every subgroup L that is commensurable with L. If Ω is a “fundamental set,” i.e., satisfies in addition (F1 ): Ω . L = H, then there also exists a fundamental set Ω for L ; it suffices to take Ω = Ω . D, where D is a system of representatives for L/(L ∩ L ). However, it is unclear whether the following holds: if Ω satisfies −   (F1 ) and (F− 2 ) for L, then we can find a set Ω that satisfies (F1 ) and (F2 ) for L . If H = GR and L is arithmetic, then CommG (L) ⊃ GQ (by 7.13), so (F2 ) is + intermediate between (F− 2 ) and (F2 ). 15.14. Proposition. Assume that G is connected and semisimple, and let Γ be an arithmetic subgroup of G. Then every normal Siegel set S of GR (see 12.3) satisfies the condition (F+ 2 ) for Γ. Proof. Let N be the largest normal Q-subgroup of G whose set of real points is compact and let π : G → G = G/N be the canonical projection. We have already noted that π(S) = S is contained in a normal Siegel set of GR (12.3). On the other hand (8.11), Γ = π(Γ) is arithmetic in G . Let c ∈ CommG (Γ). By 15.12, we have π(c) ∈ GQ . Since S is contained in a normal Siegel set, the set of x ∈ Γ , such that S. c ∩ S. x = ∅ is finite. Hence: {x ∈ Γ | S . c ∩ S . x = ∅} consists of a finite number of classes modulo N ∩ Γ. But NR is compact, so N ∩ Γ is finite.  15.15. Theorem. Assume that G is connected and semisimple. Let Γ be a subgroup of GR that is commensurable with an arithmetic subgroup of G. Then, there exists a Siegel set S of GR , relative to Q, and a finite subset C of the commensurator CommG (Γ) of Γ, such that GR = S . C . Γ. The set S . C satisfies condition (F+ 2 ) of (15.13). 163 Note that G is the almost-direct product of N with a connected semisimple Q-subgroup H. Then Γ is commensurable with H ∩ Γ (because the compactness of NR implies that N ∩ Γ is finite), which is centralized by N. Thus, we are reduced to calculating CommH (Γ), to which (15.10) applies. 164 Suppose L is a finite-index subgroup of L , and let {γ , . . . , γ } be coset representatives. It r 1 is obvious that the set for L is contained in the set for L . Conversely, if {x ∈ L | Ω . c . γi ∩Ω . x = ∅} is finite for every i, then the set for L is finite.

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Proof. If Γ is arithmetic,165 this follows from (15.7) and (15.14). From there, we pass to a subgroup of GR that is commensurable with an arithmetic subgroup by using the remarks made in (15.13).  We limited ourselves to the case where G is connected and semisimple in order to avoid some technical complications, but the statement generalizes to the case of a reductive Q-group. Verifying this is left as an exercise for the reader. 16. Fundamental sets and minima In this section, G is a connected reductive Q-group, P is a proper parabolic Q-subgroup (which is assumed to be minimal from 16.5 onwards), S is a maximal Q-split torus of P, P = M.S.U is the canonical decomposition of P relative to S, and K is a maximal compact subgroup of GR whose Lie algebra is orthogonal to that of SR . Finally, Γ is an arithmetic subgroup of G and C is a finite subset of GQ such that GQ = Γ. C . PQ (cf. 15.6). 16.1. The aim of this section is to link the fundamental sets of the second type to minimum conditions on functions of type (P, χ). The method used is the analogue “at infinity,” i.e., for GR , of the one used by Godement-Weil in the case of adelic groups [17]. However, it is technically more involved because, contrary to what happens for adelic groups, it is necessary to consider sets with more than one “cusp,” which amounts to saying we cannot always assume that C consists only of {e}. Of the material in previous sections, the following will be crucial: (a) Mahler’s criterion (1.9, 8.2); (b) The compactness criterion (§8); (c) The finiteness of PQ \GQ /Γ (15.6). In our exposition of the theory, this last fact was obtained as a consequence of the existence of fundamental sets of the form S . C, so this section does not provide an independent proof of the existence of such fundamental sets. However, (c) can be proved166 directly from the compactness criterion for adelic groups [17]. Aside from this one fact, this section does not use any of the results that were proved in §9 or §13. 16.2. Lemma. Let A be a finite subset of GQ . Let χ ∈ X(P)Q be dominant (14.4), let Φ > 0 be of type (P, χ) and let Φ be a continuous strictly positive function that is comparable to Φ. Then: inf Φ (u) > 0.

u∈Γ. A

If Φ = Φπ is standard (14.4), then Φ has a minimum > 0 on Γ. A. Proof. In view of the last remark of 14.2, it suffices to prove the second claim. Thus, let Φ(g) = π(g) . e0 , where π : G → GL(V) is a Q-morphism and e0 ∈ V is an element that spans a P-invariant line. Then, there exists a lattice L of VQ that is invariant under Γ. Since A is finite, there exists q ∈ Q∗ such that π(A) . L ⊂ q . L. 165 Γ

is arithmetic iff Γ ⊂ GQ (since Γ is commensurable with an arithmetic group). proof is in Theorem 13.26 on page 212 of the book [Ra] listed on page vii.

166 Another

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It follows that π(Γ. A) . e0 is contained in the set q . L  {0}, which is discrete in VR , and the claim follows.  16.3. Recall that if two parabolic Q-subgroups R, R of G are conjugate, then we have a canonical isomorphism between X(R)Q and X(R )Q (11.9). We will identify these two groups under this isomorphism. Let c ∈ G and f be a function on G. We will denote by rc f the image of f under the right translation: rc (x) = f (x . c) (x ∈ G). If g ∈ GR and f is of type (R, χ), then rc f is obviously of type (c R, χ). 16.4. Lemma. We assume X(G)Q = {1} and rankQ (G) = 1. Let Φ > 0 be a continuous function that is comparable to a strictly positive function of type (P, χ), where χ is dominant and non-zero. Let Ψ(g) = inf u∈Γ. C Φ(g . u), (g ∈ GR ). Then the function Ψ is bounded above on GR . Proof. It suffices to prove this for Φ of type (P, χ). For each d > 0, let: (1)

Ed = {x ∈ GR | Ψ(x) ≥ d}.

Obviously, Ed is closed, and is a decreasing function of d, and the lemma is equivalent to the existence of a constant d0 > 0 such that Ed = ∅ if d > d0 . By definition, Ψ is invariant on the right under Γ, so: (2)

Ed . Γ = E d



(d > 0).

On the other hand, it is obvious from the definition that d>0 Ed = ∅. Thus, it suffices to prove that Ed /Γ is compact in GR /Γ. We can limit ourselves to showing this for standard Φ. Let μ be the projection of G onto its adjoint group Ad G. Then μ(Γ) is arithmetic (8.11) and the canonical map GR /Γ → (Aut g)R /μ(Γ) is proper (8.5). Thus, it suffices to consider the case where the center of G consists only of {e}. We may therefore assume that G is semisimple, and is isomorphic to its adjoint group. Since Ed is closed, and invariant under Γ, its image Ed /Γ in GR /Γ is closed. On the other hand, X(G) = {1}, since G is connected and semisimple. Taking into account Mahler’s criterion, in the form of 8.2 (iv), applied to the adjoint representation, we see that the compactness of Ed /Γ will follow from the following claim: (∗) Let L be a lattice of gQ and d > 0. If zj ∈ L, gj ∈ Ed (j = 1, 2, . . .) such that lim Ad gj (zj ) = 0, then there exists j0 such that zj = 0 for j ≥ j0 . j→∞

Proof of (∗). We may assume L is invariant under Γ (7.13). Let us first show that zj is nilpotent for j sufficiently large. There exist polynomials Pi on g that are invariant under Ad(G), such that: det(Ad x − t . I) = (−t)n +

n 

Pi (x) . tn−i .

i=1

The Pi have rational coefficients, if we take coordinates with respect to a basis of L. Thus, there exists a constant a > 0 such that: ( ( (Pi (z)( ≤ a ⇒ Pi (z) = 0 (z ∈ L, 1 ≤ i ≤ n).   We have Pi (zj ) = Pi Ad gj (zj ) → 0 so, for j sufficiently large, we have Pi (zj ) = 0 (1 ≤ i ≤ n), so zj is nilpotent. It follows from 11.10 that every nilpotent element of gQ is conjugate via GQ to an element of uQ . Since GQ = Γ. C . PQ , we can find γj ∈ Γ, cj ∈ C and yj ∈ uQ

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III. FUNDAMENTAL SETS WITH CUSPS

such that zj = Ad(γj . cj ) . yj , (j ≥ j0 ). Note that167 yj ∈ uQ ∩ Ad c−1 j (L) for every j ≥ j0 . Passing to a subsequence, replacing gj . γj with gj (which is permissible, in view of (2)), and also replacing L with a a Γ-invariant lattice that contains L ∪ Ad c−1 (L), we see that we are reduced to proving: (∗∗) Suppose gj ∈ Ed , zj ∈ u ∩ L, (j = 1, 2, 3 . . .), and c ∈ C such that Ad gj . c(zj ) → 0. Then, zj = 0 for j sufficiently large. −1

Proof of (∗∗). The group Γ = c Γ is arithmetic. Therefore (see 8.7),168 there exist compact subsets η ⊂ MR and ω ⊂ UR such that: GR = K . PR = K . η . QA . ω . (Γ ∩ P). Hence, we may find σj ∈ Γ ∩ P such that: gj . c . σ−1 j ∈ K . η . QA . ω. By replacing gj with gj . c . σj . c−1 ∈ gj . Γ, we see that it suffices to establish (∗∗) under the assumption that gj . c ∈ K . η . QA . ω. Put: (kj ∈ K, mj ∈ η; aj ∈ QA; uj ∈ ω).

gj . c = kj . mj . aj . uj

Since kj . mj remains in a compact set, the assumption of (∗∗) is equivalent to: (3)

lim Ad aj . uj (zj ) = 0.

j→∞

χ

Moreover, the condition gj ∈ Ed implies aj 1. Let α be the unique169 simple Q-root of G. From (χ, α) > 0, we obtain that χ = m . α (m > 0), so: aαj 1

(4)

(j = 1, 2, . . .).

The set of positive Q-roots consists only of {α} or {α, 2α}. We have: u = gα + g2α ,

[gα , gα ] ⊂ g2α ,

[u, g2α ] = {0},

(g2α being zero if 2α is not a Q-root). These two subspaces are defined over Q. Thus, there exist lattices L ⊂ gα,Q and L ⊂ g2α,Q such that L ∩ u has finite index in L + L and such that we have: zj = zj + zj

(zj ∈ L ; zj ∈ L ; j = 1, 2, . . .).

Since U acts trivially via the adjoint representation on u/g2α , we have: Ad uj (zj ) ≡ zj mod g2α

(j = 1, 2, . . .),

so: Ad aj . uj (zj ) ≡ aαj . zj mod g2α zj

In view of (4), we therefore have → 0. But zero for j sufficiently large. We then have:

(j = 1, 2, . . .). zj

belongs to the lattice L , so it is

 Ad aj . uj (zj ) = a2α j . zj

whence also zj = 0 if j is sufficiently large. 167 We

(j ≥ j0 ), 

have corrected a minor error in the original French manuscript, which defines the 

−1 (L) in u , and erroneously states that it contains y for every lattice M = uQ ∩ j Q c∈C Ad c j ≥ j0 . This would require a union, rather than an intersection. 168 8.7 is applied to the subgroup M · U of P. 169 The root is unique because rank (G) = 1. Q

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101

16.5. Notation. Let θ ⊂ QΔ and t > 0. We put: P(θ, t) = {p ∈ PR | |pα | ≤ t, (α ∈ θ)}.

(1)

We will also write P(t) for P(QΔ, t). It is clear that:170 P(θ ∪ θ , t) = P(θ, t) ∩ P(θ , t)

(2)

(θ, θ ⊂ QΔ).

16.6. Lemma. Let θ ⊂ QΔ and let χ ∈ X(P)Q be dominant such that (χ, α) > 0 for α ∈ θ. Let Φ > 0 be a continuous function that is comparable to a function that is > 0 and of type (P, χ), and let μ > 1 be a real number. For b ∈ C, put: Ωb,μ (Φ) = Ωb,μ = {g ∈ GR | Φ(g . b) ≤ μ . Φ(g . u), (u ∈ Γ. C)}. Then there exists t > 0 such that Ωb,μ . b ⊂ K . P(θ, t) and GR = K . P(θ, t) . C−1. Γ. Proof. Let Φ > 0 be of type (P, χ) and comparable to Φ. Then, there exist constants d, d > 0 such that: d . Φ (g) ≤ Φ(g) ≤ d. Φ (g)

(g ∈ GR ).

We immediately deduce that: Ωb,μ (Φ) ⊂ Ωb,μ (Φ )

(μ = μ . d. d−1 ).

Therefore, it suffices to prove the lemma for standard Φ. Every character of P is obviously  1 on171 b K ∩ P. In view of (16.5 (2)), our proof of the first claim only needs to consider the case where θ consists of a single element, say, α. We have the decomposition Pα = Mα . Sα . Vα (11.7). Let Lα be the connected Q-group whose Lie algebra is spanned by gβ + [gβ , g−β ] + g−β (β = α, 2α). Then Lα is an almost direct factor of Mα , and we have: (1)

Mα = Lα . Qα ,

with Qα anisotropic over Q, and rankQ Lα = 1 (11.7). Moreover, Tα = (S ∩ Lα )0 is a maximal Q-split torus of Lα , whose Lie algebra is generated by an element hα that satisfies: ν(hα ) = 2(ν, α) . (α, α)−1



(ν ∈ s∗ ).

The character χ can be written in the form χ = β dβ . Λβ (dβ ≥ 0). In view of the hypothesis, dα > 0. Since the restriction of χ to Tα is equal to dα . Λα , it follows that: χ|Tα = m α|Tα

(2)

(m > 0).

Let D be a finite subset of Lα,Q such that: Lα,Q = (Lα ∩ Γ) . D . (P ∩ Lα )Q

(3)

(whose existence is assured by 15.6). Let us denote by ζ the restriction of rb Φ to b Lα . In view of (2), it is a function that is > 0 and of type b(P ∩ Lα ), m . α . We can write: g = kg . qg . g . sg . vg (kg ∈ b K, qg ∈ bQα , g ∈ b Lα , sg ∈ b Sα , vg ∈ b Vα ).

(4)

future reference, note also that K . P(θ ∪ θ , t) = K . P(θ, t) ∩ K . P(θ , t). ⊂ K . P(θ, s), for some s, and we have P(θ, s) . P(θ, t) ⊂ P(θ, s + t). Therefore, we may replace K with b K in the proof. 170 For 171b K

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III. FUNDAMENTAL SETS WITH CUSPS

We will prove below: (∗) There exists a constant μ > 0 such that, for every g ∈ Ωb,μ , we have ζ(g ) ≤ μ. ζ(g . z), (z ∈ b(Γ ∩ Lα ) . b D). Let us assume this temporarily. By 16.4, applied to b Lα , this implies there exists δ > 0 such that: (5)

ζ(g ) ≤ δ < ∞

(g ∈ Ωb,μ ). b

In view of (2) and (5), if we let a(g ) be the component of g in Tα with respect to the decomposition172 b Lα = K. M. b Tα . U , where K = b K ∩ b Lα is a maximal compact subgroup of b Lα , then there is δ > 0, such that: (6)

a(g )m α ≤ δ

(g ∈ Ωb,μ ).

The component ag of g in QbA relative to the decomposition: GR = b K . b MR . QbA . b UR is equal to the product of a(g ) by the element sg of (4). Since α is trivial on Sα , this yields a constant t > 0 such that: (7)

aαg ≤ t

(g ∈ Ωb,μ ).

By definition, this means that: g ∈ b K .b P(α, t ). Write b = k . p (k ∈ b K, p ∈ PR ). There exists t > 0 such that p . P(α, t ) ⊂ P(α, t), hence g . b ∈ b K . P(α, t), and the first claim of the lemma follows. We now prove (∗). We have:173 rb Φ(g) = rb Φ(kg . g . sg )  rb Φ(g ) . rb Φ(sg ), rb Φ(g)  η(g) . ζ(g ) (g ∈ Ωb,μ ; η(g) = rb Φ(sg )).   Let z ∈ b (Γ ∩ Lα ) . D . Then z normalizes b Vα , and centralizes b Qα and b Sα , so:

(8)

g . z = kg . g . z . sg . qg . vg

(vg = z −1. vg . z ∈ b Vα ).

It follows that: (9)

rb Φ(g . z)  η(g) . rb Φ(g . z) = η(g) . ζ(g . z)

(g ∈ Ωb,μ ).

Let us write z in the form z = σ . d (σ ∈ (Lα ∩ Γ), d ∈ D). There exist: b

γ(d, b) ∈ Γ,

c(d, b) ∈ C,

b

p(d, b) ∈ PQ ,

such that: d . b = γ(d, b) . c(d, b) . p(d, b), whence: rb Φ(g . z) = Φ(g . σ . d . b)   = Φ g . σ . γ(d, b) . c(d, b) . p(d, b)   = |p(d, b)χ | . Φ g . σ . γ(d, b) . c(d, b) . 172 The reason for replacing δ with δ is that ζ is invariant under K, not b K. But we do need to use b K, rather than K, in most of the proof, because K usually does not contain a maximal compact subgroup of b Lα . The original French manuscript erroneously works with K throughout. 173 We have  instead of =, because r Φ is probably not invariant under b K. b

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16. FUNDAMENTAL SETS AND MINIMA

103

The element b is fixed, and d runs through a finite set, so p(d, b) takes a finite number of values. Hence, there exists δ > 0 such that: (10)

inf rb Φ(g . σ . d) ≥ δ σ,d

inf

γ∈Γ; c∈C

Φ(g . γ . c)

(σ ∈ b(Lα ∩ Γ), d ∈ b D).

But by the assumption that g ∈ Ωb,μ : (11)

μ . inf Φ(g . γ . c) ≥ Φ(g . b). γ,c

Then, using (8), (9), (10), (11), we see that for μ = μ/δ > 0: (12)

η(g) . ζ(g )  Φ(g . b) ≤ μ. inf rb Φ(g . σ . d)  μ . η(g) . inf ζ(g . σ . d), σ, d

σ, d

for every g ∈ Ωb,μ . Dividing the two ends of (12) by η(g) yields (∗), which concludes the proof of the first assertion. Since the constant μ is > 1, we may find, given g ∈ GR , some σ ∈ Γ and b ∈ C such that: Φ(g . σ . b) ≤ μ . Φ(g . u)

(u ∈ Γ. C).

Then, we have g . σ ∈ Ωμ,b so g . σ . b ∈ K . P(θ, t) for suitable t, depending only on b. Since C is finite, there exists t such that this holds for every b ∈ C, and this proves the second assertion of the lemma.  16.7. Theorem. Let χ ∈ X(P)Q be a dominant character such that (χ, α) > 0 for every α ∈ QΔ, and let Φ > 0 be a function of type (P, χ). Then there exists a (normal) Siegel set S relative to P, S, such that for every g ∈ GR , the function Φg : u → Φ(g . u) (u ∈ Γ. C) attains its minimum at a point of S ∩ g . Γ. C. In particular, GR = S . C−1. Γ. Proof. Let μ be a constant > 1. Given g ∈ GR , we can find σ ∈ Γ, b ∈ C such that: Φ(g . σ . b) ≤ μ . Φ(g . γ . c)

(γ ∈ Γ, c ∈ C).

By 16.6, there exists t > 0, independent of b ∈ C, such that: g . σ . b ∈ K . P(t). Let: Γ = Γ ∩



 c−1. Γ. c .

c∈C

This is an arithmetic group (7.13) that satisfies: (1)

c . Γ ⊂ Γ. c

(c ∈ C).

In view of the compactness criterion (8.7), there exists a compact set ω ⊂ (M . U)R such that: (2)

(M . U)R = ω(Γ ∩ M . U).

We want to show that the Siegel set S = K . QAt . ω satisfies 16.7. By (2), there exists τ ∈ Γ ∩ P such that g . σ . b . τ ∈ S. But in view of (1), we can write: (3)

g . σ . b . τ = g . γ1 . b

which shows that: g ∈ S . C−1 . Γ.

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(γ1 ∈ Γ),

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III. FUNDAMENTAL SETS WITH CUSPS

Since Φ is right invariant under Γ ∩ P, we have Φ(g . γ1 . b) = Φ(g . σ . b), so: Φ(g . γ1 . b) ≤ μ . Φ(g . u)

(u ∈ Γ. C).

This shows that, given μ > 1, we can, for each g ∈ GR , find x ∈ g . Γ. C∩S such that Φ(x) ≤ μ . Φ(y) for every y ∈ g . Γ. C. Since μ is an arbitrary number > 1, the first assertion therefore follows from the fact that g . Γ. C ∩ S is a finite set (15.8).  16.8. In the following result, we no longer assume that (χ, α) > 0 for every α ∈ QΔ. Furthermore, it turns out that for at least one application, the assumptions on Φ need to be weakened slightly. Let us note that if Φ > 0 is of type (P, χ), (χ ∈ X(P)), and if Φ > 0 is comparable to Φ, then: Φ (x . p)  Φ (x) . Φ (p)  Φ (x) . |pχ |

(x ∈ GR , p ∈ PR ).

Indeed: Φ (x . p)  Φ(x . p) = Φ(x) . Φ(p) = Φ(x) . |pχ |  Φ (x) . |pχ |. 16.9. Theorem. Let χ ∈ X(P)Q be a dominant character, let θ = {α ∈ QΔ | (χ, α) > 0}, and let θ = QΔ  θ. Let Φ > 0 be a continuous function on GR that is comparable to a function that is > 0, is of type (P, χ), and is right invariant under Γ ∩ P and Lθ (cf. 11.7). Then there exists a finite subset C of GQ that contains C, and a (normal) Siegel set S relative to K, P, S, such that for every g ∈ GR , the function Φg : u → Φ(g . u), (u ∈ Γ. C ) attains it minimum at a point of g . Γ. C ∩ S. Proof. As above, let:   Γ = Γ ∩ c−1. Γ. c . c∈C

This is an arithmetic group that satisfies c . Γ ⊂ Γ. c (c ∈ C). By 15.6, we can find a finite subset D of Lθ ,Q such that: Lθ ,Q = (Γ ∩ Lθ ) . D . (P ∩ Lθ )Q . The preceding theorem implies the existence of a constant t > 0 such that   (1) Lθ ,R = (K ∩ Lθ) . (P ∩ Lθ) (t ) . D−1. (Γ ∩ Lθ ). Let μ > 1, and let σ ∈ Γ, b ∈ C such that: Φ(g . σ . b) ≤ μ . Φ(g . γ . c)

(γ ∈ Γ, c ∈ C).

We will first show: (∗) There exists y ∈ g . Γ. C . D ∩ K . P(θ , t ) such that Φ(y) = Φ(g . σ . b). We have the decomposition GR = K . Mθ ,R . Lθ ,R . Sθ ,R . Vθ ,R (11.7), which allows us to write: x = g .σ.b = kx . mx . x . sx . vx , (kx ∈ K, mx ∈ Mθ , x ∈ Lθ , sx ∈ Sθ , vx ∈ Vθ ).

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16. FUNDAMENTAL SETS AND MINIMA

105

Let174 a(x ) be the component in S ∩ Lθ of the element x ∈ Lθ ,R of the above decomposition. In view of (1), we can find τ ∈ Γ ∩ Lθ , d ∈ D such that: a(x . τ . d)α ≤ t

(α ∈ θ ).

Let y = x . τ . d. Since Lθ centralizes Sθ and normalizes Vθ , we have: a(y )α = a(x . τ . d)α ≤ t

(α ∈ θ ),

so: y ∈ K . P(θ , t ). On the other hand, since Φ is invariant on the right under Lθ ,R (by assumption), we have Φ(y) = Φ(x). Finally, we see from the relation c . Γ ⊂ Γ. c that we have y = g . σ . b . τ . d ∈ g . Γ. b . d, which proves (∗). We have Φ(y) ≤ μ . Φ(g . u) for u ∈ Γ. C, so also for u ∈ Γ. C . D, since Φ is invariant on the right under D. Lemma 16.6 then establishes the existence of a constant t > 0, independent of g, such that y ∈ K . P(θ, t ), whence, for a suitable175 t > 0: y ∈ K . P(t). Let: Γ = Γ ∩

 

 u−1. Γ. u .

u∈C . D

This is an arithmetic group that satisfies u . Γ ⊂ Γ. u (u ∈ C . D). Let ω be a compact subset of (M . U)R such that (M . U)R = ω(Γ ∩ M . U) (cf. 8.7), and let S = K . QAt . ω. We can then find τ ∈ Γ ∩ M . U such that: z = y . τ = g . σ . b . d . τ ∈ g . Γ. b . d ∩ S, and we have: Φ(z) = Φ(y) ≤ μ . Φ(g . u)

(u ∈ Γ. C . D).

Since S ∩ g . Γ. C . D is finite (15.8) and since μ is an arbitrary number > 1, this proves the theorem (with C = C . D).  16.10. Corollary. The function Ψ : g → min  Φ(g . u) is continuous on GR . u∈Γ. C

Proof. The function Ψ is invariant on the right under Γ. Thus, it suffices to study it for g running through S . C , where S is a suitable Siegel set, which we may assume to be open. Let A = {γ ∈ Γ | S . C. γ ∩ S . C −1 = ∅}. In view of 16.9, we have: Ψ(g) = min  Φ(g . u) u∈A . C

(g ∈ S . C ).

Since A is finite (15.4), we see that if g ranges over the open set S . C , then the function Ψ is the minimum of a finite number of continuous functions, so Ψ is continuous.  174 Here is a slightly different explanation of this part of the proof. From (1), we have a decomposition x = k1 . 1 . d−1. τ−1 . Let y = x . τ . d. Since Lθ centralizes Mθ and Sθ , and normalizes Vθ , we have y = 1 ∈ P(θ , t ). So y ∈ K . P(θ , t ). 175 Let t = max(t , t ).

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III. FUNDAMENTAL SETS WITH CUSPS

16.11. We will conclude with a theorem about successive minima, which generalizes (1.13), (1.14). The statement here is more complicated, largely because the equality GQ = Γ. PQ , which holds in the special case where Γ = GL(n, Z) and P is a standard parabolic subgroup of GLn , does not necessarily hold in the general situation considered here. Let: QΔ

= θ1 ∪ . . . ∪ θs

be a partition of QΔ and let ζi = θi+1 ∪ . . . ∪ θs (1 ≤ i ≤ s − 1). We put: Pi = Pζi ,

Li = Lζi

P0 = L0 = G,

(1 ≤ i ≤ s − 1),

Ls = Ps = {e},

C0 = C,

Γ0 = Γ.

By induction on i, we define an arithmetic subgroup Γi of G and a finite subset Ci of Pi,Q that contains e, on which every χ ∈ X(Pi )Q is trivial, such that:176  Γi = c−1. Γi−1 . c (1 ≤ i ≤ s − 1), c∈Ci−1

Pi,Q = (Γi ∩ Pi ) . Ci . (Pi ∩ P)Q , (cf. 15.7). We denote by χi a dominant weight such that: (χi , α) > 0

⇔

α ∈ θi

(α ∈ QΔ; i = 1, . . . , s),

and by Φi a standard function (14.4) of type (P, χi ). Finally, we put D = C0 . C1 · . . . · Cs−1 . 16.12. Theorem. We keep the notation of 16.11. (i) Let μ ≥ 1. There exists a constant t > 0 such that if: g ∈ GR ,

γi ∈ Γi ,

and

c i ∈ Ci ,

(0 ≤ i ≤ s − 1)

are such that for all ui−1 ∈ (Γi−1 ∩ Pi−1 ) . Ci−1 and 1 ≤ i ≤ s, we have: Φ(g . γ0 . c0 . . . γi−1 . ci−1 ) ≤ μ . Φi (g . γ0 . c0 . . . γi−2 . ci−2 . ui−1 ),

(1)

then g . γ0 . c0 . . . γs−1 . cs−1 ∈ K . P(t). (ii) There is a Siegel set S, relative to K, P, S, such that for every g ∈ GR , there exist γi ∈ Γi , ci ∈ Ci (0 ≤ i ≤ s − 1) satisfying (1) with μ = 1, and g . γ0 . c0 . . . γs−1 . cs−1 ∈ S ∩ g . Γ. D. Proof. Since the theorem is contained in 16.6, 16.9 for s = 1, we proceed by induction on s. (i) Let V1 be the unipotent radical of P1 and π : P1 → G = P1 /V1 be the canonical projection. The function Φi is left K-invariant and right V1 -invariant, so it defines a function Φi on G that satisfies:   (k ∈ K ∩ P1 , z ∈ P1,R ; 1 ≤ i ≤ s). Φi (k . z) = Φi π(z)   It is clear that Φi is of type π(P), χi (2 ≤ i ≤ s) and that ζ1 can be identified with the set of simple Q-roots of G , with relative to π(S) and for an ordering that is associated to π(P). Let us write g . γ0 . c0 = k . z (k ∈ K, z ∈ P1,R ). In view of 176 The

original French manuscript defines Γi =

c∈C . C1 ...Ci−1

c−1. Γi−1 . c, but that is more

complicated than necessary.

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107

the induction hypothesis, there exists a constant t , not depending on g, γi , ci such that we have:   π(z . γ1 . c1 . . . γs−1 . cs−1 ) ∈ π(K ∩ P1 ) . π(P) (t ) , which implies: g . γ0 . c0 . . . γs−1 . cs−1 ∈ K . P(ζ1 , t ).

(2)

The function Φ1 is invariant on the right under Γ ∩ P1 and under Ci (i ≥ 1). If we take into account the definition of the groups Γi , we see that we can write: g . γ0 . c0 . . . γs−1 . cs−1 = g . σ . b

(σ ∈ Γ, b = c0 . . . cs−1 ),

and that the condition: Φ1 (g . γ0 . c0 ) ≤ μ . Φ1 (g . u)

(u ∈ Γ. C),

implies: Φ1 (g . σ . b) ≤ μ . Φ1 (g . γ . d)

(γ ∈ Γ, d ∈ C0 . . . Cs−1 ). 

Then, by (16.6), there exists a constant t > 0, independent of g, γi , ci such that we have: g . σ . b ∈ K . P(θ1 , t ).

(3)

In view of (16.5 (2)), the conclusion of (i) follows from (2) and (3). (ii) The function u → Φ1 (g . u) (u ∈ Γ. C) has a minimum (16.2). Hence, we can find γ0 ∈ Γ, c0 ∈ C such that: Φ1 (g . γ0 . c0 ) ≤ Φ1 (g . u)

(4)

(u ∈ Γ. C).

Considering as above the functions Φi on G , we obtain from the induction hypothesis the existence of ci ∈ Ci , γi ∈ Γi (1 ≤ i ≤ s − 1), such that: Φi (g . γ0 . c0 . . . γi−1 . ci−1 ) ≤ Φi (g . γ0 . c0 . . . γi−2 . ci−2 . ui−1 ),

(5)

for all ui−1 ∈ (Γi−1 ∩ Pi−1 ) . ci−1 and 1 ≤ i ≤ s. Then, in view of (i), there exists a constant t > 0, independent of g, γi , ci , such that: g . γ0 . c0 . . . γs−1 . cs−1 ∈ K . P(t).

(6)

The proof then concludes similarly to the proof of (16.7). Let:177  Γ = c−1. Γs−1 . c, c∈Cs−1

let ω be a compact subset of (M . U)R such that (M . U)R = ω(Γ ∩ M . U), and let S = K . QAt . ω. There exists σ ∈ Γ ∩ M . U such that: g . γ0 . c0 . . . γs−1 . cs−1 . σ ∈ S. We have cs−1 . σ ∈ Γs−1 . cs−1 , so178 the elements:179 ci (0 ≤ i ≤ s − 1),

γi = γi (0 ≤ i ≤ s − 2),

and

γs−1 = γs−1 .cs−1σ 

satisfy our conditions.

original French manuscript defines Γ = d∈D d−1. Γs−1 . d. establish (1), note that Φs−1 (x . σ) = Φs−1 (x) for all x, because σ ∈ M . U. Since ci−1 . Γi ⊂ Γi−1 . ci−1 for all i, we have g . γ0 . c0 . . . γs−1 . cs−1 ∈ g . Γ0 . c0 . . . cs−1 ⊂ g . Γ. D. 179 We have corrected an error in the original French manuscript, which defines γ = γ . σ 0 0 and γi = σ−1. γi . σ (1 ≤ i ≤ s − 1). It also notes that σ normalizes Γi ∩ Pi and that ci . σ ∈ Γi . ci (0 ≤ i ≤ s − 1), but these facts are not needed. 177 The 178 To

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17. Groups with rational rank one In this section, we consider the space X = K\GR /Γ of double cosets of G modulo a maximal compact subgroup K and an arithmetic subgroup Γ, when the Q-rank of G is equal to one, and Γ is “net” in the sense of (17.1). This latter condition can always be satisfied by passing to a suitable finite-index subgroup (17.4). We will show that X is diffeomorphic to the interior of a compact manifold with boundary, and will also give an explicit description of the boundary (see 17.10). (17.10) answers a question asked by J.-P. Serre, who also influenced the writing of this section, notably by suggesting condition (17.1). 17.1. Let k be a commutative field, p be the characteristic of k, and Ω be an algebraic closure of k. For an element g ∈ GL(n, k), we denote by V(g) the multiplicative subgroup of Ω∗ that is generated by the eigenvalues of g. We say that g is net if V(g) is torsion-free. A subgroup of GL(n, k) is net if all of its elements are net. It is clear that g is net if and only if its semisimple part gs is net. 17.2. Proposition. Let g ∈ GL(n, k) and let A (g) be the smallest algebraic subgroup of  GL(n,  Ω) that contains g. Then g is net if, and only if, for every element χ of X A (g) , if the value of χ at g is a root of unity, then the value is equal to one. For the properties of A (g) that are used below, see for instance [5, §8, pp. 111– 127]. Proof. We have A (g) = A (gs ) × A (gu ), where g = gs . gu is the Jordan decomposition (7.3) of g. Since A (gu ) is unipotent, it lies in   the kernelof every character, whence the existence of a natural isomorphism X A (g) ∼ = X A (gs ) , and this shows that we may assume g to be semisimple. A (g) is then diagonalizable, hence, A (g) = T × H where T is a torus and H is finite and is contained in a torus. The characters of H have finite order, and separate the points of H. It immediately follows that each of the two conditions of Proposition 17.2 implies180 that g ∈ T, and hence, H = {e}. After a suitable change of coordinates, we may assume that T is the set of invertible diagonal matrices such that xii = 1 (i > d = dim T). Thus, the coordinate functions xjj (j ≤ d) constitute a basis of X(T), and their values at g generate V(g). More precisely, the elements of V(g) are the values at g of the elements of X(T), and the proposition follows.  17.3. Corollary. Assume that G ⊂ GL(n, Ω) is an algebraic group, and let f : G → GL(m, Ω) be a morphism. If g ∈ G is net, then f (g) is net.        Proof. We  have f A (g) = A f (g)  [5, §2.1(f), p. 57]. If a ∈ X A f (g) , then a ◦ f ∈ X A (g) and (a ◦ f )(g) = a f (g) . Thus, it suffices to apply 17.2.  180 Any character χ of H extends to a character of A (g) that is trivial on T. The second of the two conditions implies that χ(g) is trivial. Since this is true for all χ, we conclude that the projection of g to H is trivial; i.e., g ∈ T. Now suppose g is net. By passing to a conjugate, we may assume that g is in the group D of diagonal matrices. If a character χ of D is trivial on T, then χ(g) is a root of unity. Since χ(g) is in the subgroup generated by the eigenvalues of g, this implies χ(g) = 1. On the other hand, T is the intersection of the kernels of these characters. So we conclude that g ∈ T.

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Remark. The condition in 17.2 that is equivalent to being net may be applied in any affine algebraic group, without appealing to a linear realization. Thus, we may speak of net elements in an affine algebraic group, and 17.3 remains true for every morphism of affine algebraic groups. 17.4. Proposition. Let G be a Q-group and Γ be an arithmetic subgroup of G. Then Γ has a congruence subgroup that is net. Proof. We may assume181 that G is identified with a Q-subgroup of GL(n, C) by a morphism that maps Γ into GL(n, Z) (7.13). Thus, it suffices to prove this for GL(n, Z). Let M be the set of cyclotomic polynomials of degree ≤ n and distinct from T − 1. It is finite. Let p be a prime number that does not divide any of the numbers f (1), (f ∈ M). We claim that the congruence group: H = {g ∈ GL(n, Z) | g ≡ 1 mod p} is net. Let g ∈ H, let (si )1≤i≤n be the eigenvalues of g, and let K be the field generated over Q by the si . Since the si are the roots of the characteristic polynomial of g, which is of degree n, with rational coefficients, K has degree ≤ n over Q. Let s ∈ V(g) be a root of unity. We must show that s = 1. Assume that this is not the case. Let p be a prime ideal of K that divides p. The si reduce mod p to the eigenvalues of the reduction mod p of g, i.e., to 1, and it follows that s ≡ 1 mod p. There exists f ∈ M such that f (s) = 0. Therefore, f (1) ≡ 0 mod p, therefore also mod p, which is a contradiction.  The preceding proposition suffices for the applications that we have in mind here. For completeness, we now give a more general version, whose proof was communicated to us by Serre. We will need the following elementary lemma: 17.5. Lemma. Let L be an integral domain and let m be a maximal ideal of L such that the characteristic p of L/m is = 0. Let x be an invertible element of L, of finite order n. If x ≡ 1 mod m, then n is a power of p. Proof. It suffices to show that if (n, p) = 1, then n = 1. If this were not the case, we would have x = 1 + a (a ∈ m), whence xn = 1 + a(n + b), (b ∈ m); since xn = 1, this implies b + n = 0, n ∈ m, so n is divisible by p, contrary to the hypothesis.  Proof. It suffices to show that if (n, p) = 1, then x = 1. Write x = 1 + a (a ∈ m), so xn = 1 + a(n + b), (b ∈ m); since xn = 1, this implies n + b = 0 if a = 0, so n ∈ m, so n is divisible by p, contrary to the hypothesis.  17.6. Proposition. Let k be a field of characteristic zero and L be a subring of k that is finitely generated over Z. Let Γ be a subgroup of GL(n, L). Then Γ has a finite-index subgroup that is net. Proof. By [9, Chap. V, §3, n◦ 1, Corollary 3, p. 348] we can find a rational integer a such that if p is any prime number that does not divide a, then there exists a homomorphism fp of L into an algebraically closed field of characteristic p, 181 This proof shows there is always a subgroup of finite index that is net. In order to obtain a congruence subgroup in the case where Γ ⊂ GZ , one should work directly with Γ (which is in GQ ), rather than embedding Γ in GL(n, Z).

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such that fp (1) = 1. Since L is finitely generated, fp (L) is a finite field, therefore mp = ker fp is a maximal ideal of L, with finite residue field. It follows that Γp = {g ∈ Γ | g ≡ 1 mod mp } has finite index. Then, the same holds for the group Γpq = Γp ∩ Γq , if p and q are prime numbers that are coprime to a. Let us show that Γpq is net if, in addition, p = q. For this, it suffices to show that if g ∈ Γp and if s is a root of unity that is in the group V(g) generated by the eigenvalues si of g, then the order of s is a power of p. The si are integral over L. Therefore, the L-algebra L generated by the si is integral over L. So, there exists a prime ideal mp of L such that mp ∩ L = mp . Since g ≡ 1 mod mp , we have si ≡ 1 mod mp (1 ≤ i ≤ n) so also, s ≡ 1 mod mp . Our claim then follows from (17.5), applied to L and mp .  17.7. Corollary. Let H be a finitely generated subgroup of GL(n, k). Then H has a finite-index subgroup that is net. Proof. Let {hj } (j ∈ I) be a finite generating set of H. The ring L generated by the matrix entries of the hj and h−1 is finitely generated over Z, and we have j H ⊂ GL(n, L). Thus, we can apply (17.6).  Remark. (17.7) shows in particular that H has a finite-index torsion-free subgroup, which is a well-known result of Selberg. 17.8. In this section, G is a connected reductive Q-group, of Q-rank equal to one, Γ is an arithmetic subgroup of G, K is a maximal compact subgroup of GR , and D = K\GR is the space of right cosets of GR mod K. We denote by π the projection of GR onto the space D/Γ = K\GR /Γ of double cosets K . x . Γ of GR . The space D is canonically endowed with the structure of a C∞ -manifold, and GR acts by diffeomorphisms. If Γ is torsion free, then Γ ∩ x . K . x−1 = {e} for every x ∈ GR , so Γ acts freely on D, whence X = K\GR /Γ naturally inherits the structure of a C∞ -manifold. Let P = M . S . U be a minimal parabolic Q-subgroup of G, where M, S, U have the usual meaning (§12). We write P0 for MR . UR = (M . U)R and A for S0R . We assume that S is normal (12.3), i.e., that A is invariant under the Cartan involution of GR that is associated to K (11.17), which implies (loc. cit.) that K ∩ MR is a maximal compact subgroup of MR , so also of P0 . Let C be a system of representatives of the double cosets PQ . x . Γ in GQ . In view of (15.6), it is finite, and there exists a Siegel set S with respect to K, P, S (12.3), which we will here take to be open, such that: (1)

GR = Ω . Γ,

with Ω = S . C.

The set S has the form K . At . ω, where ω is relatively compact in P00 = M0R . UR . By enlarging ω, we may assume that ω is a fundamental set in P00 for the intersection of the groups P00 ∩ c . Γ. c−1 (c ∈ C). Then, we have in particular, (2) (3)

P00 = ω · (P00 ∩ c Γ), P0 = (K ∩

M) . ω . (P00

(c ∈ C), ∩ Γ) c

(c ∈ C).

The set QΔ of simple Q-roots consists here of a single element, say, α, and we have: (4)

At = {a ∈ A | aα < t}.

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Given s (0 < s < t), we put: (5)

As,t = {a ∈ A | s ≤ aα < t},

(6)

Ss,t = K . As,t . ω,

Ss = K . As . ω.

It is clear that Ss,t is relatively compact, and that: S = Ss,t ∪ Ss ,

Ss,t ∩ Ss = ∅.

Thus, the fundamental set Ω = S . C is the union of a relatively compact set Ss,t . C and the sets Ss . c (c ∈ C). For s small enough, the latter sets play a role analogous to that of the cusps of the classical polygonal fundamental domain of a Fuchsian group, and will also be called cusps of Ω. 17.9. Proposition. We keep the assumptions and notation of (17.8). There is some s > 0 such that X = K\GR /Γ is the union of a relatively compact set π(Ss,t . C) and disjoint open sets π(Ss . c) (c ∈ C). Moreover, the relation: K . As . P0 . c . γ ∩ K . As . P0 . c = ∅

(c, c ∈ C; γ ∈ Γ),

implies c = c , c . γ . c−1 ∈ P. Proof. By the Siegel property (15.4), the set L = {γ ∈ Γ | Ω . γ ∩ Ω = ∅} is finite. Thus, we can find s small enough such that, for every c, c ∈ C and γ ∈ Γ, the intersection Ss . c . γ ∩ Ss . c is either empty, or not relatively compact. In the second case, by (12.6), we then have c . γ ∈ PQ . c , which, in view of the choice of C, also implies c = c . So the sets π(Ss . c) (c ∈ C) are disjoint. Let c, c ∈ C, x ∈ Γ, k, k ∈ K, p, p ∈ P0 , a, a ∈ As such that: k . a . p . c . x = k. a. p. c . In view of 17.8 (3), we can write: p = k1 . q1 . c . x1 . c−1 ,

p = k1 . q1 . c. x1 . c −1 ,

with k1 , k1 ∈ K ∩ M, q1 , q1 ∈ ω, x1 ∈ Γ ∩ c−1. P . c, x1 ∈ Γ ∩ c −1 . P . c . Thus, we have: Ss . c . x1 . x . x1−1 . c −1 ∩ Ss = ∅, which, in view of what was said above, implies: c = c ,

c . x1 . x . x1−1 . c −1 ∈ P,

and consequently, c . x . c−1 ∈ P.



17.10. Theorem. We keep the assumptions and notation of 17.9, and we assume moreover that Γ is net (17.1). Then x Γ ∩ P ⊂ M . U for every x ∈ GQ . For each c ∈ C, the map π induces a diffeomorphism νc of the product (0, s) × (K ∩ M)\(M . U)R /(c Γ ∩ P) onto π(Ss . c). The space X, endowed with its natural structure of C∞ -manifold, can be identified with the interior of a compact C∞ -manifold with boundary. We call this manifold Y. There exists a bijection of C onto the set of connected components of the boundary ∂Y of Y, which maps c ∈ C to a manifold that is diffeomorphic to Ec = (K ∩ M)\(M . U)R /(c Γ ∩ P).

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Proof. The group x Γ ∩ P is arithmetic in P. It follows that, if a ∈ X(P)Q , then the group a(x Γ ∩ P) is arithmetic in GL1 (8.10), so it is contained in {±1}. If Γ is net, it follows that a(x Γ ∩ P) = {e}. Since M . U is the intersection of the kernels of the elements of X(P)Q , this proves the first assertion. Let c ∈ C. Let us put ω = (K ∩ MR ) . ω. We have:   π(Ss . c) = π(As . ω. c) = π As . ω. c . (Γ ∩ c−1. P . c) . In view of the first assertion and 17.8 (3), this gives: π(Ss . c) = π(As . P0 . c). Let m, m ∈ MR , u, u ∈ UR , a, a ∈ As be such that: π(a . m . u . c) = π(a. m. u. c). Thus, there exists k ∈ K and x ∈ Γ such that: a . m . u . c . x = k . a. m. u. c. It then follows from 17.9 that c . x . c−1 ∈ P, whence, in view of the first assertion, c . x . c−1 ∈ P0 . We then have k ∈ P, so k ∈ P ∩ K = A . M . U ∩ K = M ∩ K. Since A ∩ P0 = {e}, it follows that we have: a = a ,

m . u . c . x . c−1 = k. m. u ,

which establishes the second assertion. Since Q = π(Ss,t . C) is relatively compact, we can find r (0 < r < s) such that π(Sr . C) ∩ Q = ∅. Now, let q be such that 0 < q < r, and fix a diffeomorphism μ of (0, s) onto (q, s) that is the identity on (r, s). Let us also denote by μ the is a diffeomorphism μ × Id of (0, s) × Ec onto (q, s) × Ec . Then μc = νc ◦ μ ◦ ν−1 c diffeomorphism of π(Ss . c) onto a relatively compact open set Uc . The latter is in fact π(S(q),s . c), where we put: S(q),s = K . A(q),s . ω,

A(q),s = {a ∈ A | q < aα < s}.

The collection of the μc and the identity on Q then define a diffeomorphism of X onto a relatively compact open set Y, which is the union of Q and of the Uc (c ∈ C). The  boundary ∂Y = Y  Y of Y is the union of the disjoint subsets νc {q} × Ec . These are compact submanifolds. The union of the images of the subsets νc [q, r) × Ec is an open neighborhood182 of ∂Y that is diffeomorphic to the product of ∂Y by an interval, whence the last assertion.  Remark. Raghunathan [34] showed that if H is a connected reductive Q-group, L is a maximal compact subgroup of HR and Γ is a torsion-free arithmetic subgroup of H, then the quotient L\HR /Γ has the same homotopy type as the interior of a compact manifold with boundary. 17.11. A fibration of Ec . The group c Γ∩P acts freely and properly on (K∩M)\P0 . The group c Γ∩U is normal in c Γ∩P. Therefore, the quotient Lc = (c Γ∩P)/(c Γ∩U) acts freely and properly on: (K ∩ M)\P0 /(c Γ ∩ U). 182 This

neighborhood is open in Y, not in X.

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Moreover, since P0 is the semidirect product of MR and UR , it is clear that: (1)

(K ∩ M)\P0 /(c Γ ∩ U) ≈ (K ∩ M)\MR × U/(c Γ ∩ U),

where ≈ means “diffeomorphic.” The group UR is normal in PR , so c Γ ∩ P acts by inner automorphisms on UR . This yields a representation of Lc as group of diffeomorphisms of Fc = UR /(c Γ ∩ U). The projection σ : P0 → P0 /UR ∼ = MR induces an isomorphism of Lc onto: Lc = σ(c Γ ∩ P). In particular, Lc is discrete and torsion-free. (In fact, since σ is the restriction of a Q-morphism of algebraic groups, Lc is arithmetic, and is net, in view of (17.3).) Thus, Lc acts freely and properly on (K ∩ M)\MR , so Bc = (K ∩ M)\MR /Lc is a C∞ -manifold. It then follows from (1) that Ec is the total space of a differentiable fibration with fiber Fc , structure group Lc , and base Bc , whose projection σc is induced by σ. The group U is commutative if, and only if, 2α is not a Q-root. If U is commutative, then Fc is a compact torus (in the sense of the theory of topological groups). Let us also note that if the R-rank of G is also equal to one, then MR is compact, so Ec = Fc . If U is commutative, the boundary of Y then consists of tori. Since G is of rational rank one, X has a single “Satake” compactification, of the type described in [2, Th´eor`em 4.3]. It can be obtained by taking the quotient of the space Y of (17.10) by the equivalence relation that is the identity on the interior of Y and is defined by the projection σc on the connected component of the boundary that corresponds to c ∈ C. Therefore, this compactification is the union of X and the bases Bc (c ∈ C) of the above-mentioned fibrations.

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Bibliography [1] A. Bialynicki-Birula, On homogeneous affine spaces of linear algebraic groups, Amer. J. Math. 85 (1963), 577–582. MR186674 [2] A. Borel, Ensembles fondamentaux pour les groupes arithm´ etiques, Colloq. sur la Th´eorie des Groupes Alg´ ebriques (Bruxelles, 1962), 1962, pp. 23–40. MR0148666 , Arithmetic properties of linear algebraic groups, Proc. Internat. Congr. Mathemati[3] cians (Stockholm, 1962), 1963, pp. 10–22. MR0175901 , Density and maximality of arithmetic subgroups, J. Reine Angew. Math. 224 (1966), [4] 78–89. MR0205999 , Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer[5] Verlag, New York, 1991. MR1102012 [6] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. MR0147566 [7] A. Borel and J.-P. Serre, Th´ eor` emes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111–164. MR0181643 ´ [8] A. Borel and J. Tits, Groupes r´ eductifs, Inst. Hautes Etudes Sci. Publ. Math. 27 (1965), 55–150. MR0207712 [9] N. Bourbaki, Commutative algebra, Hermann, Paris, 1972. Translated from the French original. MR0360549 , Integration. II. Chapters 7–9, Springer-Verlag, Berlin, 2004. Translated from the [10] French original. MR2098271 , Alg` ebre. Chapitre 9, Springer-Verlag, Berlin, 2007. Reprint of the 1959 original. [11] MR2325344 [12] J. W. S. Cassels, An introduction to the geometry of numbers, Classics in Mathematics, Springer-Verlag, Berlin, 1997. Corrected reprint of the 1971 edition. MR1434478 [13] C. Chevalley, Th´ eorie des groupes de Lie, Vol. II (Groupes alg´ ebriques) and III (Th´ eor` emes g´ en´ eraux sur les alg`ebres de Lie), Hermann, Paris, 1951, 1955. MR0051242 , S´ eminaire sur la classification des groupes de lie alg´ ebriques, 1956. mimeographed [14] notes, Institut Henri Poincar´e, Paris. , Classification des groupes alg´ ebriques semi-simples, Springer-Verlag, Berlin, 2005. [15] Collected works. Vol. 3. MR2124841 [16] R. Fricke and F. Klein, Lectures on the theory of automorphic functions. Vol. 1, Higher Education Press, Beijing, 2017. Translated from the German original. MR3838411 [17] R. Godement, Domaines fondamentaux des groupes arithm´ etiques, S´ eminaire Bourbaki, Vol. 8, 1995, pp. 201–225. MR1611543 [18] C. Hermite, Œuvres de Charles Hermite. Volume 1, Cambridge University Press, Cambridge, 2009. Reprint of the 1905 original. MR2866025 [19] N. Jacobson, Lie algebras, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR559927 ´ Polytech. 29 (1880), no. 48, 111–150. [20] C. Jordan, M´ emoire sur l’´ equivalence des formes, J. Ec. Œuvres compl` etes, tome III, pp. 421–461. [21] K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. I, II, Ann. of Math. (2) 67 (1958), 328–466. MR0112154 [22] A. Korkine and G. Zolotareff, Sur les formes quadratiques, Math. Ann. 6 (1873), no. 3, 366–389. MR1509828 [23] K. Mahler, On lattice points in n-dimensional star bodies. I. Existence theorems, Proc. Roy. Soc. London. Ser. A. 187 (1946), 151–187. MR0017753

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[24] H. Matsumoto, Quelques remarques sur les groupes alg´ ebriques r´ eels, Proc. Japan Acad. 40 (1964), 4–7. MR0166297 [25] Y. Matsushima, Espaces homog` enes de Stein des groupes de Lie complexes, Nagoya Math. J 16 (1960), 205–218. MR0109854 ¨ [26] H. Minkowski, Diskontinuit¨ atsbereich f¨ ur arithmetische Aquivalenz, J. Reine Angew. Math. 129 (1905), 220–274. Ges. Werke 2, 53–100. MR1580668 [27] L. J. Mordell, The arithmetically reduced indefinite quadratic form in n-variables, Proc. Roy. Soc. London. Ser. A. 131 (1931), no. 816, 99–108. [28] G. D. Mostow, Self-adjoint groups, Ann. of Math. (2) 62 (1955), 44–55. MR0069830 , Fully reducible subgroups of algebraic groups, Amer. J. Math. 78 (1956), 200–221. [29] MR92928 [30] G. D. Mostow and T. Tamagawa, On the compactness of arithmetically defined homogeneous spaces, Ann. of Math. (2) 76 (1962), 446–463. MR0141672 [31] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Springer-Verlag, Berlin, 1994. MR1304906 [32] M. Nagata, Note on orbit spaces, Osaka Math. J. 14 (1962), 21–31. MR143767 ´ Polytech. 51 (1882), [33] H. Poincar´ e, Sur les formes cubiques ternaires et quaternaires, J. Ec. 45–91. Œuvres compl` etes, tome 5, 293–334. [34] M. S. Raghunathan, A note on quotients of real algebraic groups by arithmetic subgroups, Invent. Math. 4 (1967/1968), 318–335. MR0230332 [35] M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401–443. MR82183 , On quotient varieties and the affine embedding of certain homogeneous spaces, Trans. [36] Amer. Math. Soc. 101 (1961), 211–223. MR130878 [37] J.-P. Serre, Local fields, Springer-Verlag, New York-Berlin, 1979. Translated from the French original. MR554237 [38] C. L. Siegel, Einheiten quadratischer Formen, Abh. Math. Sem. Hansischen Univ. 13 (1940), 209–239. Ges. Werke, II, 138–168. MR0003003 , Symplectic geometry, Amer. J. Math. 65 (1943), 1–86. Ges. Werke, II, 274–359. [39] MR8094 , Zur Reduktionstheorie quadratischer Formen, The Mathematical Society of Japan, [40] Tokyo, 1959. MR0140486 ´ [41] M. Stouff, Remarques sur quelques propositions dues ` a M. Hermite, Ann. Sci. Ecole Norm. Sup. (3) 19 (1902), 89–118. MR1509009 [42] A. Weil, Discontinuous subgroups of classical groups, 1958. notes by A. Wallace, University of Chicago. , Adeles and algebraic groups, Birkh¨ auser, Boston, 1982. MR670072 [43]

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Index

Algebraic group, 31 defined over k, 31 defined over a field, 31 linear, 31 of matrices, 31 representation of an algebraic group, 32 subset, 31 Almost direct product, xi Almost-simple Q-group, 95 Anisotropic over k (reductive group), 57 Arithmetic subgroup of a Q-group, 37 of a group defined over a number field, 40 Associated ordering of roots, 62

first type, 49 in the space H(F) of positive-definite quadratic forms, 26 second type, 79 Galois cohomology, 59 Good position, 73 Group of L-units, 37 Hermite, 8 reduced, 8 Iwasawa decomposition of gR , 70 of GL(n, R), 1 of GR , 70 of SL(n, R), 4

Bounded function, xi Bruhat decomposition in GL(n, k), 16 of a reductive k-group, 61

k-group, 31 k-morphism of algebraic groups, 31 k-rank, 60 k-root, 60 k-split torus, 55 k-weight, 85 dominant, 85 fundamental, 85 Korkine-Zolotareff, 8

Cartan decomposition, 69 subgroup, 66 Center of a group (Z(G)), xii Character of an algebraic group, 32 Class of quadratic forms, 29 proper, 29 Commensurable subgroups, 37 Commensurator of a subgroup (CommH (L)), 95 Completely reducible, 36 Congruence subgroup, 38 principal, 38 Cusp, 111

Langlands decomposition, 61 M-reduced element of GL(n, R), 12 positive-definite quadratic form, 9 Mahler’s criterion, 4 Majorize, 24 Majorizer, 24 Hermite, 24 minimal, 24 Minkowski, 8 Theorem, 10 Morphism of algebraic groups, 31 defined (or rational) over k, 31

Derived group (DG), 43 Dirichlet’s Unit Theorem, 46 Finiteness lemma, 26 Flag, 13 Frattini argument, 50 Function of type (P, χ), 84 standard, 85 Fundamental set, 51, 53

Net element of GL(n, k), 108 element of an affine algebraic group, 109 117

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118

subgroup of GL(n, k), 108 Norm on Rn , 4 Parabolic subgroup, 13, 60 Quadratic form that does not represent zero rationally, 45 Quasi-split semisimple k-group, 69 Quotient X/H exists, 36

INDEX

Weyl chamber, 62 group, 60 of GL(n, k), 12 of a reductive k-group, 60

Radical of an algebraic group, 33 Rational character, 12 Reduced indefinite rational quadratic form, 25 Reduced positive-definite quadratic form in the sense of Hermite, 8 in the sense of Minkowski, 9 Reduction in GR , ix in SL(n, R), 4 Reductive algebraic group, 33 Restriction of scalars, 40 Ring of integers (Ok ), 40 Root, 60 system, 61 Self-adjoint subgroup of GL(n, R), 27, 49 with respect to a bilinear form, 49 Semisimple algebraic group, 33 Siegel domain, 1 Siegel property, 51 for indefinite quadratic forms, 26 in GLn , 17 Siegel set, 4 in the space of positive-definite quadratic forms, 8 normal, 75 of GL(n, R), 1 standard, 75 of a reductive F-group (F ⊂ R), with respect to K, P, S, 74 open, 75 over a field, 75 Simple Q-group, 95 root, 61 Split solvable k-group, 64 torus, 55 Structure theorem for algebraic groups, 40 Symmetric subset, 88 Symplectic group, 67 Torus, 33 split over k, 55 Triangularizable over k, 64 Unipotent radical, 40 Unit of a rational quadratic form, 25

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SELECTED PUBLISHED TITLES IN THIS SERIES

73 Armand Borel, Introduction to Arithmetic Groups, 2019 72 Pavel Mnev, Quantum Field Theory: Batalin–Vilkovisky Formalism and Its Applications, 2019 71 Alexander Grigor’yan, Introduction to Analysis on Graphs, 2018 70 Ian F. Putnam, Cantor Minimal Systems, 2018 69 Corrado De Concini and Claudio Procesi, The Invariant Theory of Matrices, 2017 68 Antonio Auffinger, Michael Damron, and Jack Hanson, 50 Years of First-Passage Percolation, 2017 67 Sylvie Ruette, Chaos on the Interval, 2017 66 Robert Steinberg, Lectures on Chevalley Groups, 2016 65 Alexander M. Olevskii and Alexander Ulanovskii, Functions with Disconnected Spectrum, 2016 64 Larry Guth, Polynomial Methods in Combinatorics, 2016 63 Gon¸ calo Tabuada, Noncommutative Motives, 2015 62 H. Iwaniec, Lectures on the Riemann Zeta Function, 2014 61 Jacob P. Murre, Jan Nagel, and Chris A. M. Peters, Lectures on the Theory of Pure Motives, 2013 60 William H. Meeks III and Joaqu´ın P´ erez, A Survey on Classical Minimal Surface Theory, 2012 59 Sylvie Paycha, Regularised Integrals, Sums and Traces, 2012 58 57 56 55

Peter D. Lax and Lawrence Zalcman, Complex Proofs of Real Theorems, 2012 Frank Sottile, Real Solutions to Equations from Geometry, 2011 A. Ya. Helemskii, Quantum Functional Analysis, 2010 Oded Goldreich, A Primer on Pseudorandom Generators, 2010

54 John M. Mackay and Jeremy T. Tyson, Conformal Dimension, 2010 53 John W. Morgan and Frederick Tsz-Ho Fong, Ricci Flow and Geometrization of 3-Manifolds, 2010 52 Marian Aprodu and Jan Nagel, Koszul Cohomology and Algebraic Geometry, 2010 51 J. Ben Hough, Manjunath Krishnapur, Yuval Peres, and B´ alint Vir´ ag, Zeros of Gaussian Analytic Functions and Determinantal Point Processes, 2009 50 John T. Baldwin, Categoricity, 2009 49 J´ ozsef Beck, Inevitable Randomness in Discrete Mathematics, 2009 48 Achill Sch¨ urmann, Computational Geometry of Positive Definite Quadratic Forms, 2008 47 Ernst Kunz, David A. Cox, and Alicia Dickenstein, Residues and Duality for Projective Algebraic Varieties, 2008 46 Lorenzo Sadun, Topology of Tiling Spaces, 2008 45 Matthew Baker, Brian Conrad, Samit Dasgupta, Kiran S. Kedlaya, and Jeremy Teitelbaum, p-adic Geometry, 2008 44 Vladimir Kanovei, Borel Equivalence Relations, 2008 43 Giuseppe Zampieri, Complex Analysis and CR Geometry, 2008 42 Holger Brenner, J¨ urgen Herzog, and Orlando Villamayor, Three Lectures on Commutative Algebra, 2008 41 James Haglund, The q, t-Catalan Numbers and the Space of Diagonal Harmonics, 2008 40 Vladimir Pestov, Dynamics of Infinite-dimensional Groups, 2006 39 Oscar Zariski, The Moduli Problem for Plane Branches, 2006 38 Lars V. Ahlfors, Lectures on Quasiconformal Mappings, Second Edition, 2006

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/ulectseries/.

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

Fifty years after it made the transition from mimeographed lecture notes to a published book, Armand Borel’s Introduction aux groupes arithmétiques continues to be very important for the theory of arithmetic groups. In particular, Chapter III of the book remains the standard reference for fundamental results on reduction theory, which is crucial in the study of discrete subgroups of Lie groups and the corresponding homogeneous spaces. The review of the original French version in Mathematical Reviews observes that “the style is concise and the proofs (in later sections) are often demanding of the reader.” To make the translation more approachable, numerous footnotes provide helpful comments.

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ULECT/73 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms