Smooth Compactifications of Locally Symmetric Varieties (Cambridge Mathematical Library)

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Smooth Compactifications of Locally Symmetric Varieties Second Edition The new edition of this celebrated and long-unavailable book preserves much of the content and structure of the original, which is still unrivalled in its presentation of a universal method for the resolution of a class of singularities in algebraic geometry. At the same time, the book has been completely re-typeset, errors have been eliminated, proofs have been streamlined, the notation has been made consistent and uniform, and an index and a guide to recent literature have been added. The authors begin by reviewing, in Chapter I, key results in the theory of toroidal embeddings and by explaining examples that illustrate the theory. Chapter II develops the theory of open self-adjoint homogeneous cones and their polyhedral reduction theory. Chapter III is devoted to basic facts on hermitian symmetric domains and culminates in the construction of toroidal compactifications of their quotients by an arithmetic group. The final chapter considers several applications of the general results. The book brings together ideas from algebraic geometry, differential geometry, representation theory and number theory, and will continue to prove of value for researchers and graduate students in these areas.

Smooth Compactifications of Locally Symmetric Varieties Second Edition

AVNER ASH Boston College DAVID MUMFORD Brown University MICHAEL RAPOPORT University of Bonn YUNG-SHENG TAI Haverford College With the collaboration of PETER SCHOLZE University of Bonn

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521739559 © A. Ash, D. Mumford, M. Rapoport and Y. Tai 2010 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN-13

978-0-511-67345-0

eBook (EBL)

ISBN-13

978-0-521-73955-9

Paperback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface to the second edition Preface to the first edition

page vii ix

I

Basics on torus embeddings; examples 1 Torus embeddings over the complex numbers 2 The functor of a torus embedding 3 Toroidal embeddings over the complex numbers 4 Compactification of the universal elliptic curve 5 Hirzebruch’s theory of the Hilbert modular group

1 1 7 9 14 25

II

Polyhedral reduction theory in self-adjoint cones 1 Homogeneous self-adjoint cones 2 Jordan algebras 3 Boundary components and Peirce decompositions 4 Siegel sets in self-adjoint cones 5 Cores and co-cores 6 Positive-definite forms in low dimensions

37 38 43 51 67 75 90

III

Compactifications of locally symmetric varieties 1 Tube domains and compactification of their cusps 2 The structure of bounded symmetric domains 3 Boundary components 4 Siegel domains of the third kind 5 Statement of the Main Theorem 6 Proof of the Main Theorem 7 An intrinsic form of the Main Theorem

97 97 105 123 142 159 164 176

IV

Further developments 1 Extension of differential forms to the cusps 2 Projectivity of D/Γ Supplementary Bibliography Index v

189 189 199 215 229

Preface to the second edition

When CUP approached us with the proposal of a second edition to our book, we first consulted graduate students and younger colleagues to test this idea on them. Their enthusiastic response convinced us of the soundness of the proposition. In order to keep this project within realistic bounds, we did not rewrite the book, but rather TEX-ed the original text and corrected mistakes that have come to our attention. We also smoothed somewhat the presentation and homogenized the notation. Finally, in order to increase its usability, we added an index. So, all in all, this is still essentially the same book. In particular, the text of this new edition does not reflect the developments in the field in the last 30 years. To compensate for this, we added a guide to the more recent literature at the end of the book. In this whole project we were assisted by Peter Scholze, who read the whole manuscript, corrected many mistakes, and helped us with the proof-reading. We thank him heartily. We also thank Y. May, who assisted us in TEX-problems. We also thank all those who pointed out mistakes in the first edition and often indicated to us how to correct them: we are thus grateful to C.-L. Chai, E. Looijenga, R. Pink, Y. Namikawa, and I. Satake. Finally, we thank the staff of CUP, and particularly D. Tranah, for their expert cooperation. Avner Ash, David Mumford, Michael Rapoport, Yung-sheng Tai.

vii

Preface to the first edition

Let D be a bounded symmetric domain and let Γ be a neat (see Ch. III, §7) arithmetic subgroup of Aut (D)o . The goal of this monograph is the construction of a family of non-singular† compactifications D/Γ of D/Γ. This theory was announced and described in rough outline in [2]. Very similar ideas were developed independently by Satake in [3]. Both of us were following the path indicated by the work of Igusa when Γ = Sp (2n, Z) and by Hirzebruch when Γ = SL(2, O), where O = integers in a real quadratic field. Here is an outline of the monograph. Since this work builds heavily on [1] (referred to as TE I below), we review quickly some of these results and add some comments particular to the complex case in Ch. I, §§1–3. Then, in Ch. I, §§4,5, we describe smooth compactifications of two surfaces D/Γ, in order to illustrate the general theory which follows (actually, in §4, D is not bounded – it is ∆ × C – so this is not strictly a special case). Chapter II, by A. Ash, is devoted to self-adjoint homogeneous cones. The main result is a comparison of Siegel sets and polyhedral subcones inside such homogeneous cones. These results are essential for the construction of D/Γ. The construction itself is taken up in Ch. III. The final results require considerable notation to state, but may be found in Ch. III, §§5, 7. The principal technical contribution here is M. Rapoport’s calculation of the Satake topology on
D∗ = D (rat. boundary comp.) in terms of the presentation of D as a Siegel domain of third kind, which is crucial to proving that our D/Γ is Hausdorff. The final chapter by Y. S. Tai adds two important results. Firstly, he applies the construction to prove that D/Γ is a variety “of general type” in Kodaira’s classification when Γ is small enough. Secondly, although our general D/Γ is only an analytic compactification of D/Γ, he shows that many of these D/Γ’s are indeed projective varieties. One of the main obstacles in our research was that none of us were symmetric space specialists when we began, and, of course, roots are the name of the game throughout. For our sake as well as the reader’s, we thought it useful to include a considerable amount of expository material in the hope of making † We are mainly concerned with a larger class of compactifications with “toroidal” singularities on the boundary. Within this class, there are plenty of non-singular compactifications, but these do not play any special role in our study.

ix

x

Preface to the first edition

the monograph more self-contained. We were greatly aided by similar expository projects of P. Deligne and I. Satake, who graciously lent us their notes. In general, the expository sections emphasize the geometric aspects somewhat more than the references, and, in particular, develop the ideas in the form in which we need them. Experts can skim rapidly through Ch. II, §§1–3 (note, however, the very crucial tie-up between Pierce decompositions and split tori which appears to be new) and Ch. III, §§2–4 (note here the key role played by ˇ D(F) = U(F)C · D, in the the intermediate open set D(F) : D ⊂ D(F) ⊂ D, construction of the Siegel Domain realization). I hope that the space D/Γ here constructed will have other applications in the theory of automorphic forms, e.g., to calculating invariants of the field C(D/Γ) and the dimension of the spaces of automorphic forms. Besides these applications, the theory can hopefully be pushed further in three essential directions: (i) at least for D = Sp(2n, R)/K, extend it to a construction of a scheme D/Γ over Z; (ii) to extend Hirzebruch’s proportionality theorems to the non-compact case; (iii) in view of the fact that the results describe concretely the degeneration of Hodge structures of a very special type – find an extension of them, combining the ideas of Ch. III, §7 with Schmid’s results on ˚ to describe asymptotically all families of families of Hodge structures over ∆, k ˚ Hodge structures on (∆) . David Mumford

Authorship of the various chapters Chapter I: David Mumford Chapter II: Avner Ash Chapter III: Michael Rapoport and David Mumford Chapter IV: Yung-sheng Tai References [1] G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal Embeddings I, Lecture Notes in Mathematics 339. Berlin: Springer 1972.† [2] D. Mumford, A new approach to compactifying locally symmetric varieties, in Proceedings of the Tata Institute Colloquium, Jan. 1973, Oxford Univ. Press, 1975. [3] I. Satake, On the arithmetic of tube domains, Bull. Amer. Math. Soc., 79 (1973), 1076-1094. † This is referred as ‘TE I’ throughout the monograph.

I Basics on torus embeddings; examples

1 Torus embeddings over the complex numbers We wish to review here quickly some results of TE I† and to give a more explicit description of the complex varieties obtained via certain real spaces of half the dimension. Let T be an algebraic torus, i.e., T ∼ = Gnm for some n, and let M = Hom (T, Gm ) , the character group of T , and N = Hom (Gm , T ) , the group of ‘one-parameter’ subgroups of T (in the algebraic sense). Then M ∼ = Zn and N ∼ = Zn , and there is a natural non-degenerate pairing ·, · : M × N −→ Z of determinant 1. All this is valid over any field k. When k = C, however, T can be described analytically as T/π , where T is a complex vector space and π is a discrete subgroup, generating T over C and isomorphic to Zn . Here T is the universal covering space of T and π is π1 (T ) acting on T via translations. Note, however, that for all a ∈ π the map

φa : C −→ T λ −→ λ · a induces a map

φa : C/Z −→ T/π = T , and that C/Z ∼ = Gm canonically via λ −→ e2π iλ . Thus every a ∈ π induces φa ∈ N, and this is easily checked to be an isomorphism between π and N. Thus π is just N up to a canonical identification. Since T = π ⊗ C, it follows that we have canonical maps: † Recall this reference from p. x.

1

2

I Basics on torus embeddings; examples (i) N ∼ = the usual topological π1 of T ;

(ii) N ⊗ C ∼ = the universal covering space of T ; (iii) (N ⊗ C)/N ∼ = T. We abbreviate N ⊗ C by NC and N ⊗ R by NR . Next, in the isomorphism NC /N ∼ = T , consider the subgroup corresponding to NR /N: this a compact real torus, and is the maximal compact subgroup of T . We denote it by Tc (short for “compact torus”). Moreover, NR ⊂ NC has a natural complement, viz. iNR , and, by quotienting, iNR injects into NC /N. In other words, we get a canonical decomposition NC /N ∼ = (NR /N) × (iNR ) , and hence (dividing by i in the second factor) T∼ = Tc × NR . We denote the projection T −→ NR by “ord,” which is then defined by ord(x + iy mod N) = y , for all x, y ∈ NR . If α ∈ M, and Xα : T −→ C∗ is the corresponding function (as in TE I, it is useful to think of M as an additive group, and hence to adopt exponential notation for the characters regarded as functions on T ), we obtain the formula Xα (x + iy mod N) = e2π i(α ,x+iα ,y) , for all x, y ∈ NR ; hence |Xα (z)| = e−2π α ,ord z , for all z ∈ T . Next, in TE I, Ch. I, §1, we define embeddings of T in normal affine varieties Xσ , with the action of T on itself extending to an action of T on Xσ , whenever σ ⊂ NR is a closed rational polyhedral cone not containing a line. Recall that Xσ = Spec C[. . . , Xα , . . .]α ∈M∩σˇ ; here σˇ ⊂ MR is the dual cone to σ , so M ∩ σˇ is a sub-semigroup of M. In order to study convergence in the classical topology and other details on Xσ , it will be convenient to introduce here the topological space (in the classical, not Zariski, topology) obtained by dividing Xσ by Tc . This will look like NR with points at infinity added. Let us first construct these embeddings, which we call Nσ , of NR and then show there is a map ord : Xσ −→ Nσ inducing a ∼ homeomorphism Xσ /Tc −→ Nσ .

1 Torus embeddings over the complex numbers

3

The simplest way to define Nσ is via a basis α1 , . . . , αm of the semigroup σˇ ∩ M. Then define i : NR −→ Rm >0 , x −→ (e−2π α1 ,x , . . . , e−2π αm ,x ), and let Nσ = closure of iNR in Rm ≥0 . It is very easy to see that this space is independent of the choice of basis (check that if you add to the αi one more α , then Nσ does not change). If we let NR act on Rm by x · (y1 , . . . , ym ) = (e−2π α1 ,x y1 , . . . , e−2π αm ,x ym ) , then Nσ is the closure of the orbit of (1, 1, . . . , 1). In particular, NR acts on Nσ , extending its action on itself by translation. Exactly as in the theory of torus embeddings (see TE I, Ch. I, §1, Theorem 2), we can decompose Nσ into NR -orbits; these will correspond bijectively to the faces of σ , and each one will contain a unique point (y1 , . . . , ym ) with yi = 0 or 1 for all i. Explicitly, for every face τ of σ , the corresponding orbit is:   yi = 0 if αi > 0 on Int τ O(τ ) = (y1 , . . . , ym ) ∈ Nσ | yi = 0 if αi ≡ 0 on Int τ = NR -orbit of ετ = (ε1 , . . . , εm ) , where



εi =

0 1

if αi > 0 on Int τ if αi ≡ 0 on Int τ .

This can be proven following TE I, substituting the following lemma for the use of k[[t]]. Lemma 1.1 If {xk } is a sequence in NR and S ⊂ {1, . . . , m} satisfies lim αi (xk ) = λi , i ∈ S,

k−→∞

/S, lim αi (xk ) = ∞ , i ∈

k−→∞

then (a) there is some y ∈ NR with αi (y) = 0 for i ∈ S; αi (y) > 0 for i ∈ /S; (b) there is some z ∈ NR with αi (z) = λi for i ∈ S. Proof Left to reader.

4

I Basics on torus embeddings; examples Now if we map Xσ into Rm ≥0 as follows: / NR _

ord

T _

f

:

 Nσ : Rm ≥0

⊃

 Xσ :

f (x) = (|Xα1 (x)|, . . . , |Xαm (x)|) , we get a commutative diagram. Since T is dense in Xσ , it follows that f defines a map ord : Xσ −→ Nσ and that ord(gx) = ord(x) for all g ∈ Tc . Conversely, if ord(x1 ) = ord(x2 ), it follows that |Xα (x1 )| = |Xα (x2 )| for all α ∈ σˇ ∩ M, from which it follows readily that x1 = gx2 for some g ∈ Tc . Note that if Oτ ⊂ Xσ is the orbit corresponding to τ , then ord−1 (O(τ )) = Oτ . For some purposes, it is convenient to have a coordinate-invariant way of describing Nσ as NR plus a set of ideal points at infinity. To describe Nσ this way, for every face τ of σ , let L(τ ) = smallest linear space containing τ . Then L(τ ) is the stabilizer of ετ , so we get: ∼

NR /L(τ ) −→ O(τ ) x −→ x · ετ . Let x + ∞ · τ ∈ Nσ denote x · ετ (where x1 + ∞ · τ = x2 + ∞ · τ if and only if x1 − x2 ∈ L(τ )). The reason for this notation is as follows: decompose NR = NR ⊕ L(τ ), choose any sequence xn = yn + zn ∈ NR = NR ⊕ L(τ ), and choose any y ∈ NR . Then one sees easily that 



lim xn = y + ∞ · τ in Nσ ⇐⇒

n−→∞



lim yn = y and, for every



n−→∞

w ∈ L(τ ), zn ∈ τ + w if n  0 .

Heuristically, we have added a lower-dimensional vector space isomorphic to NR /L(τ ) of ideal points x + ∞ · τ obtained by starting at x and moving out to

1 Torus embeddings over the complex numbers

5

infinity in the direction determined by the cone τ .

Our convergence condition may be rephrased by saying that a fundamental system of neighborhoods of y + ∞ · τ in NR is given by

Uε0,w (y + ∞ · τ ) = y + w + Bε + τ ,

for any w ∈ L(τ ) and any ε > 0, where Bε denotes the ε -ball around 0 (take any metric on NR ). More generally, with this notation, a fundamental system of neighborhoods of y + ∞ · τ in Nσ is given by

Uε ,w (y + ∞ · τ ) = Uε0,w (y + ∞ · τ ) ∪

τ  face of τ

(y + w + Bε + τ + ∞ · τ  ) .

For instance, if NR = R2 and σ is the positive quadrant, we get the following

6

I Basics on torus embeddings; examples

picture:

Next recall that in TE I, Ch. I, §2, we glue the affine varieties Xσ together: whenever {σα } is a rational partial polyhedral decomposition of NR , meaning (i) if σ is a face of σα , then σ = σβ , for some β ; (ii) for all α , β , the cone σα ∩ σβ is a face of σα and σβ , then we can glue the Xσα together, obtaining a scheme X{σα } . In TE I, we asked that {σα } be a finite set, so that X{σα } was a variety. This is in fact totally irrelevant: for any set {σα } as above, we get an X{σα } as before, except that it may require an infinite number of affines to cover it. Now X{σα } is always a separated normal irreducible scheme, locally of finite type over C and containing T as an open dense subset. In exactly the same way, we glue the Nσα together into a topological space N{σα } , which is NR plus a large number of ideal vector spaces situated at infinity in many different directions. Moreover, we glue the ord maps together into one map: ord : X{σα } −→ N{σα } . For instance, X{σα } , as a set, is the disjoint union of T -orbits Oσα , one for each α ; likewise N{σα } as a set is the disjoint union of NR -orbits O(σα ), one for each α , and ord−1 (O(σα )) = Oσα .

2 The functor of a torus embedding

7

2 The functor of a torus embedding In order to make some of our later constructions of compactifications D/Γ purely algebraic and valid for schemes over any ground fields, it will be useful to learn what functor a torus embedding represents. This also gives us another view of what torus embeddings are. First some notations and definitions. (1) If S is a scheme and X is a set, XS denotes the constant sheaf on S with stalk X. (2) Every semigroup or sheaf of semigroups will have an identity element e or identity section e. (3) If A1 , A2 are semigroups, a homomorphism φ : A1 −→ A2 is called strict if φ (e1 ) = e2 and φ (x) invertible implies x invertible. If A1 , A2 are sheaves of semigroups on S, we require that, for every s ∈ S, the map on stalks φs : A1,s −→ A2,s is strict. (×)

(4) If S is a scheme, then OS

will be the semigroup sheaf (OS , mult.).

The result is: Theorem 2.1 Let T be a torus over k and T ⊂ X{σα } a torus embedding, where σα ⊂ N(T )R are polyhedral cones. For any k-scheme S, let F{σα } (S) be the set of pairs (Σ, π ) consisting of a sub-semigroup sheaf Σ ⊂ M(T )S and a strict (×) homomorphism π : Σ −→ OS such that, for all s ∈ S, we have Σs = σˇ α ∩M(T ) for some α . Then there are canonical isomorphisms, functorial in k-schemes S: Hom k (S, X{σ } ) ∼ = F{σ } (S) . α

α

Proof We first show how to associate a pair (Σ, π ) to a morphism f : S −→ X{σα } . Define: Uα = f −1 (Xσα ) , Σ = the union of the subsheaves (σˇ α ∩ M(T ))Uα of M(T )S . Note that, for all s ∈ S, if f (s) ∈ Oα , then s ∈ Uβ ⇐⇒ f (s) ∈ Xσβ ⇐⇒ Oα ⊂ Xσβ ⇐⇒ σα is a face of σβ ⇐⇒ σˇ β ∩ M(T ) ⊆ σˇ α ∩ M(T ) ; hence the stalk of Σ at s is the union of the subsets σˇ β ∩ M(T ) of M(T ) for all σβ with face σα , i.e., just σˇ α ∩ M(T ). Hence if r ∈ Σs , then r ∈ σˇ α , so

8

I Basics on torus embeddings; examples

Xr is defined on Xσα and f ∗ (Xr ) is defined at s. Therefore we can define (×) π : Σ −→ OS by

π (r) = f ∗ (Xr ) . Note that

π (r) invertible in OS,s ⇐⇒ π (r)(s) = 0 ⇐⇒ Xr ( f (s)) = 0 ⇐⇒ Xr ≡ 0 on Oα ⇐⇒ r ≡ 0 on σα ⇐⇒ −r ∈ σˇ α ∩ M(T ) ⇐⇒ r invertible in Σs , hence π is a strict homomorphism. Next, let us start with (Σ, π ) and define a morphism f . Define open sets Uα by Uα = {s ∈ S | σˇ α ∩ M(T ) ⊂ Σs } . These form an open covering of S such that if σα is a face of σβ , then Uβ ⊂ Uα . Next define fα : Uα −→ Xσα = Spec k[. . . , Xr , . . .]r∈σˇ α ∩M(T ) via fα∗ (Xr ) = π (r) for all r ∈ σˇ α ∩ M(T ): this is correct since such an r is in Γ(Uα , Σ) and since π (r1 + r2 ) = π (r1 ) · π (r2 ). Now, for any α and β , let σγ = σα ∩ σβ , which is a face of σα and σβ . Then Uα ∩Uβ = {s ∈ S | σˇ α ∩ M(T ) ⊂ Σs and σˇ β ∩ M(T ) ⊂ Σs } . But if Σs = σˇ δ ∩ M(T ), then  Σs ⊃ σˇ α ∩ M(T ) and ⇐⇒ σˇ δ ⊃ σˇ α and σˇ δ ⊃ σˇ β Σs ⊃ σˇ β ∩ M(T ) ⇐⇒ σδ ⊂ σα and σδ ⊂ σβ ⇐⇒ σδ ⊂ σγ ⇐⇒ Σs ⊂ σˇ γ ∩ M(T ) , so Uα ∩ Uβ = Uγ . Finally, it is clear from the definition that fα = res fβ whenever Uα ⊂ Uβ . Therefore the fα patch together to form a morphism f : S −→ X{σα } . It is now straightforward to check that these two procedures – associating a (Σ, π ) to an f and associating an f to a (Σ, π ) – are inverse to each other: we leave this to the reader.

3 Toroidal embeddings over the complex numbers

9

For instance, we find: X{σα } (k) ∼ = {(α , π ) | π : σˇ α ∩ M(T ) −→ k(×) strict homomorphism} . If k = C, one can easily prove also that (×) N{σα } ∼ = {(α , ρ ) | ρ : σˇ α ∩ M(T ) −→ R≥0 strict homomorphism} ∼ = {(α , σ ) | σ : σˇ α ∩ M(T ) −→ R ∪ {∞} strict homomorphism} ,

where R ∪ {∞} is a semigroup via +. Here ord : X{σα } −→ N{σα } is given by

ρ (x) = |π (x)| , σ (x) = − log ρ (x) .

3 Toroidal embeddings over the complex numbers We wish to review here quickly some results of TE I, Ch. II, indicating ways to interpret them over C, and generalizing them slightly. A pair U ⊂ X, where U is a Zariski-open subset of a normal variety X, was called a toroidal embedding if, for all x ∈ X, we have that (X,U) is formally isomorphic at x to (Xσ , T ) at some t ∈ Xσ (for some torus embedding T ⊂ Xσ ). Equivalently, this means that there is an e´ tale correspondence between X and Xσ , relating x and t, with U and T corresponding open sets. Over C, a pair U ⊂X , where X is an analytic space and U is open in the complex topology, will be called a toroidal embedding if, for all x ∈ X, there exists a small neighborhood Wx ⊂ X of x such that (Wx ,Wx ∩U) is isomorphic to (Vt ,Vt ∩ T ) for some neighborhood Vt ⊂ Xσ of some t ∈ Xσ (for some torus embedding T ⊂ Xσ ). When X, U are varieties, this coincides with the previous definition. Now, this implies immediately that Wx has a canonical stratification {Yα ,x } into non-singular locally closed analytic strata with Y α ,x normal: let Ei be the irreducible components of Wx \Wx ∩U, and let the Yα ,x be the sets

i∈I

Ei \



Ei .

i∈I /

We shrink Wx if necessary, so that these Yα ,x are connected. As x varies, these strata patch up on overlaps, so we can uniquely stratify the whole of X into

10

I Basics on torus embeddings; examples

{Yα }, where the Yα are connected, locally closed, non-singular analytic strata, and where Yα ∩Wx is a union of the Yβ ,x . However, it may happen that Yα ∩Wx ⊃ more than one Yβ ,x . This means that there is a path in X starting and ending in Wx and lying all in one stratum, but linking two distinct local strata:

Since this will mean that Y α has more than one branch through x, it is equivalent to Y α being non-normal. As in TE I, p. 57, we say that (X,U) has or has not self-intersection according to whether Yα ∩Wx can be more than one local stratum, or Yα ∩Wx is always one local stratum. In TE I, we stuck with (X,U)’s without self-intersection. However, there is a class of toroidal embeddings with self-intersection that are almost as nice and that arise in the examples we will treat. Suppose Yβ1 ,x and Yβ2 ,x are part of the same global stratum Yα . Locally at x there is a unique stratum Yβ3 ,x such that Y β3 ,x = Y β1 ,x ∩Y β2 ,x . Let Yβ3 ,x define a global stratum Yγ . We say that (X,U) is without monodromy if Yγ has a neighborhood W such that Yβ1 ,x and Yβ2 ,x lie in different components of Yα ∩ W . To visualize this, note that, for every path in Yγ beginning and ending at x, we can uniquely propagate the germ of analytic space Y β1 ,x along this path. If this germ can be taken to Y β2 ,x by such a path, then, for every

3 Toroidal embeddings over the complex numbers

11

neighborhood W of Yγ , we may connect Yβ1 ,x and Yβ2 ,x within Yα ∩ W . If not, then, in some small enough W , they cannot be connected:

A toroidal embedding with monodromy If (X,U) is without monodromy, then every stratum Yα has a small complex neighborhood, which we call Star0 (Yα ), in which all the local strata Yβ ,x (where x ∈ Yα , Y β ,x ⊃ Yα ) remain distinct; i.e., Star0 (Yα ) is a union of semi-local strata (α )

(α )

(α )

Yβ such that Y β is normal or, equivalently, Yβ ∩ Wx is one local stratum. We may even assume that there is a stratum-preserving homeomorphism Star0 (Yα ) ≈ Xσ ×Yα , where T ⊂ Xσ is a true embedding. Of course, in the whole space X, we may (α ) (α ) have Yβ and Yβ as part of the same stratum Yγ . 1 2 The main point of Ch. II, §1 of TE I was to associate to each toroidal embedding without self-intersection (X,U) a conical polyhedral complex with integral structure. If we generalize slightly our definition of such a complex, we can do this for any analytic toroidal embedding without monodromy too. The following definition may be compared with the definition in TE I, p. 69. Definition 3.1 A conical polyhedral complex Σ is a topological space |Σ|, plus a stratification {Sα } of |Σ| (i.e., a partition of |Σ| into disjoint locally closed pieces Sα such that each Sα is a union of finitely many Sβ ’s), plus, for each α , a finite-dimensional vector space Vα of real-valued continuous functions on Sα such that: (a) if nα = dim (Vα ) and F1 , . . . , Fnα is a basis of Vα , then ( fi ) : Sα −→ Rnα is a homeomorphism of Sα with an open convex polyhedral cone Cα ⊂ Rnα ; (b) ( fi )−1 extends to a continuous surjective map ( fi )−1 : Cα −→ Sα

12

I Basics on torus embeddings; examples (β )

mapping the open faces Cα of Cα homeomorphically to the strata Sβ in Sα , and inducing isomorphisms res

(β )

Cα

∼

(linear functions on Rnα ) −→ Vβ .

Note that such a complex has a natural piecewise-linear (or PL) structure. Definition 3.2 An integral structure on a conical polyhedral complex is a set of finitely generated abelian groups Lα ⊂ Vα such that (i) Lα ⊗ R ∼ = Vα ; (ii) if Sβ is a face of Sα , then res Sβ Lα = Lβ . The changes from TE I are: (a) that the collection {Sα } is not supposed to be finite; and (b) we allow two faces of the same polyhedron Cα in Rn to be identified in X. We sketch how to associate a complex Σ = (|Σ|, {Sα }, {Vα }) to a toroidal embedding (X,U) without monodromy in the following. (a) For all strata Y of (X,U), let MY = group of Cartier divisors on Star0 (Y ), supported on Star0 (Y ) \U ∩ Star0 (Y ) , Y M+ NRY Y

= sub-semigroup of effective divisors , = Hom (MY , R) ,

Y σ = {x ∈ NRY | D, x ≥ 0, for all D ∈ M+ }.

(b) For all strata Z0 in Star0 (Y ), let Z be the stratum of X containing Z0 ; then we get a map

αZ0 : MY −→ M Z by restricting a divisor on Star0 (Y ) to the component of Star0 (Y )∩Star0 (Z) containing Z0 , and then extending it to Star0 (Z). This induces an isomorphism ∼

βZ0 : σ Z −→ a face of σ Y . (c) Define |Σ| =

Y



σY

equivalence relation generated by the maps βZ0

Sα = image of Int (σ ) , Yα

Vα = the functions MYα ⊗ R on σ Yα , Lα = the functions MYα on σ Yα .



∼ =

 Y

Int σ Y ,

3 Toroidal embeddings over the complex numbers

13

This is an immediate generalization of the construction of TE I, pp. 59–72. The map “ord” also has an analytic version. Define† ˚ ⊂ U}, R.S.U (X) = {φ : ∆ −→ X holomorphic such that φ (∆) where ∆ is the open unit disc and ∆˚ = ∆ \ {0}. Define ord : R.S.U (X) −→ |Σ| as follows. Let φ ∈ R.S.U (X) with φ (0) ∈ Stratum Y . Then, on some smaller disc ∆ , we have φ (∆ ) ⊂ Star0 (Y ); hence, for all divisors D ∈ MY , the pullback φ ∗ D is a divisor on ∆ with a definite multiplicity ord0 (φ ∗ D) at 0. Define ord(φ ) in σ Y by D, ord(φ ) = ord0 (φ ∗ D) , and define ord(φ ) in |Σ| as the image of this. We may also give a purely topological definition of ord via monodromy. In fact, suppose we choose a nice neighborhood Star0 (Y) so that Star0 (Y ) ≈ Xτ ×Y . Note that in this case we get isomorphisms MY ∼ = M(T ) ; hence NRY ∼ = N(T )R and

σY ∼ =τ . In particular this shows that

π1 (Star0 (Y ) ∩U) ∼ = π1 (T ×Y ) ∼ = π1 (F) × π1 (Y ) ∼ = N(T ) × π1 (Y ) ∼ = NY × π1 (Star0 (Y )) ; hence ∼

Ker [π1 (Star0 (Y ) ∩U) −→ π1 (Star0 (Y ))] −→ NY . ζ

Using this isomorphism, we find: † For the justification of the notation “R. S.” (short for ”Riemann surface” as used by Zariski), see TE I, p. 64.

14

I Basics on torus embeddings; examples

Proposition 3.3 Let φ ∈ R.S.U (X) and assume φ (0) ∈ stratum Y . For c small, we have res φ : ∆c = {z | |z| ≤ c} −→ Star0 (Y ) ; hence res φ induces

φ∗ : π1 (∂ ∆c ) −→ Ker [π1 (Star0 (Y ) ∩U) −→ π1 (Star0 (Y ))] . If 1 is the canonical generator of the left-hand side, then

ζ (φ∗ (1)) = ord(φ ) . ∼

Proof By the way that ζ : π1 (T ) −→ N(T ) is defined, it follows that, for every loop λ in T , and every character Xα , 2π α , ζ (λ ) = change in arg Xα around the loop λ . Now let T ⊂ Xσ be a torus embedding, t ∈ Xσ , let V be a neighborhood of t, and let u · Xα be a function on V , where u is a unit on V ; moreover, let λ be a loop in V arising by restricting to ∂ ∆c a holomorphic map φ : ∆c −→ V . Then 2π α , ζ (λ ) = change in arg (u · Xα ) around the loop λ . Next, go over to a toroidal embedding U ⊂ X, let x ∈ X and let V be a neighborhood of x, and let δ be a meromorphic function on V with no zeroes or poles on V ∩U. Moreover, let λ be a loop in V arising by restricting to ∂ ∆c a holomorphic map φ : ∆c −→ V . Then the principal divisor (δ ) is an element of MY , and 2π (δ ), ζ (λ ) = change in arg δ around the loop λ = change in arg (δ ◦ φ ) on |z| = c = 2π · (order of zero or pole of δ ◦ φ at 0) = 2π (δ ), ord(φ ) . Since λ = φ∗ (1), this proves what we want.

4 Compactification of the universal elliptic curve We now take up perhaps the simplest example of our theory. We deal first with this example over the complex numbers. Fix an integer k ≥ 3 once and for all. Let    a b Γ= | a ≡ d ≡ 1 mod k, b ≡ c ≡ 0 mod k, ad − bc = 1 c d

4 Compactification of the universal elliptic curve

15

act on the upper half plane H by aω + b . cω + d

ω −→ Let

ΓA = Γ
Z2 (semidirect product, where Z2 is normal with Γ acting on Z2 by (m, n) −→ (am + cn, bm + dn) ; “A” stands for “affine”). Then ΓA acts on C × H by  z + mω + n aω + b  , . (z, ω ) −→ cω + d cω + d Then M = H/Γ is the moduli space for elliptic curves with level-k structure and X = (C × H)/ΓA is the universal level-k elliptic curve over M, via the canonical projection p : X −→ M (see, for example, Lang [2]). The problem is that M, and hence X, are not compact and we seek compactifications:  / X˜ X  / M˜ .

  M

The usual procedure for M is to note that, since it is one-dimensional, there is a  such that M = M  \{finite set}. unique non-singular complete algebraic curve M  Then, from the theory of algebraic surfaces, one can also find a canonical X:  These the unique so-called non-singular relatively minimal model over M. methods do not however generalize to higher-dimensional cases, and we seek  and X by a more direct “scissors and glue” construction involvto describe M ing torus embeddings. We deal first with the cusp i∞ ∈ ∂ H. Consider the subgroup    1 b | b ≡ 0 mod k Γ1 = 0 1 and factor π : H −→ H/Γ = M via exp

H? ?? ?? π ??? 

/ ˚ ∆    π    , M

where q is the coordinate on ∆˚ and exp is defined by q = e2π iω /k .

16

I Basics on torus embeddings; examples

This makes ∆˚ isomorphic to H/Γ1 , hence π factors via exp. Moreover, define Hd = {ω | Im ω ≥ d} , ∆˚ d = {q | 0 < |q| ≤ e−2π d/k } . Then Hd = exp−1 (∆˚ d ) and ∆˚ d ∼ = Hd /Γ1 . The following lemma is easy to check. Lemma 4.1 There exists d0 such that, for all ω ∈ H, γ ∈ Γ,

ω and γω ∈ Hd0 =⇒ γ ∈ Γ1 . Therefore res π  maps ∆˚ d0 injectively to M:

Moreover, as d −→ ∞, it is well known that the sets π (Hd ) ⊂ M are a fundamental system of neighborhoods of the cusp i∞. Therefore, we find that we can glue via this map by taking M plus ∆d0 = {q | |q| ≤ e−2π d0 /k } and identifying them via res π  on ∆˚ d0 . Next, every rational point p/q ∈ ∂ H also defines a cusp of M, except that p/q and γ (p/q) for γ ∈ Γ define the same cusp. Now, a fundamental system  are the sets π (Wd (p/q)), where Wd (p/q) is of neighborhoods of p/q in M the closed disc in H of radius d, tangent to the real axis at p/q (a so-called horocycle) and, if d is small enough, the Γ-equivalence of points of Wd (p/q) becomes Γ1 (p/q)-equivalence, where Γ1 (p/q) = {γ ∈ Γ | γ (p/q) = p/q}   n 1 + pq −p2 = |n∈Z . q2 1 − pq  by glueing. But, even So we can mimic the above construction to obtain M more simply, we can use the fact that SL(2, Z) acts transitively on the set of

4 Compactification of the universal elliptic curve

17

rational points plus ∞, hence SL(2, Z)/Γ acts on M and permutes transitively all its cusps. Thus, if we know how to fill in one, we can fill in the others by acting by SL(2, Z)/Γ. Now look upstairs at C × H. Define ⎞ ⎛ subgroup of ΓA generated by ΓA1 = ⎝ (z, ω ) −→ (z + 1, ω ) and ⎠ ∼ = Z2 ; (z, ω ) −→ (z, ω + k)   subgroup of ΓA generated by ΓA1 and A . Γ2 = α : (z, ω ) −→ (z + ω , ω ) Factor π : C × H −→ (C × H)/ΓA = X via: C × HE EE EE E π EEE "

exp

/ C∗ × ∆˚ xx xx x x  |xx π , M

˚ and where exp is defined by where x is the coordinate on C∗ , and q that on ∆, x = e2π iz , q = e2π iω /k . This makes C∗ × ∆˚ isomorphic to (C × H)/ΓA1 . Now, ΓA1 is a normal subgroup ˚ The of ΓA2 and ΓA2 /ΓA1 ∼ = Z, with generator α , and ΓA2 /ΓA1 acts on C∗ × ∆. previous lemma now gives us: Corollary 4.2 There exists d0 such that, for all (z, ω ) ∈ C × H and all γ ∈ ΓA , (z, ω ) and γ (z, ω ) ∈ C × Hd0 =⇒ γ ∈ ΓA2 .

Therefore, res π  : (C∗ × ∆˚ d0 )/{α n } −→ X  it suffices to enlarge C∗ × ∆˚ d to an is injective. To compactify X over i∞ ∈ M, 0 analytic manifold Y over ∆d0 , equivariantly with respect to the action of α and so that, mod α , we get a manifold proper over ∆d0 :  /Y O C∗ × ∆˚ d0 OOOα acting OOO O' Y /{α n } o oo    wooooproper  / ∆d0 ∆˚ d0

18

I Basics on torus embeddings; examples

Here is where tori come in: think of C∗ × ∆˚ as an open subset of the twodimensional torus C∗ × C∗ (with coordinates x, q). Thus α acts on the whole torus by

(x, q) −→ (qk x, q) .

We shall construct a torus embedding C∗ × C∗ ⊂ X{σα } and the sought-for analytic manifold Y will be the closure of C∗ × ∆˚ d0 in X{σα } . In fact, identify N(C∗ × C∗ ) with Z × Z and note that α acts on N(C∗ × C∗ ) by

(a, b) −→ (a + kb, b) .

We choose {σα } to be the following infinite chain σn , n ∈ Z:

Note that α carries σn to σn+k , so that, mod α , there are only finitely many

4 Compactification of the universal elliptic curve

19

σ . The corresponding X{σn } may be pictured as follows:

Clearly α acts on X{σn } . Since each σi is generated by a basis of Z × Z, it follows that X{σn } is a manifold, i.e., smooth. Moreover, a whole neighborhood of the boundary X{σn } \ (C∗ × C∗ ) is contained in C∗ × ∆˚ d0 , so define Y to be

Y = interior of closure of C∗ × ∆˚ d0 in X{σn }   = (C∗ × ∆˚ d0 ) ∪ X{σn } \ (C∗ × C∗ ) .

What happens when we divide by α ? Clearly α does not act discontinuously on the whole of C∗ × C∗ (i.e., if |q| = 1) so we cannot form (C∗ × C∗ )/{α n }. However, it can be checked to act discontinuously on Y , and the quotient looks

20

I Basics on torus embeddings; examples

like this:

 by glueing X and We can now define that part of X that lies over i∞ ∈ M n ∗ n Y /{α } together on the common open set (C × ∆˚ d0 )/{α }. As before, we can take care of the other cusps by pushing this boundary around by SL(2, Z)/Γ.  proper over M,  with its fibers This gives us a compact non-singular surface X, elliptic curves over M and rational k-gons over the cusps. In this way, we find an analytic construction not only of M and X, but also of their natural  and X.  completions M We want to study briefly this same circle of ideas from another point of view, similar to that of the articles of Deligne and Rapoport [1] and of Mumford [3]. Suppose we are over an arbitrary ground field L with a fixed primitive kth root of unity ζ (here (char L, k) = 1; actually the same ideas work over suitable  and X over L as the schemes which ground rings too). Then we can define M represent a certain functor. Restricting ourselves to the corresponding formal  and X are quotients of schemes at the cusps, we can then see directly that M the formal neighborhood of the locus at infinity in the same torus embeddings introduced above. We need some definitions. (a) If E is an elliptic curve with given origin e, a level-k-structure on E is a pair of points x, y ∈ E of order k such that ek (x, y) = ζ (here ek is Weil’s pairing); equivalently, if L is a line bundle of degree k, and φ , ψ : L −→ L are automorphisms lifting translation by x, resp. y, then

φ ◦ ψ ◦ φ −1 ◦ ψ −1 = multiplication by ζ . (b) If E is a k-gon of rational curves with given group law on E0 ⊂ E (where E0 are the smooth points of E), a level-k-structure on E is a pair of points x, y ∈ E0 of order k such that ek (x, y) = ζ . Here, to define ek , let L be a line

4 Compactification of the universal elliptic curve

21

bundle of degree 1 on each component of E: then there exist φ , ψ : L −→ L lifting translation by x, y : E −→ E. Then

φ ◦ ψ ◦ φ −1 ◦ ψ −1 = multiplication by ζ . Then consider the following functor on schemes over L: ⎧ ⎪ set of 4-tuples (p, σ , u, v) where: ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a) p : Y −→ S is a proper flat morphism, all fibers of ⎪ ⎪ ⎪ ⎪ which are elliptic curves or k-gons of rational curves; ⎪ ⎪ ⎨ (b) σ : Y0 ×S Y −→ Y is a morphism (here Y0 is the  M(S) = ⎪ open set in Y where p is smooth), making Y0 a group ⎪ ⎪ ⎪ ⎪ scheme over S and making Y0 act on Y ; † ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (c) u, v : S −→ Y0 are sections of order k inducing a ⎪ ⎪ ⎩ level-k-structure on each fiber.

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

 be the subfunctor of (p, σ , u, v), where p is smooth. Let Let M ⊂ M   set of 5-tuples (p, σ , u, v, w), with p, σ , u, v as before  , X(S) = and w : S −→ Y any section of Y over S  of (p, σ , u, v, w) for which p is smooth. X = subfunctor of X Then the following theorem is well-known (see, e.g., [1]). Theorem 4.3  is represented by a curve M  smooth and proper over L; (a) M  \C, where C is a finite set of “cusps”; (b) M is represented by M  M)  corresponds to the identity map M  −→ M,  and (c) if (p, σ , α , β ) ∈ M(     p is the morphism from X to M, then X represents X, and X = p−1 (M) represents X;  and X are the varieties constructed above. (d) if L = C, then M Fix a cusp c ∈ C, i.e., a 4-tuple p : Y −→ Spec L,

σ : Y 0 ×Spec L Y −→ Y , u, v : Spec L −→ Y , where Y is a k-gon. These are all isomorphic modulo the natural action of  taking SL2 (Z/kZ) on M (p, σ , u, v) −→ (p, σ , au + bv, cu + dv) . † To exclude some stupid examples of σ , one must add the condition that, for every geometric point s ∈ S such that the fiber p−1 (s) is a k-gon, all translations x −→ σ (y, x), y ∈ p−1 (s) ∩Y0 , induce rotations of the graph of the k-gon p−1 (s).

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I Basics on torus embeddings; examples

So let us assume v is in the identity component of Y 0 , and u is in an adjacent one:

# (resp. (When we assume this, (p, σ , u, v) is unique up to isomorphism.) Let M −1 # denote the formal completion of M  (resp. X)  at c (resp. along p (c)). X) Now go back to the torus embeddings introduced earlier:  / X{σn } Gm × Gm p2

 / A1q ,

  Gm

where σn is the set of sectors (n, 1), (n + 1, 1), n ∈ Z. Let α : X{σn } −→ X{σn } be the automorphism (z, q) −→ (zqk , q), where z and q are coordinates on the # 1 ) be the formal completion of first and second factor Gm . Let X#{σn } (resp. A q 1 X{σn } (resp. Aq ) along the complement of the torus. Then we claim that there are canonical isomorphisms: ∼

X#{σn } /{α n }

/ X#

p2

p

 # A1q

∼

 /M #.

We will show here only how to construct: # 1 −→ M; # (a) a morphism A q (b) a map of functors # 1 (S) # M(S) −→ A q for S = Spec R, where R is an Artin local L-algebra with residue field L. We let the reader check that these are inverse to each other and that we get corresponding maps on the curves over these bases.

4 Compactification of the universal elliptic curve

23

#1 Construction (a) The point is that X#{σn } /{α n } itself is flat and proper over A q and its (one) closed fiber is indeed a k-gon. The action of Gm × Gm on X{σn } is a morphism: (Gm × Gm ) ×A1q X{σn } −→ X{σn } , and it is readily checked that this extends to

σ : (X{σn } )0 ×A1q X{σn } −→ X{σn } , where (X{σn } )0 is the open subset where X{σn } is smooth over A1q . This induces

σ : (X#{σn } /{α n })0 ×A1q (X#{σn } /{α n }) −→ X#{σn } /{α n } . Finally, the sections z = q and z = ζ of Gm × Gm over Gm define sections # 1 −→ (X#{σ } /{α n })0 . u, v : A q n # Construction (b) Now start with an element of M(S), where S = Spec R as above, i.e., (p, σ , u, v), with Y −→ S, extending the fixed (p, σ , u, v) defining the cusp. Then Yred = Y is a k-gon of rational curves, so the universal cover Y ∗ ∗ is just an infinite string of copies of Y is locally of finite type over S and Yred of P1 . If we fix an identity e : S −→ Y ∗ over e : S −→ Y , then we get a unique lifting of σ : Y0 ×Y −→ Y to

σ ∗ : Y0∗ ×Y ∗ −→ Y ∗ such that (a) σ ∗ (e, e) = e , (b) σ ∗ (x, y) = σ ∗ (y, x) if x, y ∈ Y0∗ . Next, let Y 0 (e) and Y0 (e) denote the identity components of Y 0 and Y0 . Then there are two isomorphisms Y 0 (e) ∼ = Gm,L , and we can fix one of them by requiring that the point 0 in the closure of Gm,L corresponds to the intersection point Q of the closures of Y 0 (e) and Y 0 (u) in Y . According to a result of Grothendieck [4], exp. IX, §3, tori are “rigid”, so this isomorphism lifts to a unique S-isomorphism Y0 (e) ∼ = Gm,S . But Y0∗ (e) ∼ = Y0 (e), so we get an action of Gm,S on Y ∗ ; now we begin to see why torus embeddings are involved. Let U − be the open subset of Y ∗ consisting of

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I Basics on torus embeddings; examples

“half the string” starting at Y0∗ (e) and in the direction of the limit point 0 of Gm,S :

Also lift u to u∗ : S −→ Y ∗ so as to lie in the component adjacent to Y0∗ (e). Then: Lemma 4.4 There exists a unique X ∈ Γ(U − , OY ∗ ) such that X(e) = 1, and such that, under the action σ0 : Gm,S ×S U − −→ U − of Gm,S on U − ,

σ0∗ (X) = z · X , where z is the coordinate on Gm,S . Proof Decompose Γ(U − , OY ∗ ) into eigenspaces under the action of Gm,S : Γ(U − , OY ∗ ) =

+∞ $

Wn ,

n=−∞

where σ0∗ ( f ) = zn · f if f ∈ Wn . Since U − is flat over R, it follows that Wn is a flat R-module. But Γ(U − , OY ∗ ) =

+∞ $

Wn ⊗R L ,

n=−∞ ∗

and since Y is just a string of copies of P1 , one sees immediately that W1 ⊗R L = L · X ,

5 Hirzebruch’s theory of the Hilbert modular group

25

∗

where the function X is z on Y 0 (e) and 0 on all the other components. Therefore W1 is a free rank-1 R-module, and the condition X(e) = 1 picks out a unique element. Now let σ1 (x) = σ ∗ (u∗ , x), giving us an automorphism σ1 : Y ∗ −→ Y ∗ , and a morphism σ1 : U − −→ U − . Then σ1∗ (X) is another element of Γ(U − , OY ∗ ) in W1 (notation as in the proof of the preceding lemma), so

σ1∗ (X) = q · X , for some q ∈ R . This q is the sought-for period! It defines a homomorphism L[[q]] −→ R , # 1 (S). and hence an element of A q

5 Hirzebruch’s theory of the Hilbert modular group We now investigate a second beautiful example in order to motivate and illustrate further all√ the theory which follows. Let K = Q( d) be a real quadratic number field and associate to it the following objects: O = ring of integers in K, a = a fixed ideal in O, u0 = a generator of the group of units u ∈ O, such that u ≡ 1 mod a (or of this group mod ± 1). We do not regard K as a subfield of R, but rather as an abstract field extension of Q, with two embeddings:

φ1 , φ2 : K −→ R , neither being more important than the other. Let        a b a b 1 0 Γ= ∈ SL(2, O) | ≡ mod a . c d c d 0 1 Consider the following embedding: 

SL(2, K) −→ SL(2, R × SL(2, R)      φ1 (a) φ1 (b) φ2 (a) φ2 (b) a b , . −→ φ1 (c) φ1 (d) φ2 (c) φ2 (d) c d

Then there is a unique Q-structure on SL(2, R) × SL(2, R) with SL(2, K) as its Q-rational points and Γ is an arithmetic subgroup for this Q-structure. We let

26

I Basics on torus embeddings; examples

SL(2, R) × SL(2, R) act componentwise on H × H and seek to compactify the so-called Hilbert modular surface: FK,a = (H × H)/Γ . It is known (see Shimizu [5]) that FK,a can be embedded in a compact normal analytic surface FK,a by adding only a finite number of points, called cusps F1 , . . . , FN :

Topologically, starting with the simplest cusp “i∞”, here is the picture: let Wd = {(z1 , z2 ) | Im z1 · Im z2 ≥ d} ⊂ H × H ;    1 b |b∈a , Γ1 = 0 1 and let

  b un0 Γ2 = |n∈Z, b∈a 0 u−n 0   a b = Γ∩ /Γ ∩ {±1} . 0 d 

Then Γ2 ·Wd = Wd (use the fact that φ1 (u0 ) · φ2 (u0 ) = Norm (u0 ) = ±1: see (ii) below), and it turns out that if d  0, then Γ-equivalence on H × H reduces on Wd to ±Γ2 -equivalence, i.e., z1 , z2 ∈ Wd , z1 = γ z2 for some γ ∈ Γ =⇒ γ ∈ ±Γ2 . Since ±1 acts trivially on H × H, we get Wd /Γ2 ⊂ (H × H)/Γ = FK,a . But Wd /Γ2 is easy to visualize: (i) Γ1 acts by (z1 , z2 ) → (z1 + φ1 (b), z2 + φ2 (b)) and Φ(a) = {(φ1 (b), φ2 (b)) | b ∈ a} is a lattice in R × R;

5 Hirzebruch’s theory of the Hilbert modular group

27

∼ (ii) Γ1 is a normal  Γ2 and Γ2 /Γ1 = Z with generator γ0 equal to  subgroup of 0 u0 . Now Γ1 leaves invariant Im z1 and Im z2 , and the image of 0 u−1 0 Γ2 /Γ1 acts on these by

γ0∗ (Im z1 ) = φ1 (u0 )2 · Im z1 , γ0∗ (Im z2 ) = φ2 (u0 )2 · Im z2 = φ1 (u0 )−2 · Im z2 ; (iii) hence Im z1 · Im z2 is invariant under Γ2 . Think of 1/(Im z1 · Im z2 ) as measuring the distance to the cusp. For every fixed e, if zi = xi + iyi , we get a diagram Me = {(z1 , z2 ) | Im z1 · Im z2 = e}/Γ2 fiber {(x1 ,x2 )∈R2 }/Φ(a)

 {(y1 , y2 ) ∈ R2>0 | y1 · y2 = e}/{γ0n }

≈ S1 ,

homeo

from which it follows that Me is a compact 3-manifold which is an S1 × S1 bundle over S1 . As e varies, the manifolds Me are all homeomorphic and we get (z1 , z2 ) _

∈ Wd /Γ2 fibers Me

  Im z1 · Im z2 ∈ [d, ∞) , i.e.,

Wd /Γ2 ≈ Md × [d∞) . homeo

The compactification Fk,a in the subset Wd /Γ2 simply results, topologically, by embedding Md × [d, ∞) in the cone over Md , i.e., the one-point compactification of Wd /Γ2 . Thus the subsets Md  /Γ2 , as d  −→ ∞, are a fundamental system of neighborhoods of the cusp F1 = i∞. There may be other cusps too: for every γ ∈ SL(2, K), consider   the subset a b , γ (Wd ) ⊂ H × H. If B ⊂ SL(2, K) is the subgroup of matrices 0 a−1 then γ (Wd ) is left invariant by the group Γ ∩ γ Bγ −1 , and if d  0, then, as before,

γ (Wd )/Γ ∩ γ Bγ −1 ⊂ (H × H)/Γ .

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I Basics on torus embeddings; examples

∼ =

Moreover, Γ ∩ γ Bγ −1 ∼ = γ −1 Γγ ∩ B, which is an extension     1 b −1 1 −→ b ∈ K | ∈ γ Γγ −→ γ −1 Γγ ∩ B −→ 0 1     u b −1 −→ 1 ∈ −→ u ∈ K ∗ | γ Γ γ for some b 0 u−1 Z or Z × (Z/2Z) (the 2-torsion comes from u = ±1, and if this occurs it acts trivially on H × H, so we ignore it), from which it follows, as before, that

γ (Wd )/Γ ∩ γ Bγ −1 ≈ M(γ ) × [d, ∞) homeo

for some compact 3-manifold M(γ ) which is an S1 × S1 -bundle over S1 . Again, we make a one-point compactification. Actually, there are only finitely many cusps. In fact   a b (a) if γ  = γ · δ , with δ = ∈ B, then γ  (Wd ) = γ (Wd  ), where 0 a−1 d  = d · Norm (a)2 , and Γ ∩ γ  Bγ −1 = Γ ∩ γ Bγ −1 , so we have the same cusp; (b) if γ  = ε · γ with ε ∈ Γ, then the images of γ (Wd ) and γ  (Wd ) in (H × H)/Γ are the same, hence

γ (Wd )/Γ ∩ γ Bγ −1  Yz Y YYYY, $ eeeee2 (H × H)/Γ γ  (Wd )/Γ ∩ γ  Bγ −1 Hence the cusp depends only on the double coset Γγ B ∈ Γ\SL(2, K)/B. It is easy to check that if Γγ B = Γγ  B, then, for d  0, the images of γ (Wd ) and γ  (Wd ) in (H × H)/Γ are disjoint; hence if we want FK,a to be normal (and we do), we must have different cusps here. Thus #cusps = #(Γ\SL(2, K)/B), which is finite by a classical theorem (in fact, one can check that if a = O, then Γ\SL(2, K)/B ∼ = ideal class group of K) . Now, how do we put an analytic structure on FK,a ? We shall only do this at the end and instead, by following the suggestions made by the above topological construction, plus our knowledge of toroidal embeddings, define directly K,a of FK,a . Again we start with a blown-up non-singular compactification F

5 Hirzebruch’s theory of the Hilbert modular group

29

the cusp i∞. The idea is first to factor the canonical map H × H −→ FK,a as follows: H × H −→ (H × H)/Γ1 −→ (H × H)/Γ2 −→ FK,a . We may embed (H × H)/Γ1 in a torus as follows: Γ1 acts on H × H by translations by the lattice Φ(a) in R × R: (z1 , z2 ) −→ (z1 + φ1 (b), z2 + φ2 (b)) , b ∈ a . Let T be the torus given by C × C modulo the same group of translations: T = C × C/Φ(a) , so that N(T ) = Φ(a) , N(T )R = R × R , and we get the exact sequence: 0

/ Tc

/T

R × R/Φ(a)

C × C/Φ(a)

/ R×R

(z1 , z2 ) 

/ (Im z1 , Im z2 )

ord

/ NR (T )

/0

It follows that: (H × H)/Γ1 ∼ = ord−1 (R>0 × R>0 ) , Wd /Γ1 ∼ = ord−1 ({(y1 , y2 ) | y1 y2 ≥ d, yi > 0}) . % &' ( call this Vd

The first set is open in T , the second set is closed in the first. Next, Γ2 /Γ1 ∼ = {γ0n }, and γ0 acts on H × H by

γ0 (z1 , z2 ) = (φ1 (u0 )2 z1 , φ1 (u0 )−2 z2 ) . Let v0 = φ1 (u0 )2 . Now the action of γ0 on H × H extends to C × C, hence to T and to the open subset (H × H)/Γ1 ⊂ T . In particular, γ0 acts on NR (T ) by

γ0 (x1 , x2 ) = (v0 x1 , v−1 0 x2 ) , an action which preserves the positive quadrant and the (irrational) lattice Φ(a). We have thus arrived at the following situation: T is a two-dimensional torus and γ0 : T −→ T is a hyperbolic automorphism of infinite order (in fact, up to replacing γ0 by a power or a root, we have obtained the most general hyperbolic automorphism of G2m ). At this point, the idea is to enlarge T – and its open subsets (H × H)/Γ1 and Wd /Γ1 – by adding some analytic boundary E , so that γ0 still acts on T ∪ E ,

30

I Basics on torus embeddings; examples

and then to divide by {γ0n } so that E = E /{γ0n } is the boundary that can be added to (H × H)/Γ ‘in the direction i∞’:

To enlarge T , we use the theory of torus embeddings. (i) We seek a decomposition of the positive quadrant R≥0 × R≥0 into rational sectors {σα } .


(ii) These should satisfy: γ0 σα = some σβ ; σα = R≥0 × R≥0 ; σα ∩ σβ = (0) or a common edge. Note that since (0) × R≥0 and R≥0 × (0) are irrational half-lines, all σα are in the interior of the positive quadrant, and we will need an infinite number of σα . However, we require: modulo the action of γ0 , there are only finitely many σα . (iii) In this case, if we denote one of the σα by σ0 , we can number them uniquely . . ., σ−2 , σ−1 , σ0 , σ1 , σ2 , . . . so that σi ∩ σ j is a common edge if and

5 Hirzebruch’s theory of the Hilbert modular group

31

only if i = j ± 1, and γ0 σi = σi+d for all i and some fixed d. Let
i = σi ∩ σi+1 . It looks like this:

(iv) Given such {σα }, we obtain a torus embedding T ⊂ X{σα } , where X{σα } is a scheme locally of finite type over C (in fact, locally a normal complex variety), with the action of γ0 extending to X{σα } . Moreover, X{σα } is the union of T and an infinite chain of non-singular rational curves Ei , one for each half-line
i , meeting at points Pi , one for each sector σi :

(v) Next enlarge (H × H)/Γ1 by setting (H  × H)/Γ1 = (H × H)/Γ1 ∪

+∞

Ei .

i=−∞

In fact, in the corresponding embedding NR (T ) ⊂ N{σi } , it is clear that  R>0 × R>0 = R>0 × R>0 ∪

N{σi } \ NR (T ) % &' ( chain of boundary segments

32

I Basics on torus embeddings; examples

is the interior of the closure of R>0 × R>0 in N{σi } . Therefore taking ord−1 , (H  × H)/Γ1 is the interior of the closure of (H × H)/Γ1 in X{σ } . α

(vi) It is easy to see that Γ2 /Γ1 = {γ0n } acts discontinuously on (H  × H)/Γ1 .  × R>0 with fundamental In fact, check first that it acts discontinuously on R>0   d−1
domain Ω = σi : i=0

>0

×

>0

with quotient looking topologically like this:

Note that Ω∩Vd is compact modulo R>0 , hence Vd /{γ0n } is compact modulo × H)/Γ1 and we get R>0 . Therefore ord−1 (Ω) is a fundamental domain in (H  an analytic space ((H  × H)/Γ1 )/(Γ2 /Γ1 ) , consisting of the open piece (H × H)/Γ2 and a closed analytic set * ) E=

+∞

Ei /{γ0n },

i=−∞

which is a d-sided polygon of rational curves E 0 ∪ · · · ∪ E d−1 (with E i the

5 Hirzebruch’s theory of the Hilbert modular group

33

image of Ei ):

(Here E i −→ νi and E i ∩ E i+1 −→ νi ∩ νi+1 under the ord map.) Moreover, in here, Wd /Γ2 ∪ E is compact (since ord is proper). Thus, as above, since Wd /Γ2 ⊂ FK,a , we can form FK,a ∪ E and make it into an analytic space by the above analytic structures on the two subsets FK,a and Wd /Γ2 ∪ E. To recover our previous compactification of the cusp i∞, we shall blow down E to a point. This can be accomplished by checking that E has a fundamental system of strongly pseudoconvex neighborhoods. But in fact, Vd is a strongly convex subset of R>0 × R>0 in the following sense: for all x ∈ ∂ Vd , the subset Vd is defined near x by an equation ϕx ≥ 0, where, for all t = 0 in the tangent space to ∂ Vd at x, defined by dϕx (t) = 0, we have d2 ϕx (t,t) < 0 . It is an easy lemma that, if a closed subset W ⊂ Cn is defined at a boundary point z ∈ ∂ W by

ϕz (Re z1 , . . . , Re zn ) ≥ 0 , then W is strongly pseudoconvex at z if and only if the closed set V ⊂ Rn given by ϕz (x1 , . . . , xn ) ≥ 0 is strongly convex. Since log |zi | = Re log zi , this implies that Wd is strongly pseudoconvex. Therefore d−1

E i ⊂ ((H  × H)/Γ1 )/(Γ2 /Γ1 )

i=0

is an exceptional set and can be blown down to a point. But the advantage of our boundary is that we can make it non-singular. In fact, let ei be the smallest integral point on the half-line
i , i.e., ei ∈
i ∩ Φ(a). Then from TE I, we know: X{σi } non-singular ⇐⇒ ei and ei+1 generate the lattice Φ(a) for all i .

34

I Basics on torus embeddings; examples

But it is easy to check that, for any two e, e ∈ Φ(a),  the intersection of Φ(a) and the  e, e generate Φ(a) ⇐⇒ . triangle 0, e, e contains only 0, e and e Now introduce Σ = convex hull of R>0 × R>0 ∩ Φ(a) .

It follows that if X{σi } is non-singular, then the ei must include all lattice points on ∂ Σ. On the other hand, if we take the ei to be precisely the points Φ(a)∩ ∂ Σ, and let
i = R≥0 ei , σi = ei−1 , ei , then the corresponding X{σi } is non-singular. So this is the minimal non-singular choice of X{σi } and leads to the minimal desingularization of the old boundary (H × H)/Γ2 ∪ (one point). One can, by the way, easily compute from the theory of TE I the intersection matrix (E i ·E j ) × H)/Γ1 )/(Γ2 /Γ1 ) by the following recipe: for the curves E i on ((H  (a) number the E i cyclically, i.e., E i+d = E i ; (b) then, if i = j,  0 if |i − j| > 1 (E i · E j ) = 1 if |i − j| = 1; 2

(c) (E i ) = −di , where di is the index in Φ(a) of the subgroup generated by ei−1 and ei+1 ; alternatively, di = 2 · area(0, ei−1 , ei+1 ) , if the area is normalized so that area(R × R/Φ(a)) = 1. Also, it is well-known that the vertices of Σ are nothing but the consecutive

5 Hirzebruch’s theory of the Hilbert modular group

35

best rational approximations to the irrational lines (0) × R and R × (0), generated by the continued fraction expansion of the slope of these lines (expressed via a basis of Φ(a)). In the same way, we can handle the other cusps and glue in polygons of rational curves there. For every cusp F corresponding to Γγ B, the isomorphism ∼

(H × H)/Γ −→ (H × H)/γ −1 Γγ , x −→ γ −1 x, carries F to the standard cusp i∞, so it suffices to repeat the above construction, replacing Γ by γ −1 Γγ and then to carry back the resulting boundary piece E to (H × H)/Γ via the above isomorphism.

References [1] P. Deligne and M. Rapoport, Les sch´emas de modules de courbes elliptiques, in Modular Functions of One Variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972). Lecture Notes in Mathematics 349. Berlin: Springer, 1973, pp. 143–316. [2] S. Lang, Elliptic Functions. Reading, MA: Addison-Wesley Publishing Co., Inc., 1973. With an appendix by J. Tate. [3] D. Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24 (1972), 239–272. [4] M. Demazure et A. Grothendieck (eds.), S´eminaire de G´eom´etrie Alg´ebrique du Bois Marie 1962/64–Sch´emas en groupes (SGA 3)–Vol.2. Lecture Notes in Mathematics 152. New York: Springer-Verlag, 1970. [5] H. Shimizu, On discontinuous groups operating on the product of the upper half planes, Ann. of Math. 77 (2) (1963), 33–71.

II Polyhedral reduction theory in self-adjoint cones

Minkowski was the first to demonstrate the existence of a polyhedral fundamental domain for the action of GL(n, Z) on the self-adjoint cone of all positive-definite quadratic forms in n variables with real coefficients. His work was generalized by Weyl and others to many other cases. Recently, A. Borel has produced a theory of coarse fundamental domains (called Siegel sets) for any arithmetic subgroup Γ of a reductive algebraic group with Q-structure. Using this we have gone back to exact polyhedral fundamental domains, showing their existence in a unified approach for any arithmetic subgroup acting on a self-adjoint homogeneous cone, C. More precisely, a set of closed polyhedral cones {σα }, such that σα ⊂ C and σα is spanned by a set of rational vertices for every α , is called a Γ-admissible polyhedral decomposition of C when the following hold: (1) (2) (3) (4) (5)

a face of a σα is a σβ ; σα ∩ σβ is a common face of σα and σβ ; γσα is a σβ , for all γ ∈ Γ; modulo Γ, there are only a finite number of σα ;
C = α (σα ∩C).

The result that such decompositions exist is proven at the end of Section 4. Section 1 and part of Section 2 follows closely some notes of Deligne (unpublished). The first two sections describe the work of Vinberg and Koecher on self-adjoint homogeneous cones and their connection with Jordan algebras. In Section 1 the cones appear by themselves, while in Section 2, after some background on Jordan algebras, the link between cone and algebra is explained in detail. Several small facts for later use also find proof in Section 2. Section 3 analyzes the structure of the cone, especially with respect to the way its boundary breaks up into the disjoint union of lower-dimensional selfadjoint homogeneous cones, called boundary components. The Peirce decomposition is the main tool used here. We derive a correspondence between max37

38

II Polyhedral reduction theory in self-adjoint cones

imal split tori in the automorphism group of the cone and maximal strictly commutative rational subalgebras of the Jordan algebra. An easy computation then yields an explicit description of the rational root-space structure of the group of automorphisms. We use the results of Section 3 to prove in Section 4 the main reduction theorem, roughly that Siegel sets and rational polyhedral cones are “cofinal” with respect to inclusion. In Section 5 we define “cores” and “co-cores” with two purposes: (1) the theorem near the beginning will be used later in comparing the topologies of various compactifications of the locally symmetric varieties D/Γ; and (2) the construction of explicit polyhedral fundamental domains in the spirit of Voronoi. If L is a lattice in V giving the Q-structure on V , then a typical core is the closed convex hull of C ∩ L. A typical co-core is the closed convex hull of (C ∩ L) \ {0}. By taking the cones over the faces of a co-core, one obtains a Γ-admissible polyhedral decomposition of the cone. Everything in this section uses the existence of some polyhedral fundamental domain, proved in Section 4.

1 Homogeneous self-adjoint cones 1.1 Let V be a finite-dimensional real vector space. We call C ⊂ V \ {0} a cone if C is open and if R>0C = C. We say that C is non-degenerate if C does not contain an entire straight line. Another expression for this property is given as ∗ follows. Let C ⊂ V ∗ be the set of linear forms, ≥ 0 on C. The dual cone C∗ ∗ is the interior of C \ {0}. Then a convex cone C is non-degenerate if and only ∗ / One always has C ⊂ C∗∗ with equality if and only if C is convex if C = 0. non-degenerate (or C = V \ {0}). We say that C is self-dual (or self-adjoint) if there exists a positive-definite form on V such that the resulting isomorphism between V and V ∗ transforms C into C∗ .

1.2 Let C ⊂ V be a convex non-degenerate cone, with C∗ ⊂ V ∗ its dual. Let G = Aut (C,V ) be the group of linear transformations of V which preserve C. We are going to define the characteristic function of C. Fix dual Haar measures dx, dx∗ on V and V ∗ . Proposition 1.1 The expression

ϕ (x) =

+ C∗

∗

e−x,x  dx∗

1 Homogeneous self-adjoint cones

39

defines a real-valued function on C. Proof Fix a point x ∈ C. The following condition defines a Haar measure dx1∗ on the hyperplane Hα = {x∗ | x, x∗  = α } ⊂ V ∗ : For any continuous function f of compact support on V ∗ we have +

∗

+∞

∗

f (x )dx = V∗

dα

−∞

We can thus write

ϕ (x) =

+∞

+

f (x1∗ )dx1∗ .

Hα

e−α dα

+

dx1∗ .

Hα ∩C∗

0

But, as is easily seen, Hα ∩C∗ is compact; thus the volume v(α ) of Hα ∩C∗ is finite. Since Hα is obtained from H1 by a homothety with coefficient α ,

So ϕ (x) = v(1)

+ ∞ 0

v(α ) = α n−1 v(1) . e−α α n−1 dα < ∞.

Remark 1.2 The quantity ϕ (x) is canonical up to multiplication by a constant (depending on the choice of the Haar measures). Lemma 1.3 We have ϕ (gx) =

1 det(g) ϕ (x).

In particular, the measure

µ = ϕ (x)dx on C is G-invariant. Proof Indeed,

ϕ (gx) =

+

∗

e−gx,x  dx∗ =

C∗

+

∗

e−x, gx  dx∗ . t

C∗

Now make a change of variables. In particular, for λ ∈ R>0 , we have

ϕ (λ x) =

1 ϕ (x) , λn

where n = dimV . Recall that a continuous function f on a convex subset M of affine space is (strictly) convex if, for all x1 , x2 ∈ M, for any point x on the interval joining x1 and x2 and dividing it into the ratio p : q (where p, q > 0 and p + q = 1), we have f (x) < p f (x1 ) + q f (x2 ). If M is open and f ∈ C2 , a sufficient condition

40

II Polyhedral reduction theory in self-adjoint cones

for f to be convex is that the quadratic form d2 f be positive-definite at all points of M. Proposition 1.4 (i) log ϕ is convex; (ii) ϕ is convex. Proof We have  2 d2 ϕ dϕ d log ϕ = − . ϕ ϕ

dϕ d log ϕ = , ϕ

2

From this it follows that it suffices to show that d2 log ϕ is positive-definite. To prove this, we calculate, for x ∈ C and a ∈ V = Tx (C): (dϕ (x))(a) = −

+

∗

e−x,x  a, x∗  dx∗

C∗

+

(d2 ϕ (x))(a) =

∗

e−x,x  a, x∗ 2 dx∗ .

C∗ 1

∗

1

∗

Put F(x∗ ) = e− 2 x,x  , G(x∗ ) = e− 2 x,x  a, x∗ . Then, for a = 0, ⎡ ⎞2 ⎤ ⎛ + + + 1 ⎢ ⎥ 2 ∗ (d2 log ϕ (x))(a) = G2 dx∗ − ⎝ FG dx∗ ⎠ ⎦ > 0 , ⎣ F dx (ϕ (x))2 C∗

C∗

C∗

because F and G are not proportional functions. As a consequence of this proposition we have that g=

d2 log ϕ (x) 2 dx d2 x

defines a G-invariant Riemannian metric on C. Example 1.5 Let V = R, C = R>0 ⊂ R. Then ϕ (x) = metric g on R>0 is g =

dx2 . x2

1 x

and the Riemannian

Proposition 1.6 ϕ goes to infinity upon approach of a point of the boundary of C. 

Proof Indeed, if xk , k = 1, 2, . . ., converges to x0 ∈ ∂ C, then fk (x ) = e−xk ,x   converges to f0 (x ) = e−x0 ,x  uniformly on any bounded set in V ∗ . So, it

1 Homogeneous self-adjoint cones suffices to show that

+

41

f0 (x∗ )dx∗

C∗

is a divergent integral. Let x0∗ ∈ C∗ with x0∗ = 0, x0 , x0∗  = 0. Take a small ball K lying entirely in ∗ f0 (x∗ ). Then c > 0 C and consider the set L = K + R≥0 x0∗ ⊂ C∗ . Let c = min ∗ x ∈K

f0 (x∗ ). So we have and c = min ∗ x ∈L

+ C∗

f0 (x∗ )dx∗ ≥

+ L

f0 (x∗ )dx∗ ≥ c

+

dx∗ .

L

The final expression is indeed infinite since L has infinite volume.

1.3 Proposition 1.7 Let C be a convex non-degenerate cone in V and let G = Aut (C,V ). Then G is a Lie group, the stabilizer of e ∈ C in G is compact and maximally compact if C is homogeneous. Proof The first two assertions follow from the existence of the G-invariant Riemannian metric g on C. The final one follows from the fact that any compact subgroup of G has a fixed point in C, as one proves by the usual averaging method. Proposition 1.8 Let C ⊂ V be an open set in V , such that C∗ = 0. / Let G ⊆ Aut (C,V ). If there is e ∈ C such that G · e is open in V , then G · e = C and C is a convex homogeneous cone under G. Corollary 1.9 Let C ⊂ V be a convex homogeneous cone under G. Then C∗ ⊂ V ∗ is a convex homogeneous cone under the dual group G∗ . Proof (of Proposition 1.8) First, G · e ⊂ C ⊂ C∗∗ , and C∗∗ still satisfies the hypotheses of the proposition. So we can suppose that C is non-degenerate convex. Let r > 0 such that the ball with radius r around e (with respect to the Riemannian metric g) is contained in G · e. For every sequence of points xi such that d(xi−1 , xi ) < r and x0 = e we get xn ∈ G · e (because g is G-invariant). Since C is connected, C = G · e. Proposition 1.10 Let G be a reductive connected algebraic group and let (V, ρ ) be a representation of G all defined over R. Let e ∈ V and G = G (R)o .

42

II Polyhedral reduction theory in self-adjoint cones

Suppose that the stabilizer in G of e is maximal compact and that G · e is open. Then G · e is a homogeneous self-adjoint cone. Conversely, the automorphism group of a self-adjoint cone is reductive. Proof Let K be the stabilizer of e. Let σ be the Cartan involution and let B be a positive-definite symmetric bilinear form on V such that ρ (σ (g)) = t ρ (g)−1 . The corresponding Cartan decompositions are given by Lie (G) = k ⊕ p , G = K · exp(p) = K · P . Let g1 , g2 ∈ G and let σ (g2 )−1 g1 = p2 k, with k ∈ K, p ∈ P. Then g1 e, g2 e = σ (g2 )−1 g1 e, e = p2 e, e = pe, pe > 0 . Thus, identifying V with V ∗ by B, we have G · e ⊂ (G · e)∗ . But (G · e)∗ is stable under G: for x ∈ (G · e)∗ , h ∈ G, and g ∈ G, ge, hx = σ (h)ge, x > 0 . Then Proposition 1.8 applied to (G · e)∗ shows that G · e = (G · e)∗ . Conversely, if C is self-adjoint, then Aut (C,V ) is stable under g −→ t g (transpose with respect to the positive-definite metric on V making C self-adjoint), and consequently G is reductive. Remark 1.11 It may appear at present that the concept of a self-adjoint homogeneous cone is very wide. This is not so. Any convex cone C splits in a unique way into the direct sum of indecomposable convex cones Ci . Then C is homogeneous (respectively self-adjoint) if and only if all the Ci are. Now the indecomposable self-adjoint homogeneous cones have been completely classified. They are: (1) (2) (3) (4)

the cone of positive-definite symmetric matrices; the cone of positive-definite hermitian complex matrices; the cone of positive-definite hermitian quaternion matrices; the spherical cone in Rn+1 of (x0 , . . . , xn ) with 2 x0 > x12 + . . . + xn2 ;

(5) the 27-dimensional cone of positive-definite hermitian octavic matrices of third order. Here (1)–(3) are classical, (4) is semi-classical, and (5) is exceptional.

2 Jordan algebras

43

2 Jordan algebras Jordan algebras arose in quantum mechanics from the desire to make operators commute, at the expense of their associativity. Their usefulness for our purposes comes from the one-to-one correspondence between self-adjoint homogeneous cones and formally real Jordan algebras. For comprehensive references, see [5], [7], [8], and [13]. Definition 2.1 A Jordan algebra A over a field k (of char k = 2) is a finitedimensional (non-associative) algebra with unit element p such that, for all a, b ∈ A (i) a · b = b · a; (ii) a2 · (b · a) = (a2 · b) · a, where a2 = a · a. Although not strictly necessary, we will assume for convenience that char k = 2, 3. In fact, we will only need the case char k = 0 in applications. Start with a Jordan algebra A over k. For a ∈ A, we denote by La multiplication by a: La (x) = x · a . Set S(a, b, c, d) = (a · b) · (c · d) + (a · c) · (b · d) + (a · d) · (b · c) . By (i) this is symmetric in a, b, c, d. Set T (a; b, c, d) = (a · (b · c)) · d + (a · (c · d)) · b + (a · (d · b)) · c . Again using (i), one sees that T is symmetric in b, c, d. Then S(a, b, c, d) = T (a; b, c, d) . Indeed, for b = c = d this identity reduces to (ii); the general case follows from this by polarization, since 6 is invertible in k. In particular, T is symmetric. This can be expressed as T (a; b, c, d) = T (b; a, c, d) , which can be reformulated by saying that Da,b = [La , Lb ] is a derivation, i.e., Da,b (c · d) = c · Da,b (d) + d · Da,b (c) . Theorem 2.2 (Vinberg) Let A be an algebra with unit over a field k of characteristic = 2, 3. Let t : A −→ k be a linear form, the “trace”, such that (a) A is commutative and Da,b is a derivation; (b) t([La , Lb ](c)) = 0;

44

II Polyhedral reduction theory in self-adjoint cones

(c) t(a · b) is a non-degenerate bilinear form. Then A is a Jordan algebra. Proof It suffices to prove S = T . Set {a, b, c, d; e} = t((S(a, b, c, d) − T (a; b, c, d)) · e) . We have just seen that assumption (a) is equivalent to T being symmetric. Hence {a, b, c, d; e} is symmetric in a, b, c, d. Observe that S(a, b, c, d) − T (a; b, c, d) = (Da,b·c + Db,c·a + Dc,a·b )(d) .

(2.1)

It follows with (a) and (b) that {a, b, c, d; e} is anti-symmetric in d and e: indeed, {a, b, c, d; e} + {a, b, c, e; d} = t((Da,b·c + Db,c·a + Dc,a·b )(d · e)) = 0 . But this implies that the expression { } is identically 0: indeed, {a, b, c, d; e} = −{a, b, c, e; d}; but, whereas the L.H.S. is symmetric in c and d, the R.H.S. is anti-symmetric in c and d. Since the element e is arbitrary, assumption (c) now implies the assertion.

Definition 2.3 If A is a Jordan algebra, x, y ∈ A, then x and y are said to commute strictly (or strongly) if Lx and Ly commute. A subalgebra B ⊂ A is strongly associative if any two elements x, y ∈ B strongly commute. (This makes sense because it implies that, for all x, y, z ∈ B, x(zy) = Lx Ly (z) = Ly Lx (z) = (xz)y .) Proposition 2.4 For all x ∈ A, define xn inductively by xn = x · xn−1 . Then: (a) xi , x j strongly commute for all i, j ≥ 0; (b) xi · x j = xi+ j ; (c) p, x, x2 , . . . span a strongly associative subalgebra k[x] ⊂ A. Proof Interpret S = T by taking d as a variable. It then says: La(bc) = Lab · Lc + Lac · Lb + Lbc · La − Lb · La · Lc − Lc · La · Lb .

(2.2)

In particular, if a = b = x, c = xk , this says: Lxk+2 = 2Lxk+1 · Lx + Lx2 · Lxk − Lxk · Lx2 − Lx2 · Lxk ,

(2.3)

2 Jordan algebras

45

hence all Lxk belong to the subalgebra of Hom k (A, A) generated by Lx , Lx2 . But condition (ii) in the definition of a Jordan algebra says that Lx , Lx2 commute. So all Lxk commute. Finally xi · x j = Lxi (Lx j−1 (x)) = Lx j−1 (Lxi (x)) = x j−1 · xi+1 , which, by a simple induction, proves (b), and then (c) is obvious. A useful identity is: Lemma 2.5 For all x, y ∈ A, x and y2 strictly commute if and only if x · y and y strictly commute. Proof In the equation S = T , take b = c = y, a = x to find 2[Lx·y , Ly ] = [Lx , Ly2 ]. Definition 2.6 For any x ∈ A, we say that y ∈ A is the inverse of x if x and y strictly commute and x · y = p. Note that an inverse is unique: if y1 , y2 were inverses y1 = y1 · (x · y2 ) = x · (y1 · y2 ) (x, y1 strongly commute) = y2 · (x · y1 ) (x, y2 strongly commute) = y2 . Lemma 2.7 If x has an inverse x−1 , then x−1 ∈ k[x]. Proof In fact by Lemma 2.5, x−1 and x2 strictly commute. Since Lxk , for k ≥ 3, are polynomials in Lx and Lx2 , in fact x−1 and xk strictly commute for all k ≥ 1. Then if x were a zero divisor in k[x], i.e., x · f (x) = 0, for f (x) = 0, one would get 0 = x−1 · (x · f (x)) = f (x) · (x · x−1 ) = f (x) , which is a contradiction. But k[x] is finite-dimensional, so every non-zero divisor has an inverse in k[x]. Corollary 2.8 If x has an inverse, then x2 has an inverse, and (x2 )−1 = (x−1 )2 . We now take up the Peirce decomposition of a Jordan algebra with respect to an idempotent ε ∈ A. By the recursion formula (2.3), Lε satisfies the identity ϕ (Lε ) = 0, where

ϕ (T ) = 2T 3 − 3T 2 + T .

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II Polyhedral reduction theory in self-adjoint cones

Define

ϕ0 (T ) = 2T 2 − 3T + 1 , ϕ 1 (T ) = −4T 2 + 4T , 2

ϕ1 (T ) = 2T 2 − T . Then ϕ0 + ϕ 1 + ϕ1 = 1, and ϕ divides (T − λ )ϕλ (T ) for λ = 0, 12 , 1. If x ∈ A, 2 we get x = x0 + x 1 + x1 with xλ = ϕλ (Lε )x, and ε · xλ = λ xλ . 2 Writing Aλ = ϕλ (Lε )A, we have A = A0 ⊕ A 1 ⊕ A1 2

and Aλ is the λ -eigenspace for Lε . We write Aλ (ε ) for Aλ , if necessary to avoid confusion. This is called the Peirce decomposition of A. Note that Lemma 2.5 with x ∈ A0 , y = ε , implies that ε and A0 strongly commute; hence Lx , for all x ∈ A0 , preserves the eigenspaces of ε . Similarly, Lemma 2.5 with x ∈ A1 , y = p − ε , implies that ε and A1 strongly commute; hence Lx , for all x ∈ A1 , preserves the eigenspaces of ε . In particular, if x ∈ A0 , y ∈ A1 , then x · y ∈ A0 ∩ A1 = (0). In fact, recall that S = T and hence, by (2.1), Da,b·c + Db,c·a + Dc,a·b = 0 . Take a = x, b = y, c = ε to see that x and y strongly commute. If one takes a = ε and b, c ∈ A 1 , and evaluates at d = ε , we see that ε · (ε · (b · c)) = ε · (b · c), i.e., 2 ϕ 1 (Lε )(b · c) = 0. Thus b · c ∈ A0 + A1 . We summarize this in the following 2 multiplication table:

µ \λ 0

0 0

1/2 1/2

1/2 1

1/2 product zero, strongly commute

0 + 1 1/2

1 product zero, strongly commute 1/2 1

Here if x ∈ Aµ , y ∈ Aλ , then x · y ∈ Aν , where ν is graphed as a function of µ , λ in the multiplication table above. Corollary 2.9 With respect to the inner product x, y = Tr (Lx·y ), the spaces A0 , A1/2 , and A1 are perpendicular. We call a set of idempotents ε1 , . . . , εn mutually orthogonal if εi · ε j = δi j · εi , for 1 ≤ i, j ≤ n. By Lemma 2.5 with x = εi , y = ε j , the εi all strictly commute, 3 and generate a strongly associative subalgebra W = ni=1 kεi .

2 Jordan algebras

47

We say {ε1 , . . . , εn } is a complete set of mutually orthogonal idempotents if ε1 + · · · + εn = p. In this case, we can have a Peirce decomposition for all the εi 3 at once because the Lεi commute with each other. We write A = Aν , where the Aν are simultaneous eigenspaces for Lε1 , . . . , Lεn . Because Lε1 + · · · + Lεn = 3 id , and the various eigenvalues are all 0, 12 , or 1, we end up with A = i≤ j , Ai j where, if x ∈ Ai j ,

εk · x = 12 (δki + δk j )x . Also, if x ∈ Ai j , y ∈ Ak
, and (i, j) = (k,
), then Tr (Lx·y ) = 0. We get the finest decomposition from a maximal set of mutually orthogonal idempotents, that is {ε1 , · · · , εn } such that the n is maximal for sets of orthogonal idempotents. Note that a maximal set is complete: if f = ε1 + · · · + εn = p, then {ε1 , . . . , εn , p − f } is a larger set of mutually orthogonal idempotents. Definition 2.10 If k = R, then a Jordan algebra A is said to be formally real if x2 + y2 = 0 =⇒ x = y = 0 . If B is a subalgebra of A, it too is formally real. An associative Jordan algebra is formally real if and only if it is isomorphic to a product of copies of R with componentwise multiplication. Definition 2.11 A positive-definite trace form t : A −→ R on a real Jordan algebra is a linear map such that t(x2 ) > 0 for all x ∈ A, x = 0. Proposition 2.12 Let A be a real Jordan algebra. The following are equivalent: (a) A is formally real; (b) A has a positive-definite trace form t; (c) t(x) = Tr (Lx ) is a positive-definite trace form (hence x, y = t(x · y) is a positive-definite inner product). Proof That (c) =⇒ (b) =⇒ (a) is clear. Assume (a). Then, for all x ∈ A, R[x] ∼ = 2 = λ 2ε , R ε , where ε are idempotents. If x = λ ε , it follows that x ∑ ∑ i i i i i=1 i i hence 3N

Tr (Lx2 ) = ∑ λi2 Tr Lεi , and, by the Peirce decomposition, Tr Lεi > 0. The formally real Jordan algebras were classified in 1934 by P. Jordan, Wigner, and von Neumann. This gives the classification of self-adjoint homogeneous cones in Section 1, in light of the following one-to-one correspondence between the two classes of objects. See [5], Ch. 11.

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II Polyhedral reduction theory in self-adjoint cones

Let C ⊂ V be a homogeneous self-adjoint cone and let e ∈ C, G = Aut (C,V ), g = Lie (G). Let K ⊂ G be the stabilizer of e in G and k its Lie algebra. Let σ be the corresponding Cartan involution with g = k ⊕ p being the Cartan decomposition, i.e.,  1 on k , σ= −1 on p . Now g acts on V by taking the differential at 0 of the G-action, k is the stabilizer of e, and the map

π −→ π · e from p into V is bijective. We define an algebra structure on p by setting (π · π  ) · e = π · (π  · e) . Let p ∈ p be such that p · e = e. One verifies that: (1) π · π  = π  · π (indeed (π · π  − π  · π ) · e = [π , π  ] · e = 0 since [π , π  ] ∈ k); (2) clearly p is a unit element; (3) [Lπ , Lπ  ] = ad [π , π  ]. Furthermore, let ·, · be a positive-definite form on V “which makes C selfadjoint” and such that σ is the restriction to g ⊂ End (V ) of u −→ −t u (such forms exist, see Proposition 1.10 and its proof). Define the linear form t on p by t(π ) = π · e, e . Then t([La , Lb ](c)) = [[a, b], c] · e, e (by (3) above) = [a, b] · c · e, e − c · [a, b] · e, e = −c · e, [a, b] · e − c · [a, b] · e, e (since [a, b] ∈ k) = 0 (since k · e = 0) . Also, for π = 0, t(π · π ) = π · π · e, e = π · e, π · e > 0 . We thus see that (p, ·,t) satisfies the hypotheses of Vinberg’s theorem (Theorem 2.2 above), which implies that (p, ·) is a Jordan algebra (A, ·). Going in the converse direction, the cone C can be recaptured in the following way: C = {exp(ρ (a)) · e | a ∈ A} , where ρ : A = p −→ End (V ) comes from the representation of g on V .

2 Jordan algebras

49 2

Identifying A with V , exp(ρ (a)) · e = expJ (a), where expJ (a) = 1 + a + a2 + · · · is the Jordan algebra exponential which may be calculated in the algebra R[a] generated by a in A. Since R[a] is an associative formally real Jordan algebra, it is a product of copies of R. In particular, the exponentials in A are also the squares of invertible elements. So, we can also describe the cone as the set of squares of invertible elements. Finally, starting from a formally real Jordan algebra A, define g = Der A ⊕ A (here Der A means derivations of A for its Jordan multiplication). Now g is a Lie algebra if we define [D1 , D2 ] = D1 D2 − D2 D1 , [D, x] = D(x) , [x, y] = Lx Ly − Ly Lx , for x, y ∈ A and D, D1 , D2 ∈ Der A. In fact, g is a subspace of Hom R (A, A) if (D, x) acts on A by y −→ Dy + x · y. One sees immediately that [·, ·] on g is just the commutator in Hom R (A, A). Let G ⊂ GL(A) be the corresponding Lie group, and let K ⊂ G be the subgroup corresponding to Der A. Putting the inner product x, y = Tr (Lx·y ) on A, and defining σ : g −→ g to be +1 on Der A, and −1 on A, one sees: (a) exponentiating a derivation leads to an automorphism, so K acts on A by Jordan automorphisms; in particular kx, y = x, k−1 y for all k ∈ K , and hence Dx, y = −x, Dy for all D ∈ Der A ; (b) equation (2.2) may be written as La(b·c) − Lb(a·c) = [La Lb , Lc ] + [Lc , Lb La ] , hence Tr La(b·c) = Tr Lb(a·c) , and therefore Lz x, y = x, Lz y for all z ∈ A.

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II Polyhedral reduction theory in self-adjoint cones

Thus σ g = −t g, for all g ∈ g; it follows that g is a reductive sub Lie algebra of Hom (A, A) and σ is a Cartan involution. Finally, since g · p = A, the orbit G · p is open in A; Proposition 1.8 shows that C = G · p is a homogeneous self-adjoint cone and G ⊂ Aut (C,V )o . Here, as above, G · p = (exp A · K) · p = {expJ (a) | a ∈ A}. Moreover, writing G = Aut (C,V )o , we see Lie G = k + p as usual. We have three Lie algebras: (Inner Der A) + A ⊂ Der A + A ⊂ k + p (where inner derivations are the derivations [Lx , Ly ], for x, y ∈ A). Since π −→ ∼ π · p defines p −→ A, and since, for any real reductive Lie group G without compact factors, k = [p, p], it follows that all three are equal! This proves: Theorem 2.13 Given a real vector space V and a point p ∈ V , there is a one–one correspondence between homogeneous self-adjoint cones C ⊂ V , with p ∈ C, and formally real Jordan algebra structures (V, ·) on V , with identity p, given by: (a) C = {exp(a) | a ∈ V } = {a2 | a invertible in V } and this cone is self-adjoint w.r.t. x, y = Tr (Lx·y ); (b) Lie (Aut (C,V )) = DerV ⊕ {Lx | x ∈ V } (orthogonal with respect to the Killing form on Lie Aut (C,V )). This description of C is useful for describing the symmetry on C, i.e., picking a base point p ∈ C, C ∼ = G/K, and the Cartan involution σ : G −→ G induces σ : C −→ C: we claim that, in this Jordan algebra structure on V , σ (x) = x−1 . To see this, suppose x = expJ (a). Then x = expJ a = (exp La )(p) . Now La ∈ p, hence exp La ∈ exp p ⊂ G, and hence σ (exp La ) = (exp La )−1 = exp(−La ). Thus

σ (x) = σ (exp La )(p) = exp(−La )(p) = expJ (−a) = expJ (a)−1 = x−1 . We mention one more fact about this correspondence. To derive a Jordan algebra from a cone C ⊂ V , we had to choose a basepoint p. Suppose we choose a new basepoint p = gp, with g ∈ exp p. We claim that the new Jordan algebra structure on V is isomorphic to the old one. If g = k ⊕ p is the Cartan decomposition corresponding to p, the new one for p is g = Ad g(k)⊕Ad g(p).

3 Boundary components and Peirce decompositions

51

Say u1 , u2 ∈ V and ui = πi p with πi ∈ p. Then gui = (gπi g−1 )p . Therefore, if ⊥ denotes the new Jordan multiplication, we have gu1 ⊥ gu2 = (gπ1 g−1 )(gπ2 g−1 )p = g(π1 (π2 p)) = g(u1 · u2 ) . Hence g induces an isomorphism V −→ V⊥ . Technically V⊥ is called the mutation of V by p . See [5], Ch. 5, or [8], pp. 67ff.

3 Boundary components and Peirce decompositions 3.1 Let V be a real vector space defined over Q and let C ⊂ V be a self-adjoint homogeneous cone. Let G = Aut (C,V )o . Fix once and for all a rational point p ∈ C to be the basepoint, and let K be the stabilizer of p in G. Since K is maximal compact there corresponds to it a Cartan involution σ of Lie (G), so that Lie (G) = Lie (K) + p, where p is the −1-eigenspace of σ and Lie (K) is the +1-eigenspace. There is an inner product ·, · on V with respect to which exp σ g = t (exp g)−1 , where t denotes the adjoint. Thus, elements of K are orthogonal and elements of exp p are self-adjoint w.r.t. this scalar product. As we saw in Section 1, C is self-adjoint for any such inner product. We assume that the subgroup G ⊂ GL(V ) is defined over Q. This strong assumption implies that Lie (K) and the Killing form are defined over Q, hence also p and σ . Therefore we may choose the scalar product so that it is defined over Q too. Then if L is a lattice in V giving the Q-structure, L and the dual lattice L∗ are commensurable. Regard Lie (AutV ) as End (V ), so that p ⊂ End (V ). Then we may identify p with V by identifying π and π p for any π ∈ p. Thus p acquires a Jordan algebra structure with Jordan multiplication · defined by (π · π  )p = π (π  p). We can transfer this structure to V . When we think of the Jordan algebra we will use the symbol V , reserving p for the Lie algebra guise. Now p has a Qstructure as a vector space which is identical with that of V . Therefore, V as a Jordan algebra is defined over Q, since p acts Q-morphically on V , and the ∼ Jordan multiplication is just the pullback of this action via p −→ V . Finally, V is formally real, that is, x2 + y2 = 0 =⇒ x = y = 0. Conversely, given any formally real Jordan algebra V defined over Q, its identity element p will be rational. Putting C = {x2 | x ∈ V is invertible}, we acquire a situation as described above if we let x, y = Tr (Lx·y ).

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II Polyhedral reduction theory in self-adjoint cones

3.2 Fix any subfield k of R, and take rational to mean k-rational. Let ε ∈ V be a rational idempotent. For any x ∈ V , denote by Lx the map Lx (y) = x · y, for y ∈ V . Then Lε is semi-simple and has eigenvalues 0, 12 , and 1. Let Vi (ε ), or Vi when no confusion is possible, denote the i-eigenspace. This gives the so-called Peirce decomposition V = V0 ⊕V 1 ⊕V1 ; 2

see Section 2. In particular, V0 ·V1 = 0,V0 ·V 1 ⊂ V 1 ,V1 ·V 1 ⊂ V 1 ,V 1 ·V 1 ⊂ V0 ⊕V1 , 2

2

2

2

2

2

and V0 , V1 are sub Jordan algebras. Since p is the identity in V , also p − ε is an idempotent, and V0 (ε ) = V1 (p − ε ). The Peirce decomposition is defined over k since ε is rational. Thus if Ci = Ci (ε ), i = 0, 1, is the cone of squares of invertible elements in Vi , then Ci is a self-adjoint homogeneous cone since Vi inherits formal-reality from V , and everything is compatibly defined over k. We call Ci a rational boundary component of C. Note that C and {0} are, by definition, boundary components; we call them improper boundary components. Note† that C, the closure of C, is simply the set of squares in V . To persuade ourselves that this is indeed a good notion of boundary component, we prove: Proposition 3.1 The closure C of C is the disjoint union of the real boundary components. Proof Say y ∈ C and y = x2 for some x ∈ V . Then x generates a sub Jordan algebra W , consisting of polynomials in x with no constant term. Note that W ⊂ R[x] and, since R[x] is formally real, R[x] is isomorphic to a product of copies of R with componentwise multiplication. Hence also any (not necessarily unital) R-subring of R[x] is isomorphic to a product of copies of R and, in particular, possesses a unit. Therefore W has a unit e. In V itself, e is an idempotent, and W ⊂ V1 (e). Now e ∈ W means that e = a1 x + a2 x2 + · · · + an xn for some ai ∈ R. Square both sides to get a21 x2 + 2a1 a2 x3 + · · · + a2n x2n = e , † In fact, let S = (V \ {0}) modulo homotheties, and ϕ : S −→ S be ϕ (x) = x2 . Because V is formally real, ϕ is well-defined. The image of ϕ is closed since S is compact. Thus the set of squares Σ in V is closed, so Σ ⊃ C. On the other hand, the set of invertible elements in V is an open dense set, so Σ ⊂ C.

3 Boundary components and Peirce decompositions

53

and therefore (a21 x + 2a1 a2 x2 + · · · )x = e . Now a21 x + 2a1 a2 x2 + · · · is in W , so x is invertible in V1 (e). Thus y = x2 is the square of an invertible element in V1 (e) = V0 (p − e) and thus is in some boundary component. To show the boundary components do not overlap, suppose y is an invertible element in both V1 (ε1 ) and V1 (ε2 ). There exists xi ∈ V1 (εi ) with xi · y = εi , i = 1, 2. In general, if y is an invertible element in a Jordan algebra, then its inverse is an element in the subalgebra W generated by y and p. Note that W is associative. Applying this to V1 (ε1 ) and V1 (ε2 ) gives x1 , x2 ∈ W . Therefore xi · y = εi ∈ W for i = 1, 2, and so ε1 = ε1 · ε2 = ε2 . Recall that we have an inner product on V , defined by x, y = Tr (Lx·y ). The Peirce eigenspaces of Lε are mutually orthogonal with respect to it. Thus in some orthogonal basis, we can write the matrix for Lε as ⎞

⎛

1

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ Lε = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

..

.

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0 1 1 2

..

. 1 2

0 ..

0

. 0

which we will always write using block notation: ⎛ Lε = ⎝

⎞

1

⎠.

1 2

0 Recall the identification of p and V and note that, for any element x, exp x = p + x + 12 x2 + 16 x3 + · · · is the same in the Jordan algebra as in the Lie algebra p ⊂ End (V ). Thus ⎞ ⎛ et 1 ⎟ ⎜ Lexpt ε = ⎝ e2t ⎠. 1

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II Polyhedral reduction theory in self-adjoint cones 1

Introducing the new parameter s = e 2 t , we have ⎞ ⎛ 2 s ⎠ = a(s) , Lexpt ε = ⎝ s 1 where a(s) is a one-parameter subgroup contained in exp p ⊂ G. Clearly, a(s) is an algebraic one-parameter subgroup, defined over k if ε is. Also, ε = ε p = 1  d 2 a (1)p, where the prime denotes ds . In particular, if we choose for ε the identity p, we get a(s) = h(s2 ), where h(u) is the one-parameter subgroup of homotheties. Since ε · ε = ε , we have ε (ε p) = ε p, and thus π 1 p = 0, where πi is the 2

orthogonal projection onto Vi (ε ), for i = 0, 12 , 1. Moreover, if a(0) = lim a(s) which exists in End (V ), then a(0) = π0 and V0 (ε ) = a(0)V .

s−→0

Remark 3.2 More generally, let a(s) be any one-parameter subgroup in G. There exists a unique n ∈ Z such that lim h(sn )a(s) exists and is nonzero. s−→0

Denote the limit by a(0). We will show later that a(0)V is a sub Jordan algebra of V , and a(0)C is a boundary component, k-rational if a(s) is.

3.3 Corresponding to any one-parameter subgroup a in G there is a parabolic subgroup P(a) = {x ∈ G | lims−→0 a(s)−1 xa(s) exists}, see [11], §2.2. In particular, let a(s) be the one-parameter subgroup defined over k corresponding, as above, to a rational idempotent ε . On Lie (G), a(s) acts semi-simply in the 3 adjoint representation, so that Lie (G) = χ Uχ , where χ is a character of a(s) and Uχ = {u ∈ Lie (G) | a(s)−1 ua(s) = χ (s)u} . Since a is one-dimensional, χ (s) = sm for some m ∈ Z. Let Um = Uχ in this case. Let Z(a) be the centralizer of a in G and let U be the subgroup of G with 3 Lie algebra equal to m>0 Um . Then Z(a) normalizes U, and P(a) = Z(a)U is the parabolic subgroup corresponding to a(s). Clearly, P(a) is k-parabolic since a(s) is defined over k. Write Norm (C0 ) = {g ∈ G | gC0 ⊂ C0 }, where C0 is any boundary component. First we establish a lemma. Lemma 3.3 π0 (C) = C0 .

3 Boundary components and Peirce decompositions

55

Proof Since π0 = lim a(s), we see that π0 (x) ∈ C for any x ∈ C. Now, by the s−→0

self-adjoint property of C and C0 , we know that 7 6 C = z ∈ V | z, y2  > 0 for all y ∈ V, y = 0 , 6 7 C0 = z ∈ V0 | z, y2  > 0 for all y ∈ V0 , y = 0 . Because π0 is an orthogonal projection onto V0 , we have π0 (x), y2  = x, y2  > 0 for any y ∈ V0 , since x ∈ C. Therefore π0 (x) ∈ C0 . Conversely, suppose x0 ∈ C0 . Let ε  = p − ε , so that ε  is the identity in V0 . Now x0 ∈ C0 implies there exist y, z ∈ V0 (ε ) with y2 = x0 and y · z = ε  . Then (y + ε )2 = y2 + 2y · ε + ε 2 = x0 + ε and (y + ε ) · (z + ε ) = y · z + ε = ε  + ε = p . Thus x0 + ε ∈ C and π0 (x0 + ε ) = x0 . Proposition 3.4 P(a) = Norm (C0 ). Proof Let u be an element of Um for m > 0. Then sm a(s)u = ua(s). Taking the limit as s −→ 0, we have uπ0 = 0. Thus uC0 = uπ0 (C) = 0. Hence exp u normalizes C0 , and in fact restricts to the identity on C0 . If z ∈ Z(a), then z commutes with a(s) for all s, hence also commutes with π0 , and z normalizes C0 as well. Since P(a) = Z(a)U, we see that P(a) ⊂ Norm (C0 ). If Norm (C0 ) is actually a larger parabolic than P(a), then there must be some m < 0 with Um ∩ Lie Norm (C0 ) = (0) (since Ad a(s) is semi-simple). Pick v = 0 in this intersection. We have a(s)v = s−m va(s), and, letting s −→ 0, we see π0 v = 0. Therefore π0 (exp v) = π0 . Since exp v ∈ Norm (C0 ), and p − ε ∈ C0 , we get (exp v)(p − ε ) = (p − ε ). On the other hand, we can apply the first paragraph to the one-parameter subgroup b(s) = a−1 (s)h2 (s) corresponding to p− ε and to C1 , the corresponding boundary component in V0 (p − ε ). This shows that exp v restricts to the identity on C1 . Because ε ∈ C1 , we get (exp v)ε = ε . Adding these together gives (exp v)p = p, which says exp v ∈ K. But K is compact and Um is nilpotent so v = 0, a contradiction. Proposition 3.5 For any boundary component C0 and any g ∈ G, gC0 is also a boundary component. Proof Let P = Norm (C0 ). Then G = KP, so it suffices to prove that gC0 is a boundary component if g ∈ K. But g ∈ K implies that g fixes p, so it is easy to see that g : V −→ V is an automorphism of Jordan algebras. Thus, if ε is an idempotent, so is gε , and gV0 (ε ) = V0 (gε ).

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II Polyhedral reduction theory in self-adjoint cones

3.4 Any semi-simple Jordan algebra W over k has a unique decomposition (up to permutation of factors) W = W1 ⊕· · ·⊕Wr , where Wi are simple Jordan algebras over k, ideals in W , and Wi ·W j = 0 if i = j. In our set-up, we say that the cone C is k-reducible if and only if C can be written as C = C1 +C2 (or equivalently C = C1 +C2 ), where the Ci are open cones in k-rational linear subspaces Vi ⊂ V such that V1 ∩V2 = (0); otherwise C is called k-irreducible. Lemma 3.6 The cone C decomposes uniquely into C = C1 + · · · +Cn with each Ci being k-irreducible in Vi and V = V1 ⊕ · · · ⊕Vn . Proof Suppose C = C1 +C2 and C = D1 +D2 are two decompositions of C. We claim that C1 = (C1 ∩D1 )+(C1 ∩D2 ). Indeed, let x ∈ C1 and write x = d1 +d2 , with di ∈ Di . Write di = ei1 + ei2 , ei j ∈ C j . Then x − (e11 + e21 ) = (e12 + e22 ) is in the linear span of C1 and of C2 . Thus e12 + e22 = 0; hence e12 = e22 = 0 and d1 , d2 ∈ C1 . By a dimension argument, C has at least one decomposition as desired in the lemma, and the claim shows that it is unique up to permutation of the factors. Proposition 3.7 The cone C is k-irreducible if and only if V is k-simple as a Jordan algebra. Proof If V is k-reducible, then C obviously is also. Now suppose C is kreducible. Write C = ∑ Ci as a sum of k-irreducible factors. Since this decomposition is unique, any g ∈ G(k) acting on C must permute the Ci . Since G(k) is Zariski-dense in G, we conclude that each g ∈ G permutes the factors Ci ; and because G is connected, it actually takes each Ci into itself, i.e., G = ∏ Aut (Ci )o . Now it is obvious that V = ∏ Vi as Jordan algebras and V is not k-simple.

3.5 Proposition 3.8 Suppose that ε1 , . . . , εn are a set of k-idempotents which are mutually orthogonal, i.e., εi is k-rational for each i and εi · ε j = δi j ε j . Then ∑ki=1 Rεi = Lie (A), where A is a k-split torus of rank k and A ⊂ exp p. Proof This follows from the following three facts:

3 Boundary components and Peirce decompositions

57

(a) In any Jordan algebra, Lu and Lu·v commute if and only if Lu2 and Lv do. This implies that ∑ Rεi is a strictly commutative sub Jordan algebra. (b) [Lπ , Lπ  ] = ad [π , π  ] for π , π  ∈ p. (c) For any x ∈ Lie (K), ad x(p) = 0 =⇒ x = 0. This is because x −→ ad x ∼ defines Lie (K) −→ DerV as we have seen in Section 2, cf. Theorem 2.13. Thus A = ∑ Rεi is a commutative sub Lie algebra of p. Each Rεi = Lie ai (s) is an algebraic Lie algebra, where ai (s) is the one-parameter subgroup corresponding to εi , so A is algebraic too. Then A = exp A is a connected diagonalizable algebraic group and Lie A is defined over k, so A is a k-torus. Since each ai (s) is a k-one-parameter subgroup, we see that A is k-split. The rank of A is n, because εi = 12 ai (1), and the εi are linearly independent. Note that, denoting Y (A) = Hom (Gm , A), there is a canonical isomorphism k

∼

∑ Qεi −→ Y (A) ⊗ Q

i=1

taking εi to ai ∈ Y (A).

3.6 Proposition 3.9 Every maximal R-split torus A, such that A ⊂ exp p, arises as in Proposition 3.8, i.e., A = exp(∑ni=1 Rεi ), where εi are mutually orthogonal idempotents of V . Proof First we establish the following lemma: Lemma 3.10 If dimV > 1, then V possesses an idempotent different from 0 and p. Proof Choose u ∈ V linearly independent from p, and let W be the sub Jordan algebra generated by u. Then W is totally real and strictly commutative, so W ∼ = Rs , where the Jordan multiplication in Rs is just componentwise multiplication. If s > 1, then W has at least two non-trivial idempotents, and, if s = 1, some multiple of u works. Since any two maximal R-split tori contained in exp p are conjugate by an element of K which also acts as an automorphism of V , it suffices to take a maximal set ε1 , . . . , εn of mutually orthogonal idempotents and check that the resulting A = exp(∑ Rεi ) is maximal. We will prove this by induction on dimV . If dimV = 1, the result is obvious. If dimV > 1, then n ≥ 2 by the lemma.

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II Polyhedral reduction theory in self-adjoint cones

Let V = V0 ⊕ V 1 ⊕ V1 be the Peirce decomposition of V with respect to ε1 . 2 Then {ε2 , . . . , εn } are a maximal set of mutually orthogonal idempotents in V0 and similarly {ε1 } in V1 . For i = 0 or 1, let Ci ⊂ Vi be the boundary components, Gi = Aut (Ci )o , A1 = exp Rε1 , and A0 = exp ∑ni=2 Rεi . By induction, Ai is a maximal R-split torus in Gi . If A = A1 × A2 is not maximal, let A  A be a maximal R-split torus with A ⊂ exp(p). Since A commutes with a1 , we have A ⊂ Z(a1 ) ⊂ Norm (C0 ) ∩ Norm (C1 ), and we get restriction maps ϕ : A −→ G1 × G2 . By the maximality of Ai in Gi , the image ϕ (A ) has to be all of A1 × A2 . So ϕ has a non-trivial one-parameter subgroup b(s) in its kernel. But then b(s) is the identity on Ci ; hence b(s)(ε1 ) = ε1 and b(s)(p − ε1 ) = p − ε1 , hence b(s)(p) = p, and therefore b(s) ∈ K. Since K is compact, this is a contradiction, so A = A.

3.7 Proposition 3.11 For every k ⊂ R, there exist maximal k-split tori A such that A ⊂ exp p and all such arise as ∑ni=1 Rεi , where εi are mutually orthogonal k-idempotents of V . Proof The existence of such A follows from: Lemma 3.12 If G is a reductive algebraic group defined over k ⊂ R, σ : G −→ G is a Cartan involution over R, and P ⊂ G is a minimal k-parabolic of G, then there is a unique torus S ⊂ P such that (i) σ (x) = x−1 , for all x ∈ S; (ii) S is a lifting into P of the unique maximal k-split torus T in P/Ru (P). If σ is defined over k, then so is S. Proof By Proposition 1.8 in [4], there exists a unique Levi subgroup L of P which is stable under the Cartan involution σ . The lifting S1 of T into this L has the required properties. Conversely, for any such torus S, we know that P = ZG (S)
Ru (P), and S stable by σ implies ZG (S) stable by σ , i.e., ZG (S) = L. Thus S = S1 . By the uniqueness of S, σ defined over k implies S is Gal (C/k)invariant, hence S is defined over k. Going back to the proposition, let A ⊂ exp p be a maximal k-split torus. Then A = Lie A ⊂ p is defined over k, so that A generates a sub Jordan algebra A0 ⊂ V defined over k (identifying p and V as usual).

3 Boundary components and Peirce decompositions

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Now A is contained in some maximal R-split torus B. We may assume B ⊂ exp p. Proposition 3.9 implies that Lie B is a strictly commutative sub Jordan algebra. Therefore A0 is as well, since A0 ⊂ Lie B. So we have A0 ∼ = ∑ri=1 Rεi (over R) and, if A0 is the k-torus defined as exp(A0 ), ∼ Y (A0 ) ⊗ Q . ∑ Qεi = Now Gal(C/k) acts on both A0 and on A0 . Acting on A0 , it consists of Jordan isomorphisms, and hence is given by permutations of the εi . The above isomorphism is Gal(C/k)-equivariant. If {εi(1,1) , . . . , εi(1,
1 ) }, {εi(2,1) , . . . , εi(2,
2 ) }, . . . are the Gal(C/k)-orbits among the {εi }, the subspace (∑ Qεi )Gal(C/k) of invariants under Gal(C/k) in ∑ Qεi is generated by

ε1 =


1

∑ εi(1, j) , ε2 =

j=1


2

∑ εi(2, j) , . . . .

j=1

But then A, which is the maximal k-split torus in A0 , is given by  Gal(C/k)    = exp ∑ Rεi , A = exp ∑ Rεi and, since εi are mutually orthogonal idempotents, A has the required form.

3.8 Choose a maximal set {ε1 , . . . , εn } of mutually orthogonal k-idempotents of V . Let A be the corresponding maximal k-split torus, so that A = Lie (A) ∼ = ∑ Rεi , by canonical k-isomorphism. We propose to compute the root structure of G with respect to A. To do this, we use a Peirce decomposition for all the εi at once. Since Lεi and Lε j commute, and because Lε1 + · · · + Lεn = id , it is easy to see that 3 V decomposes into simultaneous eigenspaces V = r≤s Vrs with eigenvalues given by 1 εt · v = (δtr + δts )v 2 if v ∈ Vrs . For convenience of notation, write Vsr = Vrs if s ≥ r, and xrs = xsr for an element of Vrs . For instance, if x = ∑k≤
xk
, then x · εm = 12 ∑k =m xkm + xmm . Note that if r = s, Vrs = V 1 (εr ) ∩V 1 (εs ), and Vrr = V1 (εr ). 2 2 Let g = Lie G. By Theorem 2.13, we know that g = DerV ⊕V . Here DerV

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II Polyhedral reduction theory in self-adjoint cones

are the derivations of V as a Jordan algebra. Hence DerV ⊂ End (V ), it is defined over k, and DerV ∼ = Lie K over k. If D, D ∈ DerV and x, y ∈ V , the Lie bracket in g is given by [D, x] = Dx , [D, D ] = DD − D D , [x, y] = Lx Ly − Ly Lx . To find the root structure, we are looking for elements 0 = (D, x) ∈ g such that [εi , (D, x)] = λi (D, x), for i = 1, . . . , n, and some λi ∈ C. Thus we require [εi , (D, x)] = ([Lεi , Lx ], −Dεi ) = λi (D, x). That is, Dεi = −λi x , [Lx , Lεi ] = −λi D . Applying the lower equation to ε j and using the top gives y · (εi · ε j ) − εi · (x · ε j ) = −λi Dε j = λi λ j x . Write x = ∑k≤
xk
, with xk
∈ Vk
. We can deduce the following. In the case i = j, 1 1 xi
+ xii − ∑ xi
− xii = λi2 x . ∑ 2 i =
4 i =
That is, 1 xi
= λi2 x . 4 i ∑ =
Hence, 3

(i) either x ∈
=i Vi
and λi = ± 12 ; (ii) or xi
= 0 for
= i and λi = 0. In particular, λi ∈ {0, ± 12 } for all i. In the case i = j, 1 − xi j = λi λ j x . 4 Hence, (i) either x ∈ Vi j and −λ j = λi = ± 12 ; (ii) or xi j = 0 and λ j = 0 or λi = 0. Putting this information together shows that at most two λk can be non-zero. There are three possibilities:

3 Boundary components and Peirce decompositions

61

(1) λi = 12 , λ j = − 12 , x ∈ Vi j , and all other λk are 0, where 1 ≤ i ≤ n, 1 ≤ j ≤ n and i = j; 3 (2) all λk = 0 and x ∈ Vrr ; (3) λi = ± 12 , all other λk are 0, where 1 ≤ i ≤ n. Possibility (3) is actually impossible. Since λ
= 0 for
= i, we know xi
= 0 3 for i =
, but λi = ± 12 implies x ∈
=i Vi
. So x = 0, and then λi D = −[Lx , Lεi ] makes D = 0 also. Possibility (2) just gives elements that centralize A. Possibility (1) is in fact realized, as long as Vi j = 0. To check this, take, without loss of generality, i = 1, j = 2, pick x ∈ V12 , and let D = −2[Lx , Lε1 ]. Claim (a) (b) (c) (d) (e)

[Lx , Lε2 ] = −[Lx , Lε1 ] and [Lx , Lε j ] = 0 for j ≥ 3; Dε1 = 12 x; Dε2 = − 12 x; Dε j = 0 for j ≥ 3; (D, x) is in a root space for g. 8

Proof Since x ∈ V12 = V 1 (ε1 ) ∩V 1 (ε2 ) ∩ j≥3 V0 (ε j ), we see that (b), (c), and 2 2 (d) are trivial. Clearly (e) follows from (a), (b), (c), and (d). So it remains to prove (a). Let fi (y) = x · (εi · y) − εi · (x · y). Then we must check that: (1) f1 (y) + f2 (y) = 0 for all y ∈ V ; (2) f j (y) = 0 for all y ∈ V , j ≥ 3. Since each fi is linear, it suffices to do this for y ∈ Vrs for various 1 ≤ r, s ≤ n. We have several cases. 8

(i) y ∈ V11 ⊕ V12 ⊕ V22 . This is true if and only if y ∈ j≥3 V0 (ε j ). Since V0 (ε ) ·V0 (ε ) ⊂ V0 (ε ) for any idempotent ε , we see that x · y ∈ V11 ⊕V12 ⊕ V22 . Thus (1) follows since ε1 + ε2 is the identity on V11 ⊕V12 ⊕V22 , and (2) follows since ε j is zero on V11 ⊕V12 ⊕V22 if j ≥ 3. (ii) y ∈ V13 . Then y ∈ V0 (ε2 ) ∩V 1 (ε3 ). Since V0 (ε ) ·V 1 (ε ) ⊂ V 1 (ε ), we have 2

2

2

x · y ∈ V23 . Then (1) follows since ε1 + ε2 = 12 on V23 ⊕V13 , and (2) follows since ε3 is 12 on V23 ⊕V13 and ε j = 0 on V23 ⊕V13 for j ≥ 4. (iii) y ∈ V1k or V2k for any k ≥ 3 is similar to (ii). (iv) y ∈ V jk with j, k ≥ 3. Then x · y = 0: if j = k, this follows from y ∈ V1 (ε j ) and x ∈ V0 (ε j ); if j = k, this follows from x·y ∈ V 1 (ε1 )∩V 1 (ε2 )∩V 1 (ε j )∩ 2 2 2 V 1 (εk ). Hence also x · (εi · y) = 0, whence (1) and (2). 2

62

II Polyhedral reduction theory in self-adjoint cones Summarizing this discussion, we find g = Z(A) ⊕

$

gi j ,

i = j

where gi j = {(D, x) | x ∈ Vi j , D = −2[Lx , Lεi ]} . Applying the above formulae, one readily checks: Z(A) = (Z(A) ∩ DerV ) ⊕ (Z(A) ∩V ) = {D ∈ DerV | Dεi = 0 for all i} ⊕

n $

Vii ,

i=1

and that, if we define for i < j Der (V )i j = {D ∈ DerV | D = [Lx , Lεi ], for some x ∈ Vi j } , then gi j ⊕ g ji ∼ = Vi j ⊕ Der (V )i j , dim gi j = dim g ji = dimVi j = dim Der (V )i j . Finally, we study the set of (i, j) such that Vi j = (0). Proposition 3.13 Let γi ∈ Hom (∑ni=1 Rεi , R) be the dual basis to εi . Let V = ∏α Vα be the decomposition into k-simple Jordan algebras and let {εi }i∈Iα be the idempotents in Vα . Then: (i) Vi j = (0) if and only if i = j and i, j are in the same Iα ; (ii) the k-roots are 12 (γi − γ j ), where i = j and i, j are in the same Iα ; (iii) the Weyl group is the group of all permutations of the {εi } preserving the partition {Iα }. Proof Since every element of the Weyl group is represented by an element of K, it acts on A by a Jordan isomorphism, hence it acts by a permutation of the εi . Now if Vi j = (0), then 12 (γi − γ j ) is a root and the reflection wi j in the hyperplane γi − γ j = 0 is in the Weyl group. Since (γi − γ j )(εk ) = 0, for k = i, j, this reflection must be the permutation fixing εk for k = i, j and interchanging εi , ε j . Now any subgroup of the permutation group generated by transpositions is the group of all permutations preserving some partition {Jα }. Thus if Vi j = (0), then wi j lies in the Weyl group and hence i, j are in the same Jα . Conversely, since the wi j for i, j such that Vi j = (0) generate the Weyl group, if Jα has at least two elements, then Vi0 , j0 = (0) for some i0 , j0 ∈ Jα . Then for any i = j with i, j ∈ Jα , there exists σ ∈ Norm K (A) such

3 Boundary components and Peirce decompositions

63

that σ εi0 = εi , σ ε j0 = ε j . Then Vi, j = σ (Vi0 , j0 ) = (0). Thus Vi j = (0) if and only if i = j and i, j are in the same Jα . Finally, V =∏

∏

α i, j∈Jα

Vi j ,

and ∏i, j∈Jα Vi j is k-simple. Hence Vα = ∏i, j∈Jα Vi j and Iα = Jα . 3.9 Let ε be an idempotent in V , and let Ci = Ci (ε ), and Vi = Vi (ε ) for i = 0, 1. Let a(s) be the one-parameter subgroup corresponding to ε . Denoting by Z(a) the centralizer of a in G, we know that Z(a) is the Levi subgroup of Norm (C1 ), and also the Levi subgroup of Norm (C0 ). In fact, Z(a) = Norm (C1 ) ∩ Norm (C0 ). Setting Gi = Aut (Ci ,Vi )o for i = 0, 1, we get a map

ϕ = res V0 × res V1 : Z(a)o −→ G0 × G1 . Proposition 3.14 We have a decomposition: Z(a)o = G0 · G1 · K0 (a direct product, modulo a finite abelian subgroup), where K0 = ker ϕ is compact, res ϕ : Gi −→ Gi is surjective with finite kernel, and Lie Gi = [Vi ,Vi ] +Vi ⊂ Lie G . If ε is k-rational, then so is this decomposition. Proof By Section 2, we know that Lie G = [V,V ] ⊕ V . For i = 0, 1, define the subalgebra gi = [Vi ,Vi ] ⊕ Vi ⊂ Lie G. Now any element of Vi strictly commutes with ε , so that actually gi ⊂ Lie Z(a). There are restriction maps Ψi : Lie Z(a) −→ Lie Gi . We claim that Ψi |gi is an isomorphism. We know by Section 2 that Lie Gi = [Vi ,Vi ] ⊕ Vi . By definition, Ψi is just the identity on Vi and, since it preserves the Lie bracket, it is also the identity on [Vi ,Vi ], hence the claim is proven. Let K0 = Ker ϕ , k0 = Lie K0 . Note that if g ∈ K0 , then gε = ε and g(p − ε ) = p − ε , hence gp = p; thus K0 ⊂ K and is compact. Now because Lie Z(a)/k0 ∼ = Lie G0 × Lie G1 ∼ = g0 × g1 , it follows that we get a vector space decomposition: Lie Z(a) = k0 ⊕ g0 ⊕ g1 . We claim these factors commute. (i) To show [g0 , g1 ] = (0), it suffices to show [V0 ,V1 ] = (0), but this just says that V0 , V1 strongly commute, which was shown in Section 2.

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II Polyhedral reduction theory in self-adjoint cones

(ii) To show [k0 , gi ] = (0), it suffices to show [k0 ,Vi ] = (0). But k0 = Ker (dϕ ) is an ideal so [k0 ,Vi ] ⊂ k0 ; while [k, p] ⊂ p implies [k0 ,Vi ] ⊂ p.

Corollary 3.15 res V0 : Z(a)o −→ G0 is surjective. Corollary 3.16 ker(res V0 |Z(a)o )o = G1 · K0 . 3.10 In this subsection, we assume G is k-simple. Let n be the k-rank of G and let ε1 , . . . , εn be a maximal set of mutually orthogonal k-idempotents. Call a j (s) the one-parameter subgroup corresponding to the idempotent f j = ε1 +· · ·+ ε j . Note that a j (0), defined as lim a j (s), exists in End (V ). s−→0

Lemma 3.17 V0 ( f j+1 )  V0 ( f j ). Proof Pick any x ∈ V0 ( f j+1 ). Then 0 = f j+1 · x = f j · x + ε j+1 · x. But ε j+1 ∈ V1 ( f j+1 ) so ε j+1 · x = 0. Thus x ∈ V0 ( f j ). Meanwhile ε j+1 ∈ V0 ( f j ) \V0 ( f j+1 ). Thus if we write C j = C0 ( f j ) = a j (0)C, we have 0 = Cn  Cn−1  Cn−2  · · ·  C1  C0 = C . We call this a flag of k-boundary components. If we fix ε1 , . . . , εn once and for all, we call it the standard flag, and its members the standard rational boundary components. Let A be the maximal k-split torus with Lie A = ∑ni=1 Rεi . Proposition 3.18 If b(s) is any k-split one-parameter subgroup in G such that b(0) = lim b(s) exists in End (V ) and is non-zero, then b(0)C is the image by s−→0

some g ∈ G(k) of a standard rational boundary component. In particular, it is a boundary component. Proof By conjugating by some g ∈ G(k) we may assume that b(s) is a oneparameter subgroup of A. We may always replace b by bn for some positive integer n, since b(0) does not change. From the explicit description of the root mn 1 m2 structure in Section 3.8, it is easy to check that {am 1 a2 · · · an | mi ≥ 0 for i = 1, . . . , n − 1} is a k-Weyl chamber in A. Thus conjugating with some n ∈ N(k), i where N is the normalizer of A, we may assume b(s) = ∏ni=1 am i (s) with mi ≥ 0 2m n for i = 1, . . . , n − 1. Note that b(s)εn = s εn . Since b(0) exists, it follows that

3 Boundary components and Peirce decompositions

65

mn ≥ 0 as well. Recall that ai (0) is the orthogonal projection onto V0 ( fi ) and that 0 = V0 ( fn )  V0 ( fn−1 )  V0 ( fn−2 )  · · ·  V0 ( f1 )  V . mn 1 Then am 1 · · · an (0)C = C j , where m j+1 = · · · = mn = 0 and m j > 0.

Note that b(0)C is always a boundary component, even if G is not k-simple. Every k-boundary component is the translate of a standard one. If the standard flag were a subflag of some larger flag of k-rational boundary components, any non-standard boundary component would have the same dimension as a standard one and thus be equal to it. So, the standard flag is a maximal flag of k-rational boundary components. Proposition 3.19 Any flag F of k-rational boundary components is the image by some g ∈ G(k) of some subflag of the standard flag. Proof The proof is similar to that of the previous proposition. Let Vs  Vs−1  · · ·  V1  V be the flag of k-boundary components in question. Let di be the Jordan identity in Vi . Then E = {p − d1 , d1 − d2 , . . . , ds−1 − ds , ds } is a set of mutually orthogonal k-idempotents, and they generate an associative sub Jordan algebra. The corresponding torus B is k-split. So, after conjugating by an element g ∈ G(k), we may assume that B ⊂ A. Now, Lie A = ∑ni=1 Rεi and Lie B ⊂ Lie A. Therefore, E = {ε11 + · · · + ε1 j1 , . . . , εs1 + · · · + εs js } . Since the Weyl group acting on {ε1 , . . . , εn } is the full group of permutations, after conjugating by some n ∈ N(k) we may assume that di = fϕ (i) for some increasing map ϕ : {1, . . . , s} −→ {1, . . . , n − 1}. (Here, again, N is the normalizer of A.) However, fϕ (i) is the projection onto the ϕ (i)’th member of the standard flag. Since the idempotent determines the boundary component, we now have our flag as a subflag of the standard flag. Proposition 3.20 There is a bijection between the set of flags F of k-boundary 8 components and k-parabolics P ⊂ G given by Φ : F −→ Ca ∈F Norm (Ca ). Proof From the root structure, we see easily that the minimal k-parabolic corresponding to the set of simple roots { 12 (γi − γi+1 ) | i = 1, . . . , n−1} normalizes 8 the standard flag. Thus Ca ∈F Norm (Ca ) actually is a k-parabolic for any flag F, by use of the preceding corollary.

66

II Polyhedral reduction theory in self-adjoint cones

Define a standard k-parabolic to be one of the form Φ(F), where F is a subflag of the standard flag. Then Norm (Ci ) for standard Ci , i = 1, . . . , n − 1, are the maximal standard k-parabolics. The fact that any k-parabolic is conjugate via some g ∈ G(k) to an intersection of maximal standard ones implies that Φ is surjective. We show now that Φ is injective. Since any k-parabolic is uniquely the intersection of maximal k-parabolics, it suffices to show that, for any two kboundary components, Ca and Ca , Norm (Ca ) = Norm (Ca ) =⇒ Ca = Ca . By Proposition 3.18, there exist g, g ∈ G(k) and standard k-boundary components Ci , C j such that gCi = Ca and gC j = Ca . Therefore, gNorm (Ci )g−1 = Norm (Ca ) = Norm (Ca ) = g Norm (C j )g−1 . So we get g−1 gNorm (Ci )(g−1 g)−1 = Norm (C j ) . However, the standard k-parabolics are not conjugate to each other, so i = j. Then, since the normalizer of any parabolic is equal to itself, we get that g−1 g ∈ Norm (Ci ). Thus Ca = gCi = g (g−1 g)Ci = gCi = Ca .

Corollary 3.21 A set of k-parabolics intersect in a k-parabolic if and only if their corresponding boundary components can be arranged into a flag. Corollary 3.22 A real boundary component C of C is k-rational if and only if Norm (C ) is k-rational. Proof Combine the theorem over k and the theorem over R. Lemma 3.23 A is generated by a1 (s), . . . , an (s). 1

Proof ai (s) = expt(ε1 + · · · + εi ) where s = e 2 t , so a1 , . . . , an generate a torus with Lie algebra ∑ni=1 Rεi . What is the orbit Ap ⊂ C? We compute: for any q ∈ V , which strongly commutes with the εi ,     exp ∑ t j ε j q = ∏ expt j ε j q = ∑(expt j )ε j q − ∑ ε j q + q = ∑(expt j )ε j q .

4 Siegel sets in self-adjoint cones

67

Here we used εi · ε j = δi j εi , ∑ ε j = p and t2 ε + · · · = et ε − ε + 1 . 2 We are thinking of elements π of V ∼ = p as being in End (V ). Thus   exp ∑ ti εi p = ∑(expti )εi p = ∑(expti )εi . expt ε = 1 + t ε +

Hence: Proposition 3.24 The orbit of A is “linear,” namely, Ap = ∑ni=1 R>0 εi . We may use ε1 , . . . , εn as orthogonal coordinates in Ap, since εi , ε j  = Tr Lεi ·ε j = 0 if i = j. Now 12 aj (1) = ε1 + · · · + ε j and ⎧ ⎫   7 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V1 ( f j )   2 ⎪ ⎪ ⎪ ⎪ s ⎪ ⎪ ⎪ ⎪   ⎪ _ _ _ _ _ _ _⎪ ⎪ ⎪ 7 ⎪ ⎪   ⎪ V1 ( f j) ⎪ ⎪ a j (s) =⎪ s  ⎪ ⎪ ⎪ ⎪ 2  ⎪ ⎪ _ _ _ _ _ _ _ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ 7 ⎪  1 ⎪  ⎪ ⎪ V0 ( f j ) . ⎩ ⎭ Clearly, εk ∈ V0 ( f j ) if k > j and εk ∈ V1 ( f j ) if k ≤ j. Thus  εk if k > j a j (s)εk = s2 εk if k ≤ j . 4 Siegel sets in self-adjoint cones 4.1 First we recall the general theory of Siegel sets. Let G be a reductive algebraic group defined over Q and G = G (R)o and let X be the associated non-compact symmetric space (i.e., the homogeneous space G/K, for a maximal compact subgroup K ⊂ G). For every parabolic subgroup P ⊂ G defined over Q and minimal among such subgroups (“minimal Q-parabolic” for short), we wish to define a class of subsets S ⊂ X called the Siegel sets associated to P = P(R)o . Definition 4.1 Choose a basepoint p ∈ X. Let A ⊂ P be the unique torus that is a conjugate of the maximal Q-split torus of P and such that Lie A ⊥ Lie (Stab p) (cf. Lemma 3.12). Let ∆ ⊂ Hom (A, Gm ) be the simple positive roots (i.e., the minimal roots in the adjoint action of A on Lie P); let A+ = {g ∈ A | β (g) ≥ 1 for all β ∈ ∆}. Then the Siegel sets are the subsets of X of the form Sω = ω A+ p ,

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II Polyhedral reduction theory in self-adjoint cones

where ω ⊂ P = P(R)o is a compact subset. We make three remarks. First of all, this definition does not depend on the choice of p. In fact, let p, p be two basepoints, let K = Stab (p). Since G = P · K, P acts transitively on X and so we may write p = gp for some g ∈ P. Let A, A ⊂ P be the tori corresponding to p, p . Then A = gAg−1 ; hence, for all ω ⊂ P compact, Sω = ω (A )+ p = ω (gA+ g−1 )(gp) = Sω g . Secondly, this definition is slightly different from the usual one (see [3], pp. 85ff.). However, we claim that all “usual” Siegel sets are Siegel sets as above, and each of the above is contained in a usual Siegel set. In all applications, it is not the exact shape of the Siegel sets that is important, but rather the way they grow. To be precise, any class of subsets {Xα } cofinal with Siegel sets in the sense: (i) for all α , there exists ω such that Xα ⊂ Sω ; (ii) for all ω , there exists α such that Sω ⊂ Xα , would do just as well. In our case, the “usual” Siegel sets are defined by decomposing P into P = MAN , where N is the unipotent radical of P, A is some maximal Q-split torus in P, and M is the anisotropic part of Z(A), and by choosing p such that Lie (Stab p) ⊥ Lie A. Then one takes the set S∗ω ,t = ω At p, where ω ⊂ MN is compact and At = {g ∈ A | β (g) ≥ t for all β ∈ ∆}. But if at ∈ A satisfies β (at ) = t for all β ∈ ∆, then S∗ω ,t = Sω at . In the other direction, if ω ⊂ P is compact, then ω ⊂ ω1 · ω2 , where ω1 ⊂ MN and ω2 ⊂ A are compact. Let t = inf β (g) ; β ∈∆ g∈ω2

then Sω ⊂ S∗ω1 ,t .

4 Siegel sets in self-adjoint cones

69

The third remark is that if G is not Q-simple, but rather G = G1 × · · · × Gk over Q, then all our sets decompose: X = X1 × · · · × Xk , P = P1 × · · · × Pk , A = A1 × · · · × Ak , + A = A+ 1 × · · · × Ak . +

So, among all Siegel sets, those with ω = ω1 × · · · × ωk are cofinal and these decompose as Sω = Sω1 × · · · × Sωk . The virtue of Siegel sets lies in the following two fundamental results of reduction theory. Let Γ ⊂ G(Q) be an arithmetic subgroup. Then: (i) there exist ω ⊂ P compact and a finite set F ⊂ G(Q) such that X = Γ · F · Sω ; (ii) for all ω compact and all g1 , g2 ∈ G(Q), the set / {γ ∈ Γ | g1 Sω ∩ γ g2 Sω = 0} is finite.

4.2 Now return to the case of cones: let G = Aut (C,V )o , where C is a homogeneous self-adjoint cone, and suppose V and G are defined over Q. Choose a rational basepoint p, let K = Stab (p), so that V is a Jordan algebra, and choose a maximal Q-split torus A perpendicular to K. We also assume that G is Qsimple. Let A = Lie A, A = ∑ Rεi , where ε1 + · · · + εn = p as in Section 3. Let Ci be the boundary component containing pi = εi+1 + · · · + εn so that 7 6 F = C ⊃ C1 ⊃ · · · ⊃ Cn−1 is a maximal flag of Q-rational boundary components corresponding to a minimal Q-parabolic P containing A. Let 1

ai (e 2 t ) = exp(t(ε1 + · · · + εi )) be the one-parameter subgroup such that ai (0)C = Ci and  ε j if j > i ai (s)ε j = sε j if j ≤ i .

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II Polyhedral reduction theory in self-adjoint cones

Let C = C ∪C1 ∪C2 ∪ · · · ∪Cn−1 ∪ {0} . From our explicit description of the root structure, we see that the set ∆ of simple roots consists in our case of β1 , . . . , βn−1 , where on the Lie algebra level ⎧ 1 ⎨ − 2 if i = j 1 dβ j (εi ) = if i = j + 1 ⎩ 2 0 otherwise . Since 12 ai (1) = ε1 + · · · + εi , it is easy to see that

β j (ai (s)) = s−δi j . Lemma 4.2 A+ p is the closed polyhedral cone D+ generated by the points p, and p1 , . . . , pn−1 .

Proof Since βi (a j (s)) = s−δi j , we have A+ = {a1 (s1 ) · · · an (sn ) | si ≤ 1 for i = 1, . . . , n − 1} . Therefore A+ p is the set of points (s1 · · · sn )2 ε1 + (s2 · · · sn )2 ε2 + · · · + s2n εn , with 0 < si ≤ 1 for i = 1, . . . , n − 1 and 0 < sn . Moreover, A+ p is the set of such points where now 0 ≤ si ≤ 1 for i = 1, . . . , n − 1 and 0 ≤ sn . The lemma follows easily.  Corollary 4.3 Sω ⊂ C.  Proof Note that ω ⊂ P, hence ω fixes C. We want to know how the Siegel sets look in the boundary components. By induction, it will be apparent that it suffices to do this for the largest boundary

4 Siegel sets in self-adjoint cones

71

component, C1 . Let p1 = p − ε1 = ε2 + · · · + εn be the basepoint in C1 and write G1 = Aut (C1 ,V1 )o . Let ψ : P(a1 ) −→ G1 be the restriction map. We know ψ (A)p1 = ∑nk=2 R>0 εk . We know that P is the normalizer of the standard flag, and P is a minimal Q-parabolic in G. Let P1 be the normalizer of the inherited standard flag in C1 . By Proposition 3.14, we also have a subgroup G1 ⊂ Z(a1 ) ⊂ P(a1 ), where ψ  = ψ |G is an isogeny onto G1 . Let 1

P1

= ψ −1 (P1 ) = {g ∈ G1 | ψ (g) normalizes C1 , . . . ,Cn−1 } = P ∩ G1 .

Proposition 4.4 For any g ∈ G1 , there exists g ∈ Z(a1 ) such that ψ (g ) = g and g ε1 = ε1 . If g ∈ P1 , we may take g to be in P. Proof This follows from the above discussion and from the fact that G1 is in the kernel of the restriction of Z(a1 ) to V1 (ε1 ). Proposition 4.5 For any Siegel set Sω , we have Sω ∩C1 = Sψ (ω ) . Proof We simply compute: Sω ∩C1 = ω A+ p ∩C1 = ω A+ p ∩C1 (since ω is compact) = ω (A+ p ∩C1 ) (since ω normalizes C1 ) = ψ (ω )A+ 1 p1 (by Lemma 4.2) = ψ (ω )A+ 1 p1 (since ψ (ω ) is compact) = Sψ (ω ) .

4.3 We now compare Siegel sets with polyhedral cones. First of all, by a polyhedral cone π ⊂ V we mean a closed set defined equivalently as {x ∈ V |
i (x) ≥ 0 for all i = 1, . . . , k} , for some finite set of linear functions
i , or as  k



∑ λi yi ∈ V | λi ≥ 0

i=1

,

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II Polyhedral reduction theory in self-adjoint cones

for some finite set of vectors yi ∈ V . A word of caution: by the conventions we have been forced into a polyhedral cone is, by definition, closed; a homogeneous self-adjoint cone is, by definition, open. Hopefully there are only a few places where this might cause confusion. A polyhedral cone is called Q-rational if one can choose the
i (or the yi ) to be Q-rational. We assume, as above, that V is defined over Q, and that C is a homogeneous self-adjoint cone in V , and that G = Aut (C,V )o is defined over Q and is Qsimple. Let P ⊂ G be a minimal Q-parabolic, let F = {C ⊃ C1 ⊃ · · · ⊃ Cn−1 } be the associated flag, and let C = C ∪C1 ∪· · ·∪Cn−1 ∪{0}. Finally, let D+ = A+ p. The main result will be as follows. Theorem 4.6 The closures of the Siegel sets Sω in C and the polyhedral cones π ⊂ C are cofinal, i.e., every Sω is contained in some π and every π is contained in some Sω . Proof To prove that every Siegel set is in a polyhedral cone is easy. Start with ω D+ with ω ⊂ P compact. Then, for i = 1, . . . , n − 1, let πi be a polyhedral cone generated by a finite set of elements of Ci such that ω pi ⊂ πi , and let π0 be a polyhedral cone generated by elements of C such that ω p ⊂ π0 . (This exists because ω pi is a compact subset of Ci .) Then (writing p0 for p for simplicity of notation) *   )

ω D+ =

n−1

b 

⊂

| λi ≥ 0, b ∈ ω 

n−1

∑ λi (bi pi ) | λi ≥ 0, bi ∈ ω

 ⊂

∑ λi pi

i=0

i=0

n−1



∑ qi | qi ∈ πi

= π0 + π1 + · · · + πn−1 ,

i=0

as asserted. The other inclusion is harder: as it stands, it is not well adapted to proof by induction and, instead, we prove by induction the following stronger fact.  there is a compact set ω ⊂ P Proposition 4.7 For every polyhedral cone π ⊂ C, such that p + π ⊂ ω (p + D+ ) .

4 Siegel sets in self-adjoint cones

73

Multiplying both sides by homotheties R>0 , it follows that π ⊂ ω D+ as asserted. Proof (of Proposition 4.7) We proceed by induction on dimC. Separating the generators of the π into those in C and those in C1 = C1 ∪C2 ∪ · · · ∪Cn−1 ∪ {0}, we write π = π0 + π1 , where π0 ⊂ C, π1 ⊂ C1 . Let R≥1 be the reals [1, ∞). We claim that, for suitably large ω , p + π0 ⊂ R≥1 ω p . Let H be the hyperplane of points of the form p + q, where p, q = 0. Then π0 ∩ H is compact and, if π0 ∩ H ⊂ ω p, it follows that p + π0 ⊂ R≥1 (π0 ∩ H) ⊂ R≥1 ω p:

Then p + π = (p + π0 ) + π1 ⊂ R≥1 ω p + π1 ⊂ R≥1 ω (p + ω −1 π1 ) = R≥1 ω (ε1 + (p1 + π1 )) , where, in the last step, π1 is a larger polyhedral cone in C1 containing ω −1 π1 . Now use the induction hypothesis. Hence, there exists a compact subset

ω1 ⊂ P1 = {g ∈ Aut (C1 )o | g fixes the flag C1 ⊃ C2 ⊃ · · · ⊃ Cn−1 } , such that p1 + π1 ⊂ ω1 (p1 + D+ 1), + where D+ 1 = A1 p1 is the polyhedral cone generated by p1 , p2 , . . . , pn−1 . Let P1 ⊂ P be the subgroup defined in Section 4.2 such that P1 −→ P1 is surjective

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II Polyhedral reduction theory in self-adjoint cones

with finite kernel and P1 acts identically on the boundary component containing ε1 . Let ω1 be the inverse image of ω1 in P1 . Then p + π ⊂ R≥1 ω (ε1 + ω1 (p1 + D+ 1 )) = R≥1 ωω1 (ε1 + p1 + D+ 1) = ωω1 [R≥1 (p + D+ 1 )] . But + R≥1 (p + D+ 1 ) = {(1 + λ )(p + a) | a ∈ D1 , λ ≥ 0}

= {p + (λ p + a) | a ∈ D+ 1 , λ ≥ 0} = p + D+ , so p + π ⊂ ωω1 (p + D+ ), as asserted. If we now drop the assumption that G is Q-simple, we immediately get the following generalization. Theorem 4.8 Let G = G(1) × · · · × G(k) , C = C(1) × · · · ×C(k) , P = P(1) × · · · × P(k) , where G(i) are Q-simple, G(i) = Aut (C(i) ,V (i) )o , and P(i) ⊂ G(i) is a minimal Q-parabolic. Let P(i) correspond to a flag F(i) = {C

(i)

(i)

(i)

(i)

= C0 ⊃ C1 ⊃ · · · ⊃ Cni −1 } ,

and let i −1 k n

C = ∏

(i)

Cj .

i=1 j=0

Then the closures of the Siegel sets Sω in C and the polyhedral cones π ⊂ C are cofinal. Now let C∗ be the union of C and all its Q-rational boundary components. Then, combining Theorem 4.8 with the main results of reduction theory (via Siegel sets), we find: Corollary 4.9 (i) For every arithmetic subgroup Γ ⊂ G and every pair of closed polyhedral cones π1 , π2 ⊂ C∗ , the set {γ ∈ Γ | γπ1 ∩ π2 ∩C = 0} /

5 Cores and co-cores

75

is finite. (ii) For every arithmetic subgroup Γ ⊂ G, there exists a closed polyhedral cone π ⊂ C∗ such that (Γπ ) ∩C = C.

Moreover it is now an easy matter to construct a polyhedral fundamental domain for the action of an arithmetic group Γ on C in the following sense. Definition 4.10 A decomposition of C into rational polyhedral cones {σα } is called a Γ-admissible polyhedral decomposition of C if the following properties are satisfied: (1) (2) (3) (4) (5)

a face of a σα is a σβ ; σα ∩ σβ is a common face of σα and σβ ; γσα is a σβ , for all γ ∈ Γ; mod Γ, there are only finitely many σα ;
C = α (σα ∩C).

In fact, just choose a Siegel set S and a finite set F ⊂ G(Q) such that C = ΓFS. Let π be a polyhedral cone with rational vertices in C such that S ⊂ π . If H1 , . . . , Hm are the hyperplanes defining π , consider H = {γ f Hi | γ f π ∩ π = 0, / where γ ∈ Γ, f ∈ F, i = 1, . . . , m} . Since π is contained in some Siegel set, the property of Siegel sets guarantees
that the images by Γ of the connected components of π \ H∈H H and their faces solve the problem.

5 Cores and co-cores 5.1 We have C ⊂ V , with a compatible lattice L ⊂ V as usual. We also have the characteristic function ϕ : C −→ R>0 , which is defined up to a constant (see Section 1). The following proposition will enable us to make a sensible normalization of ϕ . Proposition 5.1 The function ϕ is bounded on C ∩ L. Proof For any open cone A in V , let A∗ denote its dual in V ∗ , as in Section 1. Let π be an open simplicial cone in C with vertices x1 , . . . , xr ∈ C ∩ L. Then π ∗ = {
∈ V ∗ |
(xi ) > 0 for all i = 1, . . . , n}. If we choose
1 , . . . ,
r ∈ V ∗ to be the dual basis to x1 , . . . , xr , then π ∗ is simplicial with vertices
1 , . . . ,
r and
1 , . . . ,
r ∈ L∗ ⊗ Q.

76

II Polyhedral reduction theory in self-adjoint cones Now, for any cone A, let

ϕA (x) =

+

e−x,
 d
.

A∗

Thus our ϕ is just ϕC . Since π ⊂ C implies that C∗ ⊂ π ∗ , we know

ϕC (x) ≤ ϕπ (x) = m

+

e−x,∑ ai
i  da ,

Rr>0

where m is some constant. Let d be the volume of the parallelopiped spanned by x1 , . . . , xr . If now x = ∑ bi xi ∈ π ∩ L, then bi ≥ d1 for all i, and

ϕC (x) ≤ ϕπ (x) = m

+

e− ∑ bi ai da ≤ m

Rr>0

+

1

e− d ∑ ai da < M ,

Rr>0

where M is some constant depending only on π .
By the reduction theory, we have C = σα , where the σα are polyhedral rational cones open in their linear span, and there are only a finite number of them modulo Γ. Since any polyhedron can be written as the union of simplices, we may assume each σα is simplicial. Since ϕ (γ x) = ϕ (x) for all γ ∈ Γ and x ∈ C, we have shown that

ϕ is bounded on L ∩ (the union of the top-dimensional simplices). We continue by induction. Assume that

ϕ is bounded on L ∩ (the union of the σα with dim σα ≥ s). Suppose σβ has dimension s − 1. Let Star σβ be the union of all simplices σα having σβ as a face. Then Star σβ is a neighborhood of σβ . By induction, ϕ is bounded on L ∩ (Star σβ \ σβ ), say by the constant B. There exists a finite set {λi } ⊂ L∗ such that, for any y ∈ ∂ Star σβ , we have λi (y) = 0 for some i, and λi (z) > 0 for every z ∈ σβ and all i. Because λi (x) ≥ 1 for each i and each x ∈ σβ ∩ L, we see that dist (L ∩ σβ , ∂ Star σβ ) > δ > 0 , where dist denotes the distance between the two sets. Choose v ∈ (Star σβ \ σβ ) ∩ L, and pick m > |v| δ . We claim that, for any x ∈ L ∩ σβ , both mx + v and mx − v are in Star σβ \ σβ . For mx + v, the claim is trivial, but, since the mδ -ball about mx is contained in Star σβ and |v| < mδ , the claim is true for mx − v as well. Thus, ϕ (mx) = ϕ ( 12 (mx − v) + 12 (mx + v)) ≤ B since ϕ is convex. So ϕ (x) ≤

5 Cores and co-cores

77

mr B for any x ∈ L ∩ σβ , where dimV = r. As there are only finitely many σβ modulo Γ, we see that ϕ is bounded on L ∩ (union of all σβ with dim σβ ≥ s − 1) .

We normalize ϕ so that max{ϕ (x) | x ∈ C ∩ L} = 1. More generally, for any rational boundary component C1 , we have an orthogonal projection πC1 onto C1 and we normalize ϕC1 such that max{ϕ (πC1 x) | x ∈ C ∩ L} = 1. Now consider subsets (called kernels) K ⊂ C such that R≥1 K ⊂ K, and C ⊂ / K. We say that two kernels K1 , K2 are comparable if there exist R>0 K, and 0 ∈ λ1 , λ2 ∈ R>0 with λ1 K1 ⊂ K2 ⊂ λ2 K1 . Let L denote L \ {0}. Theorem 5.2 The following kernels are comparable: (a) (b) (c) (d) (e)

Γ(p +C); the convex hull of C ∩ L; {x ∈ C | ϕC1 (πC1 x) ≤ 1 for any rational boundary component C1 }; {x ∈ C | x, y ≥ 1 for all y ∈ C ∩ L }; the set ) * Γ·



ωi (A+ i p + p)

i




for any finite collection {Si = ωi A+ i p} of Siegel sets with Γ · (

i Si ) = C.

Note that in (c) we count C as a rational boundary component. Definition 5.3 This class of comparable kernels are the cores of C. In order not to interrupt the proof of the theorem, we first prove two lemmas. If C1 is a rational boundary component corresponding to an idempotent ε , we denote by C1⊥ the boundary component corresponding to p − ε . Lemma 5.4 Let C1 be a rational boundary component and set C0 = C1⊥ . Take γ ∈ Γ arbitrary and let C1 = (γ −1C0 )⊥ . Then

ϕC1 πC1 γ = ϕC1 πC1 . Proof First we show that, for all g ∈ Norm (C0 ), there exists g ∈ Norm (C1 ) ∩ Norm (C0 ) such that πC1 g = gπC1 . Let a(s) be the one-parameter subgroup exp(t ε ), where ε is the identity

78

II Polyhedral reduction theory in self-adjoint cones 1

in the Jordan algebra of C0 and s = e 2 t . Then πC1 = lim a(s) and hence s−→0

Norm (C0 ) = P(a−1 ). Then

πC1 (gx) = lim a(s)gx s−→0

= lim a(s)ga−1 (s)a(s)x s−→0

= g(πC1 x) , where x ∈ C1 is arbitrary and g = lim a(s)ga−1 (s) exists by the definition of s−→0

P(a−1 ) and is in Z(a). Further, if g ∈ Stab (p), then πC1 g = gπg−1C1 . This is simply because g is an orthogonal transformation. Now, Norm (C0 ) and Stab (p) generate G, so write γ = gh with g ∈ Norm (C0 ) and h ∈ Stab (p). Then

πC1 γ = πC1 gh = gπC1 h = ghπh−1C1 . Meanwhile, gh : h−1C1 −→ C1 is an isomorphism of homogeneous cones, and, because the characteristic function is unique up to a constant,

ϕh−1C1 = µϕC1 gh , for some µ ∈ R>0 . Thus µϕC1 πC1 γ = µϕC1 ghπh−1C1 = ϕh−1C1 πh−1C1 . So, by the way we normalized ϕC1 and ϕh−1C1 , and since γ fixes the lattice, it is clear that µ = 1. It remains to check that h−1C1 = (γ −1C0 )⊥ . But γ −1C0 = h−1 g−1C0 = −1 h C0 , so if ε is the idempotent of C0 , it follows that h−1C0 has h−1 ε0 as its idempotent, and thus (γ −1C0 )⊥ has p − h−1 ε0 = h−1 (p − ε0 ), which is the idempotent of h−1C1 . Corollary 5.5 The set in part (c) in the theorem is Γ-invariant. Proof This follows from the lemma and the fact that application of any γ ∈ Γ sets up a bijection of rational boundary components. Lemma 5.6 Let ω A+ p be a Siegel set. For a maximal set ε1 , . . . , εn of orthogonal idempotents of Lie A, let
(y) =

εn , y εn , εn 

5 Cores and co-cores

79

for y ∈ C, where the inner product is chosen as usual so that A consists of self-adjoint transformations. Then there exists a constant m > 0 such that
(ghp) ≥ R for some g ∈ ω , h ∈ A+ =⇒
(hp) ≥ Proof We know that to define




−1 h∈A+ {h ω h}

R . m

is relatively compact, so it makes sense

m = sup{
(h−1 ghp) | g ∈ ω , h ∈ A+ } . Now, if q ∈ C is arbitrary, we show next that
(hq) =
(hp)
(q). In fact, write q = λ εn + f with λ ∈ R>0 and f orthogonal to εn . Since h ∈ A is self-adjoint, and εn is an eigenvector for anything in A, it follows that h f is orthogonal to εn . Thus,
(hq) =
(hλ εn + h f ) = λ
(hεn ) . Apply this to q = p, remembering that the εi are mutually orthogonal:
(hp) =
(hεn ). By definition,
(q) = λ . Thus
(hq) =
(hp)
(q). We conclude that if
(hp) < mR , then
(ghp) =
(h(h−1 gh)p) < mR
(h−1 ghp) < R, this being the contrapositive of what the lemma states. Proof (of Theorem 5.2) We will prove the theorem in a chain, showing each set in parts (a) through (e) is contained in some dilation of the subsequent set. (1) Let H be the convex hull of C ∩ L. Choose M ∈ Z>0 such that M p = w for some lattice point w. Since H is Γ-invariant, it suffices to show that p + C ⊂ 1 M H. By the reduction theory, C is covered by the Γ-translates of a finite union of rational polyhedra σα . So it is enough to show, given γ ∈ Γ and σα , that p + γ −1 σα ⊂ M1 H, or, equivalently, that γ p + σα ⊂ M1 H. If x is any point of σα , we can write x = ∑ki=1 ti vi , where vi ∈ L ∩ C and ∑ ti ≤ 1. Then   M(γ p + ∑ ti vi ) = ∑ ti (Mvi + γ w) + 1 − ∑ ti (γ w) ∈ H because Mvi + γ w and γ w are in L ∩C. We conclude that γ p + σα ⊂

1 M H.

(2) Let J be the set described in part (c) of the theorem. Then H ⊂ J simply by the way we normalized the ϕC1 s, and by their convexity. (3) We must show that there exists λ ∈ R>0 such that, if x ∈ C and ϕC1 (πC1 x) ≤ 1 for every rational boundary component C1 , then x, y ≥ λ for all y ∈ C ∩ L.

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II Polyhedral reduction theory in self-adjoint cones

We will prove this by induction on dimC. If dimC = 0, there is nothing to prove. Now suppose dimC > 0. If C1 is a rational boundary component, the hypothesis for x is inherited by πC1 x and so, by induction, there exists λ1 (depending on C1 ) such that πC1 x, y ≥ λ1 for all y ∈ C1 ∩ L. But x, y = πC1 x, y, so x, y ≥ λ1 in addition. Now there are only a finite number of t Γ-orbits of rational boundary components (see Proposition 15.6 in [3]). Let C1 , . . . ,Cs be representatives for the t Γ -orbits of rational boundary components and let λ = min (λ1 , . . . , λs ). If C0 is another rational boundary component, pick γ ∈ Γ such that t γ C0 = Ci for some i between 1 and s. If y ∈ C0 ∩ L , then t γ y ∈ Ci ∩ L and γ −1 x ∈ K by Lemma 5.4 so that x, y = γ −1 x, t γ y ≥ λ . It remains to consider y ∈ C ∩ L. We need only show that there exists some λ  > 0 such that ϕC (x) ≤ 1 =⇒ x, y ≥ λ  > 0 for all y ∈ σ ∩ L ∩ C, where σ is spanned by y1 , . . . , yr ∈ L ∩ C. If y = ∑ ai yi and d = det(y1 , . . . , yr ), then y ∈ L ∩C =⇒ ai ∈ d1 Z and ai ≥ 0 for each i. Furthermore y = ∑i|ai >0 yi ∈ L ∩C, as otherwise {yi | ai > 0} would be contained in a proper boundary component and hence so would y. But x, y ≥ d1 x, y  and x, y  is bounded away from zero because (i) the set {x | ϕC (x) ≤ 1} is in the complement of some ball B around 0 by Proposition 1.6; (ii) {x, y  | x ∈ C \ B} is bounded away from zero because y ∈ C; (iii) there are only finitely many possible y . This gives the desired λ  > 0 with the property that ϕC (x) ≤ 1 =⇒ x, y ≥ λ  for all y ∈ σ ∩ L ∩C.


(4) We know that C = Γ · ni=1 ωi A+ i p, so, for any x ∈ C, we have x = γ gap with + γ ∈ Γ, g ∈ ωi and a ∈ Ai for some i. We must show that there exists M > 0 such that x, y ≥ 1 for all y ∈ C ∩ L implies that M · ap ∈ A+ i p + p. Let ε1 , . . . , εn be the maximal set of orthogonal idempotents of Lie A, so that A+ i p is the convex cone spanned by ε1 , ε1 + ε2 , . . . , ε1 + · · · + εn = p. Then it is + easy to see that, if x ∈ A+ i p, then x ∈ Ai p + p if and only if εn , x ≥ εn , εn . t Choose M1 ∈ Z>0 so that γ M1 εn is a lattice point for all γ ∈ Γ. (This is possible because L and L∗ are commensurate.) The hypothesis on x says that t γ −1 M1 εn , x ≥ 1. That is, M1 εn , gap = M1 t γ

−1

εn , γ gap ≥ 1 .

Letting m be the constant in Lemma 5.6, we get M1 εn , ap ≥

1 m.

If M =

5 Cores and co-cores

81

M1 mεn , εn , then εn , Map ≥ εn , εn , and we are done. (5) For this last step, it clearly is enough to show M · ω (A+ p + p) ⊂ C + p for some M > 0. Now, A+ p + p ⊂ C + p, so that ω (A+ p + p) ⊂ ω C + ω p = C + ω p. Since ω is compact, we may choose M > 0 so that M · ω p ⊂ C + p. Indeed, just pick M such that q − M1 p ∈ C for all q ∈ ω p. Then M · ω (A+ p + p) ⊂ M ·C + M · ω p ⊂ C +C + p ⊂ C + p. Corollary 5.7 Any core is comparable to its closed convex hull. Proof Some of the cores in the list of the theorem are closed and convex. Let A be one of them and let B be any core. Then there are λ1 , λ2 , µ1 , µ2 ∈ R>0 such that λ1 A ⊂ B ⊂ λ2 A and µ1 B ⊂ A ⊂ µ2 B. The first implies that

λ1 A ⊂ closed convex hull of B ⊂ λ2 A , and so

λ1 µ1 B ⊂ closed convex hull of B ⊂ µ2 λ2 B . Thus the closed convex hull of any core is again a core.

5.2 We need another class of comparable kernels that stand in duality to the cores of C. Recall that R≥1 denotes the set of real numbers [1, ∞). We call A ⊂ V a semi-conical convex set if A is convex and R≥1 A = A. For any set A ⊂ V , define: semi-hull(A) = closed convex hull of R≥1 A ; Aˇ = {x ∈ V | x, a ≥ 1 for all a ∈ A} . Note that Aˇ = 0/ if 0 ∈ A. Proposition 5.8 If semi-hull(A) does not contain 0, then Aˇˇ = semi-hull(A) . ˇˇ Let b ∈ Aˇˇ and suppose b ∈ Proof Clearly A ⊂ A. / semi-hull(A). Then by the separating hyperplane theorem, there is a y ∈ V such that y, b < λ and y, a ≥ λ for all a ∈ semi-hull(A), for some λ ∈ R. If λ > 0, replace y by λ1 y. This gives y, b < 1 and y, a ≥ 1 for all a ∈ ˇˇ a contradiction. semi-hull(A), which implies that y ∈ Aˇ and so b ∈ / A,

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II Polyhedral reduction theory in self-adjoint cones

Meanwhile, if y, a < 0 for some a ∈ A, then y,ta −→ −∞ as t −→ +∞, which is impossible. Thus only the case λ = 0 remains. Now we have y, a ≥ 0 for all a ∈ semi-hull(A) and y, b < 0. By the separating hyperplane theorem applied to 0 and semi-hull(A), there exists z ∈ V with z, a ≥ µ for all a ∈ semi-hull(A) and 0 = z, 0 < µ . That is, µ > 0. Pick a small positive number δ so that y + δ z, b < 0 still holds. But y + δ z, a ≥ δ µ > 0 for all a ∈ semi-hull(A). So we are back to the previous case. ˇ Proposition 5.9 If K is a kernel of C, then so is K. Proof By definition, K is a kernel if and only if K ⊂ C, and R≥1 K ⊂ K, and ˇ it is automatic that R≥1 Kˇ ⊂ Kˇ and 0 ∈ / K. Now, for K, C ⊂ R>0 K, and 0 ∈ / ˇ Also, C ⊂ R>0 K implies that, for any x ∈ Kˇ and any y ∈ C, we have Kˇ = K. x, y > 0, so that Kˇ ⊂ C. Finally, for any y ∈ C, y, x is bounded away from 0 ˇ as x runs through K because 0 ∈ / K. Thus C ⊂ R>0 K. Proposition 5.10 If two kernels K1 and K2 are comparable, then so are Kˇ 1 and Kˇ 2 . Proof This follows because, if µ > 0, then (µ K)∨ = µ −1 Kˇ for any set K, and, if K1 ⊂ K2 , then Kˇ 2 ⊂ Kˇ 1 for any two sets K1 and K2 . So K → Kˇ sets up a duality among closed convex kernels, and even among equivalence classes of comparable kernels which have the property of being comparable with their closed convex hulls (e.g., cores). Thus the duals of the cores (see Section 5.1) are a new class of comparable kernels, called co-cores. In particular, denote two of the cores by Σ1 = closed convex hull of C ∩ L , Σ2 = {x ∈ C | x, y ≥ 1 for all y ∈ C ∩ L } .

(5.1)

Then the corresponding co-cores are Σˇ 1 = {x ∈ C | x, y ≥ 1 for all y ∈ C ∩ L} , Σˇ 2 = closed convex hull of C ∩ L . Using these co-cores we will construct some new Γ-admissible polyhedral decompositions of C. First we must prove a few general propositions. Recall that e is an extreme point of a closed convex set A if and only if, for all x, y ∈ A, if e = 12 (x + y), then x = y = e. We denote by E(A) the set of all extreme points of A. The Krein–Milman theorem says that any compact convex set Σ is the closed

5 Cores and co-cores

83

convex hull of its extreme points. This fails if Σ is not compact, but we can say the following: Proposition 5.11 If Σ is a closed convex kernel, then Σ is the closed convex
hull of e∈E(Σ) (e +C). Proof For any e ∈ E(Σ), r ∈ C, there is a λ ∈ R>0 such that λ r ∈ Σ, since Σ is 1 1 (λ r) ∈ Σ; that is, λ +1 (e + r) ∈ Σ. Since Σ a kernel. Then we have λ λ+1 e + λ +1
is a semi-cone, e + r ∈ Σ. Thus Σ contains the closed convex hull of (e +C). Conversely, suppose q ∈ Σ, but q is not contained in the closed convex hull
of (e +C). By the separating hyperplane theorem, there exists λ ∈ R, w ∈ V such that w, e + r ≥ λ for all e ∈ E(Σ), r ∈ C, but w, q < λ . If we had w, r < 0 for some r ∈ C, then w, e + tr −→ −∞ as t −→ ∞, which is impossible. Therefore w ∈ C, and then w, q < λ implies that λ > 0. In the case when w ∈ ∂ C, choose y ∈ C of small enough norm so that w + y, q < λ . Of course, w + y, e + r ≥ λ still, so, replacing w by w + y, we may always assume w ∈ C. Let H = {z ∈ V | z, w ≤ λ }. Then H ∩ Σ is closed and convex. It is also compact, since w ∈ C. By the Krein–Milman theorem, since q ∈ H ∩ Σ and q, w < λ , there must exist an extreme point e0 of H ∩ Σ not on the hyperplane {z | z, w = λ }. Then e0 is an extreme point of Σ and w, e0  < λ , a contradiction. Corollary 5.12 If Σ is a closed convex kernel, Σˇ = E(Σ)∨ ∩C. Proof Clearly, E(Σ) ⊂ Σ, so Σˇ ⊂ E(Σ)∨ . Conversely, suppose w, e ≥ 1 for all e ∈ E(Σ), for some w ∈ C. Then w, e + r ≥ 1 for all e ∈ E(Σ), r ∈ C, so, by the previous proposition, w, s ≥ 1 for all s ∈ Σ. Next we show that among Γ-invariant kernels, cores and co-cores represent the two extremes. Precisely, we have: Proposition 5.13 If Σ is a Γ-invariant closed convex kernel, then there is a core K1 and a co-core K2 such that K1 ⊂ Σ ⊂ K2 . Proof Since Σ is a kernel, we have λ p ∈ Σ for some λ ∈ R>0 . Then λ (Γp) ⊂ Σ. Dualizing gives Σˇ ⊂ [λ (Γp)]∨ .

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We know a priori that Σˇ ⊂ C, so, just as in the proof of the corollary above, we have Σˇ ⊂ [λ (Γp)]∨ ∩C = [λ (Γp) +C]∨ = [λ Γ(p +C)]∨ . Thus K1 = [λ Γ(p +C)]∨∨ is a core and K1 ⊂ Σ. Now Σˇ is itself a t Γ-invariant ˇ closed convex kernel, so, by what we have just proved, there is a core K2 ⊂ Σ. ˇ Then K3 = Kˇ 2 is a co-core and Σ = Σˇ ⊂ K3 .

5.3 The idea is to make a Γ-admissible polyhedral decomposition by taking the cones over the faces of some Γ-invariant kernel Σ. Roughly speaking, Σ must be “locally polyhedral”, so that its faces will be polygons. We also want the number of faces to be finite modulo Γ. For example, take C = R2>0 , L = Z2 . In this case the cores Σ1 , Σ2 in (5.1) coincide.

One sees that the problem with a core is that a face can be parallel to the boundary of the cone, and so contain an infinite number of vertices. We will deal with co-cores instead. Let C∗ be the union of C with all rational boundary components. Definition 5.14 † A closed convex kernel Σ is called rationally locally polyhedral if, for any rational polyhedral cone Π with vertices in C∗ , there exist x1 , . . . , xs ∈ V (Q) ∩C and λ1 , . . . , λs ∈ Q>0 such that Π ∩ Σ = {y ∈ Π | xi , y ≥ λi for i = 1, . . . , s} . Definition 5.15 We shall call a Γ-invariant rational locally polyhedral closed and convex kernel a Γ-polyhedral kernel. Proposition 5.16 If Σ is a Γ-polyhedral kernel, then there exists M ∈ Z such that E(Σ) ⊂ M1 L. † This definition differs from the corresponding definition in the first edition.

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85

Proof Using our reduction theory, choose Π1 , . . . , Πn , a fundamental set of rational polyhedra for Γ. It is easy to see that they may be chosen so that their projections into any standard boundary component form a fundamental set in the boundary component. So,

γ Πi = C∗ .

γ ∈Γ i=1,...,n

Clearly, E(Σ) ∩ Πi ⊂ E(Σ ∩ Πi ) for i = 1, . . . , n. By the definition of Γ-polyhedral, Σ ∩ Πi is cut out of V by a finite number of rational hyperplanes. Therefore, E(Σ ∩ Πi ) is a finite set of rational points, and there exists M ∈ Z such that E(Σ ∩ Πi ) ⊂ M1 L, for i = 1, . . . , n. For any e ∈ E(Σ) ∩C∗ there is a γ ∈ Γ and i ∈ {1, . . . , n} such that γ e ∈ Πi .
Since Σ is Γ-invariant, we have γ e ∈ E(Σ). Thus E(Σ)∩C∗ = Γ[E(Σ)∩Πi ] ⊂ 1 M L. Furthermore, Σ∩C is contained in the closed convex hull Σ of the set E(Σ)∩ ∗ C +C, since, clearly, Σ ∩ Πi is contained in Σ and Σ is Γ-invariant. Because Σ ∩ C is dense in Σ, also Σ ⊂ Σ . But Σ is convex and closed, so that Σ = Σ . Now we need the following lemma†: Lemma 5.17 Let D be a discrete subset of C and let X be the closed convex hull of D +C. Then every extreme point of X belongs to D. Proof Recall the definition of an exposed point: a point p in a closed convex set Y is said to be exposed if there exists a linear function φ such that φ (p) = 0 but φ |Y \{p} > 0. Then Straszewicz’s theorem (see [12], Theorem 18.6) states that the exposed points are dense in the extreme points. Since D is discrete, it is enough to see that every exposed point p of X belongs to D. Since p +C ⊂ X, we have φ |C ≥ 0 and φC\{0} > 0. But then

φ (p) = inf(φ |X ) = inf(φ |D+C ) = inf(φ |D ) is assumed in some point p ∈ D and hence p = p ∈ D. Taking D = E(Σ) ∩C∗ in the lemma, we have X = Σ = Σ. We conclude that E(Σ) = E(X) ⊂ D = E(Σ) ∩ C∗ , and we have already seen that E(Σ) ∩ C∗ ⊂ 1 M L. For any y ∈ C, write Hy = {z ∈ V | z, y = 1} . † This lemma is taken from [9] and replaces an incorrect argument in the first edition.

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II Polyhedral reduction theory in self-adjoint cones

The following proposition will give us some explicit Γ-admissible decompositions of C. Recall that if X is a closed convex set and H is a hyperplane, then H is said to be a supporting hyperplane of X if X lies entirely in one of the two closed half-planes defined by H and X ∩ H = 0. / In particular, X ∩ H is a face of X. Proposition 5.18 Let Σ be a Γ-polyhedral co-core, and let Y be the set of y ∈ C such that Hy is a supporting hyperplane of Σ which meets E(Σ) in a set of points spanning V . Let σy be the cone over Hy ∩ Σ. Then the set S of all faces of σy (including σy ), as y ranges through Y , is a Γ-admissible polyhedral decomposition of C, cf. Definition 4.10. Proof First, we claim that in fact Y ⊂ C. Indeed, assume y ∈ Y ∩ ∂ C. By Proposition 5.16, y must be rational. Hence there exists some z ∈ C \ {0} rational with y, z = 0. But then some multiple of z lies in C ∩ L \ {0}, and hence some multiple of z lies in the co-core Σ. But this contradicts the assumption that Hy is a supporting hyperplane of Σ. We have several things to check. (1) Since y ∈ C, it follows that Hy ∩ Σ is compact and so is supported by its extreme points. Since Hy is a supporting hyperplane, E(Hy ∩ Σ) ⊂ E(Σ). By Proposition 5.16, E(Hy ∩ Σ) ⊂ M1 L for some M ∈ Z, and so Hy ∩ Σ is the closed convex hull of a finite set of points. Therefore σy is truly a polyhedral cone, and so is each of its faces. (2) By definition, S is Γ-invariant and, if σ ∈ S , then any face of σ is in S . (3) We show that if σ , τ ∈ S , then σ ∩ τ is a face of both σ and τ . First, suppose both σ and τ are top-dimensional, so σ = σy and τ = σz for some y, z ∈ Y . Then σy = R≥0 (Hy ∩ Σ) and σz = R≥0 (Hz ∩ Σ). Clearly, R≥0 (Hz ∩ Hy ∩ Σ) ⊂ σy ∩ σz . If 0 = w ∈ σy ∩ σz , there exist λ , µ ∈ R>0 with λ w ∈ Hy ∩ Σ and µ w ∈ Hz ∩ Σ. We must have λ = µ , for, if λ > µ , say, then λ w, y = 1 =⇒ / Σ. Thus σy ∩ σz = R≥0 (Hz ∩ Hy ∩ Σ). Since Hz ∩ Hy ∩ Σ µ w, y < 1 =⇒ µ w ∈ is a face of Hy ∩ Σ, we conclude that σy ∩ σz is a face of σy , and similarly of σz . Now suppose σ and τ are arbitrary, so that σ = σy ∩K1 ∩K2 ∩· · ·∩Kn , where K1 , . . . , Kn are supporting hyperplanes of σy , and τ = σz ∩ Kn+1 ∩ · · · ∩ Km , where Kn+1 , . . . , Km are supporting hyperplanes of σz . Then σ ∩ τ = σy ∩ σz ∩ K1 ∩ · · · ∩ Km . We have just seen that there is a hyperplane K supporting σy , so that σy ∩ σz = σy ∩ K. So

σ ∩ τ = σy ∩ K ∩ K1 ∩ · · · ∩ Km = σ ∩ K ∩ Kn+1 ∩ · · · ∩ Km .

5 Cores and co-cores

87

Since σ ∩ K ⊂ σy ∩ K = σy ∩ σz , we know that Kn+1 , . . . , Km all support σ ∩ K, and K supports σ ⊂ σy . Therefore σ ∩ τ is a face of σ . Similarly, it is a face of τ .


(4) Finally we show that C ⊂ σ ∈S σ and that S is a finite set modulo Γ. Let Π1 , . . . , Πn be a fundamental set of rational polyhedra, as in the proof of the previous proposition. For any i ∈ {1, . . . , n}, we know that Σ ∩ Πi is cut out by a finite number of half-spaces. In fact, there are w1 , . . . , wm ∈ V such that Πi = {z ∈ V | wk , z ≥ 0 for k = 1, . . . , m} , and x1 , . . . , xt ∈ V , λ1 , . . . , λt ∈ R such that Σ ∩ Πi = {z ∈ Πi | x
, z ≥ λ
for
= 1, . . . ,t} . Clearly, Σ ∩ Πi is the union of the semi-hulls over its faces of top dimension. A face giving such a top-dimensional semi-hull will be cut out by a hyperplane K which supports Σ ∩ Πi such that there are v1 , . . . , vN ∈ K ∩ Σ ∩ Πi spanning V and either (i) wk0 , va  = 0 for some k0 ∈ {1, . . . , m} and all a = 1, . . . , N , or (ii) x
0 , va  = λ
0 for some
0 ∈ {1, . . . ,t} and all a = 1, . . . , N. Obviously (i) is impossible because the va span V , and (ii) is possible only if λ
0 = 0 and K = {z ∈ V | x
0 , z = λ
0 }. Letting y = λ1 x
0 , we have K = Hy .
0 By definition, y ∈ Y . Let Y0 ⊂ Y be the set of all y obtained in this way. Clearly, Y0 is finite. For any x ∈ C, we have λ x ∈ Σ for some λ ∈ R>0 . Thus γλ x ∈ Σ ∩ Πi for some γ ∈ Γ and i ∈ {1, . . . , n}, or, in other words, γ x ∈ σy for some y ∈ Y0 . Since Σ is Γ-invariant, this says x ∈ τ for some τ ∈ S . Meanwhile, for any τ ∈ S , let x ∈ τ be in the interior. Then again γ x ∈ σy for some y ∈ Y0 and some γ ∈ Γ. By (3) above, γτ ∩ σy is a face of both γτ and σy . Since γ x lies in the interior of γτ and in σy , this implies that γτ is a face of σy . Since Y0 is a finite set, we are done.

5.4 Let Σ be a Γ-polyhedral kernel. It is easy to see from the definition and the proof of Proposition 5.18 that there are a finite number of points x1 , . . . , xn ∈ V (Q) ∩C such that Σ = {y ∈ C | y, t γ xi  ≥ 1 for all γ ∈ Γ, i = 1, . . . , n} .

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II Polyhedral reduction theory in self-adjoint cones

In general, let N be a positive integer and T ⊂ C ∩ N1 L (recall L = L \ {0}). Denote ΣT = {x ∈ C | x, y ≥ 1 for all y ∈ T }. Sometimes we will simply write Σ if the T is understood. Proposition 5.19 For any t Γ-invariant subset T ⊂ C ∩ N1 L , ΣT is a Γ-polyhedral kernel. Proof First, ΣT is clearly closed, convex, and Γ-invariant. Also R≥1 ΣT ⊂ ΣT , 0∈ / ΣT , and C ⊂ R>0 ΣT . In other words, ΣT is a kernel. Let Π be a polyhedral cone with rational vertices contained in C∗ . Since Π can be written as the union of simplicial cones with rational vertices, we may assume that Π is spanned by z1 , . . . , zr ∈ NL∗ ∩C, where r = dimV . Then 6 7 Π = ∑ ai zi | ai ≥ 0 for i = 1, . . . , r . We want to show that there exists a finite set t1 , . . . ,tk ∈ T such that, if z ∈ Π and t j , z ≥ 1 for all j = 1, . . . , k, then t, z ≥ 1 for all t ∈ T . For then we would have, Π ∩ ΣT = {z ∈ Π | z,t j  ≥ 1 for all j = 1, . . . , k} . To do this we use the following. Let Z≥0 denote the non-negative integers. For a = (a1 , . . . , am ) ∈ (Z≥0 )m and b = (b1 , . . . , bm ) ∈ (Z≥0 )m , we write b ≥ a if and only if bi ≥ ai for all i = 1, . . . , m. Lemma 5.20 For any set S ⊂ (Z≥0 )m , there exists a finite subset S0 ⊂ S such that, for all b ∈ S, there exists some a ∈ S0 with b ≥ a. Proof For m = 1, it is clear. We proceed by induction on m, assuming the lemma for m − 1. Let π j : (Z≥0 )m −→ (Z≥0 )m−1 be the projection omitting the j’th coordinate. Fix some c = (c1 , . . . , cm ) ∈ S. For each j = 1, . . . , m and each q = 0, . . . , c j − 1, write S j,q = {s ∈ S | s j = q}. By induction applied to π j (S j,q ), we get a finite set S0j,q ⊂ S j,q such that, for all b ∈ S j,q , there exists
some a ∈ S0j,q with b ≥ a. Now S0 = j,q S0j,q ∪ {c} satisfies the lemma. We continue with the proof of the proposition. Define Φ : L ∩C −→ (Z≥0 )r by Φ(y) = (y, z, . . . , y, zr ). Let S0 ⊂ Φ(T ) be as in the lemma, and choose a section {t1 , . . . ,tk } of Φ over S0 . For any z = ∑ ai zi ∈ Π, suppose t j , z ≥ 1 for all j = 1, . . . , k. Then, for any t ∈ T , there is a j ∈ {1, . . . , k} such that t, zi  ≥ t j , zi  for all i = 1, . . . , r. Thus t, z = ∑ ai t, zi  ≥ ∑ ai t j , zi  = t j , z ≥ 1 because ai ≥ 0 for all i. So t1 , . . . ,tk do the trick.

5 Cores and co-cores

89

Proposition 5.21 If Σ is a Γ-polyhedral kernel, then Σˇ is a t Γ-polyhedral kernel. Proof By Corollary 5.12, we have that Σˇ = {y ∈ C | y, ξ  ≥ 1 for all ξ ∈ E(Σ)}. So what we want to show is that, given Π, there exists a finite number of extreme points ξ1 , . . . , ξt ∈ E(Σ) such that if z ∈ Π and ξi , z ≥ 1 for i = 1, . . . ,t, then ξ , z ≥ 1 for every extreme point ξ . This will follow precisely as in the proof of Proposition 5.19, if we use Proposition 5.16. We can summarize all this as follows: Proposition 5.22 The following three statements are equivalent: (a) Σ is a Γ-polyhedral kernel; (b) Σ = {y ∈ C | y, t γ xi  ≥ 1 for all γ ∈ Γ, i = 1, . . . , n} for some x1 , . . . , xn ∈ V (Q) ∩C; (c) Σ is the closed convex hull of

(γ xi +C)

i=1,...,n γ ∈Γ

for some x1 , . . . , xn ∈ V (Q) ∩C. Apply this to Σ1 , Σ2 , Σˇ 1 , Σˇ 2 of (5.1): Σ2 = {x ∈ C | x, y ≥ 1 for all y ∈ C ∩ L } and Σˇ 1 = {x ∈ C | x, y ≥ 1 for all y ∈ C ∩ L} are rationally locally polyhedral by Proposition 5.19. Then, by Proposition 5.21, the same is true of Σ1 = Σˇˇ 1 = closed convex hull of C ∩ L and Σˇ 2 = closed convex hull of C ∩ L . To obtain Γ-admissible decompositions of the cone, we need only take into account Proposition 5.18. Now Σˇ 2 is Γ-invariant, but Σˇ 1 is only t Γ-invariant. We may use instead ˇΣ∗ = {x ∈ C | x, y ≥ 1 for all y ∈ C ∩ L∗ }. Then we may summarize: 1 Corollary 5.23 Taking cones over the faces of the closed convex hull of C ∩ L (resp. {x ∈ C | x, y ≥ 1 for all y ∈ C ∩ L∗ }) yields a Γ-admissible polyhedral decomposition of C.

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II Polyhedral reduction theory in self-adjoint cones

The first kind is called a Voronoi decomposition (of the first type) and the second is called a decomposition into central cones, since in each case they generalize known constructions for the cone of positive-definite quadratic forms in a given number of variables.

6 Positive-definite forms in low dimensions We consider the most classical example of the above theory: Vn = vector space of symmetric n × n real matrices ; Cn = cone of positive definite elements in Vn . Then Vn has a natural Q-structure given by the rational n × n matrices and is a Jordan algebra via X ·Y = 12 (XY +Y X). So Aut (Cn ) is just GL(n, R)/(±In ) acting via X −→ t AXA,

A ∈ GL(n, R) ,

and the characteristic function of the cone is given by

ϕ (X) =

1 . det(X)(n+1)/2

There are two natural lattices in Vn : Ln = integral n × n-matrices X , Ln∗ = semi-integral n × n-matrices X , i.e.,

: 9 1 Ln∗ = X ∈ Vn | Xii ∈ Z, Xi j ∈ Z if i = j . 2

If In ∈ Cn is taken as the basepoint, we get the inner product X,Y  = Tr (XY ) on Vn for which: (i) Cn is self-adjoint; (ii) Stab (In ) = O(n, R)/(±In ) acts orthogonally; (iii) its polar complement P, the set of symmetric matrices in GL(n, R), acts by self-adjoint maps; and (iv) the notation Ln∗ is justified: Ln∗ = {x ∈ Vn | x, y ∈ Z for all y ∈ Ln } .

6 Positive-definite forms in low dimensions

91

The arithmetic group Γn = GL(n, Z)/(±In ) preserves both lattices. The rational boundary components of Cn correspond one-to-one with the subspaces W ⊂ Rn defined over Q, the correspondence being given by   ; ; X positive semi-definite W ←→ C(W ) = X ∈ Vn ; . with null space W Thus



C(W ) Cn∗ = Cn ∪ W   ; ; X positive semi-definite . = X ∈ Vn ; with rational null space W The problem of finding an explicit decomposition of Cn∗ into rational polyhedral cones {σα } invariant under Γn , and hence of finding a fundamental domain for Γn , is a very old one. For all n, an important role is played by the fundamental cone φn ⊂ Cn∗ :   ; ; off-diagonal entries Xi j non-positive . φn = X ∈ Vn ; row sums ∑ni=1 Xi j non-negative In fact, φn is a simplicial cone, being expressible also as   ; n 2 2; φn = ∑ λi xi + ∑ λi j (xi − x j ) ;λi ≥ 0, λi j ≥ 0 . i=1

1≤i< j≤n

In some areas of applied mathematics, the cone φn plays a role – basically because the inequalities xi j ≤ 0 for i = j and ∑ni=1 xi j ≥ 0 are the simplest linear way to force a matrix to be positive semi-definite. A basic result is: Cn∗ =



γφn ⇐⇒ n ≤ 3 .

γ ∈GL(n,Z)

This illustrates the interesting fact that only in 4-space or higher do lattice packing problems and related geometry of numbers problems get interesting. For n = 2 or 3, however, all “reasonable” admissible polyhedral decompositions of Cn∗ are based on {γφn } – either {σα } equals {γφn }, or it equals {γψi }, where {ψi } is a decomposition of φn . Thus the standard fundamental domain for n = 2,  ;  a −b ; φ˜2 = ;0 ≤ 2b ≤ a ≤ c , −b c

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II Polyhedral reduction theory in self-adjoint cones

comes about by barycentric subdivision of φ2 :

Since φ2 is mapped into itself by the group of order six,           0 1 0 1 1 −1 1 −1 1 I2 , , , , , 1 0 −1 1 1 0 0 −1 1

0 −1

 ,

we see that φ˜2 ∩ γ φ˜2 is at most a face of φ˜2 , if γ = id . For n ≥ 4, we need more cones. To find them systematically, we use a cocore. In fact, there are apparently two very natural co-cores in Cn , which we call the perfect and central co-cores: Kperf = the closed convex hull of Ln∗ ∩Cn \ {0} ; Kcent = {X ∈ Cn | Tr (XY ) ≥ 1 for all Y ∈ Ln∗ ∩Cn } . The dual cores are: Kˇ perf = {X ∈ Cn | Tr (XY ) ≥ 1 for all Y ∈ Ln∗ ∩Cn \ {0}} ; Kˇ cent = the closed convex hull of Ln∗ ∩Cn . According to a result of Barnes and Cohn [2], min

Y ∈Ln∗ ∩Cn \{0}

Tr (XY ) =

min

m∈Zn \{0}

t

mXm ,

and hence we get a second definition of the perfect co-core and core: Kˇ perf = {X ∈ Cn | t mXm ≥ 1 for all m ∈ Zn \ {0}} ; Kperf = closed convex hull of the rank-1 matrices Xi j = mi m j with m ∈ Zn \ {0} . Furthermore,

µ (X) =

min

m∈Zn \{0}

t

mXm

is the piecewise-linear function which is 1 on ∂ Kˇ perf .

6 Positive-definite forms in low dimensions

93

Finally, we say that a form X ∈ Cn is perfect, (resp. central), if it is a vertex of Kˇ perf , (resp. Kˇ cent ). By the general theory, perfect (resp. central) forms correspond one-to-one with the top-dimensional faces of Kperf , (resp. Kcent ), hence to the n(n+1) 2 -dimensional cones in the decomposition of Cn defined by Kperf , (resp. Kcent ), which we call the perfect cones (resp. central cones). In the case of perfect forms, we see easily that X is perfect if and only if µ (X) = 1 and X is the only solution to the equations t

mY m = 1 for all m ∈ Zn such that t mXm = 1 ,

and that the corresponding perfect cone is the convex hull of the rank-1 forms Xi j = mi m j for all m ∈ Zn such that t mXm = 1. Central cones, on the other hand, are given by

σY = {X ∈ Cn∗ | Tr (XY ) ≤ Tr (XZ) for all Z ∈ Ln∗ ∩Cn } , where Y is a central form. For n ≤ 6, all perfect forms have been listed; see [1] and [6]. For example, consider the form

αn =

 1 2 x1 + · · · + xn2 + (x1 + · · · + xn )2 2

corresponding to the matrix ⎛

⎜ 1 ⎜ 2 ⎜ . ⎝ ..

1 .. .

··· ··· .. .

1 2

1 2

···

1

1 2

1 2 1 2

⎞

⎟ ⎟ .. ⎟ . . ⎠ 1

Note that αn , up to unimodular equivalence, may be described as the Killing form for the root system An , written with respect to a basis of the lattice generated by the roots. This may be shown to be both perfect and central and it corresponds to a common face of Kcent and Kperf , namely the simplex in Cn∗ with vertices xi2 , (xi − x j )2 , where 1 ≤ i, j ≤ n. This shows that φn is both a central and a perfect cone. Another important form is

δn =

 1 (x1 − x2 )2 + x32 + · · · + xn2 + (x1 + · · · + xn )2 2

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II Polyhedral reduction theory in self-adjoint cones

corresponding to the matrix ⎛

1 ⎜ 0 ⎜ 1 ⎜ ⎜ 2 ⎜ . ⎝ ..

0 1 1 2

.. .

1 .. .

··· ··· ··· .. .

1 2

1 2

1 2

···

1 2 1 2

1 2 1 2 1 2

⎞

⎟ ⎟ ⎟ ⎟. .. ⎟ . ⎠ 1

Note that δn , up to unimodular equivalence, may be described as the Killing form for the root system Dn , written with respect to a basis of the lattice generated by the roots. This form is also both perfect and central, but it does not seem to be known whether it defines identical faces of Kcent and Kperf . Note that Kcent ⊇ Kperf ; Kˇ cent ⊆ Kˇ perf . It may be shown that equality holds if and only if n ≤ 5. In particular, for n = 4, Kcent = Kperf , and forms are perfect if and only if they are central. It turns out that, in this case, α4 , δ4 , and their images under Γ4 are the only perfect forms; hence, if ψ4 is the perfect cone corresponding to δ4 , we conclude that, for n = 4, the cones γφ4 and γψ4 for γ ∈ Γ4 form an admissible polyhedral decomposition of C4∗ .

References [1] E. S. Barnes, The complete enumeration of extreme senary forms, Phil. Trans. Royal Soc. London 249 (1957), 461–506. [2] E. S. Barnes and M. J. Cohn, On the inner product of positive quadratic forms, J. London Math. Soc. 12 (2) (1975/76), 32–36. [3] A. Borel, Introduction aux Groupes Arithmetiques. Paris: Hermann, 1969. [4] A. Borel and J.-P. Serre, Corners and arithmetic groups, Comm. Math. Helv. 48 (1973), 436–491. [5] H. Braun and M. Koecher, Jordan-Algebren. Berlin: Springer-Verlag, 1966. [6] H. S. M. Coxeter, Extreme forms, Canad. J. Math. 3 (1951), 391–441. [7] N. Jacobson, Structure and Representations of Jordan Algebras. Providence, RI: American Mathematical Society, 1968. [8] M. Koecher, Jordan Algebras and their Applications. Lecture Notes. University of Minnesota, 1962.

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95

[9] E. Looijenga, Semi-Toric Partial Compactifications I. Preprint series of Department of Mathematics, Catholic University, Nijmegen, The Netherlands, June 1985. [10] O. Loos, Symmetric Spaces I: General Theory. New York: Benjamin, 1969. [11] D. Mumford, Geometric Invariant Theory. Berlin: Springer-Verlag, 1965. [12] R. T. Rockafellar, Convex Analysis. Princeton, NJ: Princeton University Press, 1970. [13] T. A. Springer, Jordan Algebras and Algebraic Groups. Berlin: SpringerVerlag, 1973. [14] G. Voronoi, Nouvelles applications des param`etres continus a` la th´eorie des formes quadratiques I, J. Reine Angew. Math. 133 (1907), 97–178.

III Compactifications of locally symmetric varieties

This chapter presents the main results of this book. In order to make the book reasonably self-contained, and because of the difficulty in assembling the known results from a very diffuse and often hard-to-read literature, we have tried to include, with at least sketches of proof, a large proportion of the needed background on hermitian symmetric spaces. We have profited greatly from lectures on hermitian symmetric spaces given by P. Deligne in Paris in 1973. Large parts of Sections 2 and 3 are no more than an elaboration of his work. One of us (Rapoport) profited also from conversations with R. P. Langlands. Section 1 explains the method of compactification in the case of a tube domain; some of the facts established there are used later. This is also done in Satake [10]. Sections 2, 3, and 4 review facts about hermitian symmetric domains and their boundary theory. In addition to help from Deligne and Langlands, we have used principally Helgason [6] and Wolf [12], but also Harish-Chandra [5], Langlands [8], Koranyi and Wolf [7], and Satake [11]. Section 5 states the Main Theorem; Section 6 contains its proof. Here basic facts from the fundamental paper by Baily and Borel [1] are used. For the reduction theory, we have found the book by Borel [2] very useful. Finally Section 7 contains a more intrinsic formulation of the Main Theorem.

1 Tube domains and compactification of their cusps Let NR be a finite-dimensional R-vector space and let C ⊂ NR be a homogeneous self-adjoint cone. Set G = Aut (C, NR )o (this notation thus differs from that in Chapter II). 97

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III Compactifications of locally symmetric varieties

Definition 1.1 The open subset U = NR + iC = {x + iy | x ∈ NR , y ∈ C} of NC is called a tube domain. Let G be the group of complex-analytic automorphisms of U. Proposition 1.2 (1) U is a bounded symmetric domain. (2) G, equipped with the compact open topology, is a Lie group. Proof (2) follows from (1) by general theory: U, as a bounded domain, is equipped with the Bergman metric. Moreover, G is a closed subgroup of the Lie group of isometries with respect to this metric. Thus G itself is a Lie group. To see that U is a bounded domain, choose a basis of NR such that C ⊂ {y1 > 0, . . . , yn > 0} (this is obviously possible). This embeds U into Hn , where as usual H denotes the upper half-plane. Since H is a bounded domain (isomorphic to the unit disc), so is Hn and thus also U. To see that U is a homogeneous domain, note that the real translations by NR and the complex-linear extensions of the linear maps in G take U to U and act transitively on U. Finally, to see that U is symmetric, it suffices to construct an involutive symmetry around the point ie ∈ U (here e = a, the basepoint in C). As in Chapter II, Section 2, we describe C as the set of exponentials for a real Jordan algebra structure on NR with identity e; this algebra structure extends to a complex Jordan algebra structure on NC . We claim that, in this algebra structure, every x ∈ U is invertible, that −x−1 ∈ U, and that the map x −→ −x−1 is the required symmetry. But for any algebra structure with identity e, we have that −(ie)−1 = ie, and x −→ −x−1 is an involution near ie with no other fixed point. It remains to show that every x ∈ U is Jordan-invertible and that −x−1 ∈ U. First of all, for every g ∈ G, we claim that x invertible implies gx invertible, and (gx)−1 = σ (g) · x−1 ,

(1.1)

where σ : G −→ G is the Cartan involution fixing G ∩ Stab (e). In fact, we saw in Chapter II, Section 2, that (g · e)−1 = σ (g) · e, hence if g, g1 ∈ G, then (g · g1 · (ie))−1 = −iσ (g · g1 ) · e = −iσ (g) · σ (g1 ) · e = σ (g) · (g1 · (ie))−1 , i.e., (1.1) holds for x ∈ iC. By analytic continuation it holds for all invertible x ∈ NC . This reduces our problem to showing that, for all x ∈ NR , x+ie is invertible and −(x + ie)−1 ∈ U. To see this, let R[x] be the real Jordan subalgebra of NR generated by x and e. As in Chapter II, R[x] ∼ = ∑ni=1 Rεi , where εi are

1 Tube domains and compactification of their cusps

99

idempotents, and R[x] ∩ C ∼ = ∑ni=1 R>0 εi . Indeed, C is self-adjoint w.r.t. the pairing y, z = Tr (Ly·z ) by Chapter II, Theorem 2.13. Then, for y = ∑ni=1 yi εi ∈ R[x] ∩C, we have yi εi , εi  = yεi , εi  = y, εi2  > 0 , and hence y ∈ ∑ni=1 R>0 εi . Conversely, any such element is a square of an invertible element and thus lies in C. Since x + ie lies in the subalgebra C[x], we are reduced to checking the one-dimensional case NR = R. But it is obvious that, for all a ∈ R, a + i ∈ C is invertible and −(a + i)−1 ∈ H, the upper halfplane. Now start with (1) an algebraic group G over Q such that (2) its associated Lie group G = G (R)o is the connected component of the group of complex-analytic automorphisms of U. We also assume that (3) the subgroup P ⊂ G, which is the semi-direct product of G = Aut (C, NR )o by the group of real translations NR , is defined over Q, i.e., P = P(R)o , where P ⊂ G is an algebraic subgroup defined over Q. This defines a rational structure on NR and G such that G acts rationally on NR . Let Γ ⊂ G (Q) ∩ G be an arithmetic subgroup. Let NZ be the group of translations contained in Γ: this is a lattice in NR . We set Γ = (Γ ∩ P)/NZ , which is the image of Γ via P −→ G. Here is an example of the situation we consider. Example 1.3 (Siegel case) NR = Sym2 (Rn ) ; C is the cone of positive-definite quadratic forms on Rn ; U is the Siegel upper half space in Cn(n+1)/2 ; G = Sp (2n, R)/{±1}, Γ = Sp (2n, Z)/{±1} ; G = GL(n, R)/{±1}, Γ = GL(n, Z)/{±1} ;   a b ∈ G acts on U by z −→ (az + b) · (cz + d)−1 ; a d ) * a 0 ∈ G acts on NR by x −→ a · x · t a ; 0 t a−1   ∗ ∗ P= ∩ Sp (2n, R)/{±1} . 0 ∗

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Let {σα } be a Γ-admissible polyhedral decomposition of C. We now come to the main idea of this whole volume. Let T = NC /NZ : this is a torus in the algebraic group sense with NZ ∼ = Hom (Gm , T ), the group of one-parameter subgroups of T . As in Chapter I, we have an exact sequence: ord

0 −→ Tc −→ T −→ NR −→ 0 . Since U is NZ -invariant, it is the inverse image of an open subset U  ⊂ T ; alternatively, U  = ord−1 (C). For every ε ∈ C, set Cε = C + ε , Uε = NR + iCε , Uε = ord−1 (Cε ) = Uε /NZ . (The Uε are nothing but Piatetskii-Shapiro’s cylindrical sets.) Next, by the theory of TE I†, {σα } defines a T -equivariant embedding: T ⊂ X{σα } . Let U  = interior of the closure of U  in X{σα } ; Uε = interior of the closure of Uε in X{σα } . Now, Γ acts on {σα }, as well as on T . So Γ acts on X{σα } , prolonging its action on T ; this action preserves U  . Theorem 1.4 (i) Γ acts properly discontinuously on U  . (ii) (Γ ·Uε )/Γ is open and relatively compact in U  /Γ. We use the commutative diagram introduced in Chapter I, Section 1: X{σα } ∪ T ∪ U

ord

−−−−−→ N{σα } ∪ ord

−−−−−→ −−−−−→

NR ∪ C

Proof of (i) We know that ord is a continuous Γ-equivariant map. It thus suffices to show that Γ acts properly discontinuously on ord(U  ). Because ord is a quotient map by the compact group Tc by Chapter I, and, in particular, an open map, U  = ord−1 (C ), where C is the interior of the closure of C inside † Recall this reference from p. x.

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101

N{σα } . As in Chapter I, Section 1, we can describe the points of N{σα } by symbols x + ∞ · σα , for x ∈ NR , and a fundamental system of neighborhoods of x + ∞ · σα meets C in the sets x + y + Bε + σα , where Bε is the ε -ball around 0, and y ∈ σα . It follows that if x + ∞ · σα ∈ C , then x + y + Bε + σα ⊂ C for suitable y, ε . But x + y + ∞ · σα = x + ∞ · σα , so all points of C are represented by symbols x + ∞ · σα with x ∈ C. Conversely, if x ∈ C, then, for small enough ε , we have x + Bε + σα ⊂ C; hence, x + y + z + ∞ · σβ ∈ closure of C in N{σα } , for all y ∈ σα and z ∈ Bε , and for all faces σβ of σα . This means that x + ∞ · σα ∈ C . Thus C =



α x∈C

{x + ∞ · σα } .

Next, for all x + ∞ · σα , we claim that there is a finite set σα1 , . . . , σαn of polyhedral cones such that

n

i=1

σαi is a neighborhood of x + ∞ · σα .

(1.2)

In fact, let y1 , . . . , ym ∈ C have the property that x is in the interior of the polyhedral cone y1 , . . . , ym  spanned by the yi ; more precisely, suppose x + Bε ⊂ y1 , . . . , ym . Let τ be the polyhedral cone spanned by σα and the yi . We apply the main theorem of reduction theory (Chapter II, Corollary 4.9), plus the fact that mod Γ there are only finitely many σα , to conclude that {α | σα ∩ τ ∩C = 0} / is finite. Since the σα cover C, it follows that

τ ∩C ⊂ (σα1 ∪ · · · ∪ σαn ) for suitable σαi . Therefore x + Bε + σα ⊂ (σα1 ∪ · · · ∪ σαn ) . This proves (1.2). Now let xi ∈ C be a sequence of points converging to x ∈ C and let γi ∈ Γ be such that yi := γi · xi converges to a point y ∈ C . We have to show that the set of γi for i  0 consists of only finitely many elements {γ 1 , . . . , γ n } and that γ j · x = y, for j = 1, . . . , n. Since C is a Γ-invariant open dense subset of C , we may suppose that xi ∈ C. Now, by the preceding discussion, there exist finitely many polyhedra σαi and σα j such that xk ∈ σαi , yk ∈ σα j . By taking a subsequence if necessary, we may suppose that xi ∈ σ , yi ∈ σ  . It

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follows that γi σ ∩ σ  ∩C = 0. / By Corollary 4.9 of Chapter II, this implies that the set of the γi , for i  0, consists of only finitely many elements {γ 1 , . . . , γ n } which necessarily satisfy γ j · x = y, for j = 1, . . . , n. Proof of (ii) The openness is clear. Since C is the quotient of U  by a compact group we have to show the following statement: (ii ) Let z1 , z2 , . . . be a sequence of points in Cε ; then there exist elements γi ∈ Γ, such that, after passing to a subsequence, the sequence γi zi converges to a point in C . Now, by conditions (4) and (5) of a Γ-admissible decomposition, applied to {σα } (see Definition 4.10 in Chapter II), we can find zi = γi · zi such that, after passing to a subsequence, all zi lie in one and the same σα . It follows from the description of the topology of Nσα given in Chapter I that a subsequence of the {zi } converges to a point z ∈ Nσα . It remains to show that z ∈ C . This is clear if σα \ {0} ⊂ C. In general, σα ∩ ∂ C is contained in a finite union of rational boundary components Ck = Nk,R ∩ C of C for k = 1, . . . , n. Here Nk,R ⊂ NR is a rational subspace and we may choose a rational linear functional
k on NR such that
k ≥ 0 on C and (
k = 0) ∩ C = Ck . We may assume that ε ∈ L, where L is a lattice fixed by Γ and commensurable with NZ . It follows that γi ε ∈ L. Furthermore,
k on L ∩C is bounded away from zero, say by ck , as the image of L in NR /Nk,R is a lattice. Since zi ∈ γi ε +C , it follows that
k (zi ) >
k (γi ε ) ≥ ck . Since this is true for all k, the limit z of the sequence {zi } lies in C .  Now, the basic idea to compactify  U/Γ is to glue U/Γ and (Γ ·Uε )/Γ along  the set (Γ · Uε )/Γ = (Γ ∩ P) · Uε /(Γ ∩ P). Roughly speaking, this will add to U/Γ points at infinity at the cusp i∞ · C of U. To glue, we must know that (Γ · Uε )/Γ is a subset of U/Γ, i.e., that if ε is large enough, two Γ-equivalent points of Uε are in fact Γ ∩ P-equivalent. This is a consequence of “PiatetskiiShapiro’s lemma.” We will not pursue the construction further at this stage because we will also have to treat the “higher-dimensional cusps,” i.e., the cusps at ∞ in Siegel domains of the third kind. And we will then use Siegel sets instead of PiatetskiiShapiro’s cylindrical sets Uε to carry out the details. However, before launching into the morass of the general cusps, it is nice to look at the case where the above, relatively simple, construction is sufficient – the case of Q-rank 1. This is what we will do in the following appendix.

1 Tube domains and compactification of their cusps

103

Appendix: Groups of Q-rank 1 acting on tube domains We keep the notations from Section 1. Proposition 1.5 The following conditions are equivalent: (i) (ii) (iii) (iv) (v)

C/Γ · R>0 is compact; the Q-rank of G is 1; the only rational point on ∂ C is 0; the Q-rank of G is 1; the (proper) rational boundary components of the hermitian symmetric domain U are all zero- dimensional.

Proof That (ii) ⇐⇒ (iv) follows from the facts that, first, a Q-parabolic contains a maximal Q-split torus of G and, second, that G is the reductive part of the parabolic subgroup P of G; hence G contains a maximal Q-split torus of G. The equivalence of (ii) and (iii) comes from the correspondence between parabolic subgroups and rational boundary components, see Chapter II, Section 3.10. Similarly, the equivalence of (iv) and (v) comes from the general theory of bounded domains (see Section 3, Proposition 3.9, below). Finally, the equivalence (iii) ⇐⇒ (i) is a consequence of the reduction theory for cones.

Example 1.6 Let k be a totally real extension of degree n of Q. Let C = Rn>0 ⊂ Rn ; G = Rk/Q (Gm ) (Weil’s restriction of scalars) ; G = Rk/Q (SL(2)) . In this case U is the n-fold product of H with itself. Also Γ is commensurable with SL(2, O), where O is the ring of integers in k, and Γ is commensurable with the group of units of k. This example is called the Hilbert case, and for n = 2 was treated in Chapter I, Section 5. In this appendix, we want to look more closely at the Q-rank 1 case. In this case, we see immediately that U  \U  ⊂ Uε for every ε .   Therefore, U  /Γ = (U  /Γ) ∪ (Γ ·Uε )/Γ . Let E = (U  /Γ) \ (U  /Γ) be the locus at infinity that is added on. Then E ⊂ (Γ ·Uε )/Γ for every ε , so, by Theorem 1.4, E is relatively compact in U  /Γ. But E is closed in U  /Γ, so

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E is itself compact. Put another way, we have an analytic space U  /Γ and a compact analytic subset E, and U  /Γ is just (U  /Γ) \ E:

The main result of this appendix is as follows. Theorem 1.7 The set E is exceptional, i.e., can be blown down in U  /Γ to a point. 

Proof We may, if we like, pass to a subgroup of finite index Γ of Γ and prove  that (U  \U  )/Γ can be blown down. So we can assume from the beginning that Γ acts freely on U  ; in particular,  U /Γ is a manifold. We are going to apply the following fact (cf. [4]). Let A ⊂ X be a compact analytic subset of an analytic space, where X \ A is a manifold. If A possesses arbitrarily small strictly Levi-pseudoconvex neighborhoods, then A can be blown down to a point. (Recall that an open subset U ⊂ X is called strictly Levi-pseudoconvex if, for all y ∈ ∂ U, there exists a real C2 -function ϕ defined in a neighborhood V of y such that (i) U ∩V = {x ∈ V | ϕ (x) < 0} ; (ii) (convexity condition) for 0 = t ∈ Ty with dϕ (t) = 0, one has ∂ ∂¯ ϕ (t) > 0.) As in the previous section, let C be the interior of the closure of C in N{σα } . We will use the characteristic function ϕ of the cone C introduced in Chapter II, Section 1. It is easy to see that ϕ extends to a continuous function ϕ on C with ϕ ≡ 0 on C \ C. Moreover, ϕ is Γ-invariant, and hence comes from a function ϕ on C /Γ. Let f = ϕ ◦ ord be the induced function on U  . Then f comes from a function f on U  /Γ. Since ϕ (x) = 0 if and only if x ∈ C \ C, we get f (x) = 0 if and only if x ∈ E. Therefore the open sets Vc = {X ∈ U  /Γ | f (x) < c} form a family of arbitrarily small neighborhoods of E. Hence it suffices to

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105

show that Vc is pseudoconvex, or to check the Levi condition at the points of the boundary of Vc . We can choose coordinates zi = xi + i · yi on U  such that f becomes f (z1 , . . . , zn ) = ϕ (y1 , . . . , yn ) . Since ϕ is strictly convex (see Chapter II, Proposition 1.4), f is strictly Leviconvex, and the theorem is proven. Remarks (i) In the Hilbert case, the function ϕ : Rn −→ R considered in the above proof is simply ϕ (x1 , . . . , xn ) = 1/(x1 · · · xn ). For n = 2, the constructed neighborhoods coincide with the ones considered in Chapter I, Section 5. (ii) By applying the above procedure to all cusps we obtain in the end a compactification of U/Γ, which is just U/Γ with a finite number of points added: we recover in the Q-rank 1 case the so-called Baily–Borel compactification.

2 The structure of bounded symmetric domains In this section we summarize the standard theory of symmetric spaces. One of its purposes is to introduce notation.

2.1 A Riemannian symmetric space is a connected Riemannian manifold D such that, for every point x ∈ D, there exists an involutive automorphism sx which has x as an isolated fixed point. If M is a complex hermitian manifold, then D is a hermitian symmetric space if, for every point x ∈ D, there exists an involutive automorphism sx which has x as an isolated fixed point (here, of course, the condition that sx is an automorphism means that sx is holomorphic as well as isometric). If D is a hermitian symmetric space, then the Riemannian manifold D decomposes† as D = D0 × D1 × · · · × Dn , where: • D0 is the quotient of a complex vector space with a translation-invariant metric by a discrete group of translations (such a hermitian symmetric space is said to be of euclidean type); • Di , i = 0, is an irreducible and non-euclidean hermitian symmetric space. † If D is simply connected, this is well known and can be found, for instance, in Helgason [6]. In the hermitian case, simply connectedness is not needed: see Wolf [13], p. 490, Lemma 1.

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III Compactifications of locally symmetric varieties

Those factors Di , for i = 0, which are compact are said to be of compact type and are rational projective varieties; a non-compact factor Di with i = 0 is said to be of non-compact type and is a bounded domain in Cn (we will prove this later). The space D is called non-euclidean, resp. symmetric domain, resp. of compact type, if, respectively, the factor D0 is absent in the above decomposition, or all the Di are of non-compact type, or all Di are of compact type. In the latter two cases D is simply connected. If, in addition, there is only one factor present, D is called simple. Let D be a non-euclidean hermitian symmetric space and denote by G the identity component of its (Lie) group of automorphisms. Then the group G acts transitively on the space D and, choosing a basepoint o ∈ D, we may write D = G/K , where K ⊂ G is a compact subgroup. The symmetry so induces an automorphism σ of the group G such that K = σ K . Letting g = Lie (G) , k = Lie (K) = subspace of g, where σ = Id , p = subspace of g, where σ = −Id , we get a decomposition g = k⊕p . ∼

Note that there is a canonical isomorphism p −→ To , the tangent space to D at o. If D is now a hermitian symmetric domain, then the group G is semi-simple and adjoint, K is a maximal compact subgroup, and the above direct sum decomposition is a Cartan decomposition. Note that G is the connected component G (R)o of the set of real points of a unique algebraic group G defined over R: via the adjoint representation Ad : G −→ GL(g) , G is the Zariski closure of Im(Ad). Set kc = k , pc = ip(⊂ gC ) , gc = k c ⊕ p c . Then gc is a compact real form of gC . Passing to the group level, we can define

2 The structure of bounded symmetric domains

107

the compact dual of D by Dˇ = Gc /Kc . ˇ Conversely, starting with a hermitian symmetric space of compact type D, one gets back to the space D by the same construction. Let D be a non-euclidean hermitian symmetric space. Then there exists a morphism uo : U 1 −→ G from the circle group U 1 into G such that uo (z) ∈ K for any z ∈ U 1 and uo (z) induces the multiplication by z on the tangent space To of D at o. The group K ⊂ G is the centralizer of uo (U 1 ) in G, and hence is connected. Further, if D is simple, uo (U 1 ) is the center of K ⊂ G. We set ho = u2o . ∼

Noting that, via the isomorphism p −→ To the adjoint action of K on p corresponds to the action of K on To , we see that ;   J = Ad ho e2π i/8 ;p defines the given complex structure on To , whereas

σ = Ad (ho (i)) . Let pC = p+ ⊕ p− be the decomposition into ±i-eigenspaces for J. Note that these are abelian subalgebras of gC since, e.g., [p+ , p+ ] ⊂ {x ∈ g | Jx = −x} = (0) . Denote by P± the subgroup of GC generated by exp(p± ). Then KC normalizes P± and KC · P− is a parabolic subgroup of GC with unipotent radical P− ; hence GC /KC · P− is a projective algebraic variety, which we call for the moment X. Theorem 2.1 (Borel and Harish-Chandra embedding theorem) (a) The map P+ × KC × P− −→ GC given by multiplication is injective, G is contained in the image, and (KC · P− ) ∩ G = K. (b) Hence we have the following maps:

III Compactifications of locally symmetric varieties −−−−−→

GC /KC · P−

P+ ↑ exp p+

X

∼

D

−−−−−→ P+ × KC × P− /KC · P−

∼ =

∼ =

G/K

∼ =

108

These maps are holomorphic open immersions, the image of D in p+ is a bounded domain, and the image of p+ in X is a dense Zariski open set. (c) Finally, the compact form Gc ⊂ GC of G acts transitively on X; furthermore, Gc ∩ (KC · P− ) = K so that X∼ = Dˇ := Gc /K , i.e., X is the compact dual of D. For a full proof of this, we refer the reader to Helgason [6], Ch. 8, §7, and we give here only a brief indication; thus, on the Lie algebra level, we have gC = kC ⊕ p+ ⊕ p− , k = g ∩ (kC ⊕ p− ) = gc ∩ (kC ⊕ p− ) , and dim D = dim Dˇ = dim p+ = dim P+ = dim GC /KC · P− , from which we deduce that all the assertions are “true locally,” e.g., the natural maps D −→ X, and P+ −→ X, and Gc /K −→ X are local immersions. The main step is to check G ⊂ (bounded subset of P+ ) · KC · P− . This can be done by using Theorem 2.4 below to reduce to the simplest case G = SL(2, R)/{±1}, where it follows by an explicit explicit calculation. The rest is straightforward.

2.2 We want next to look at holomorphic maps between bounded symmetric domains f : D1 −→ D2 . The maps which have good Lie-theoretic meaning are the symmetric maps; this means that, for every x ∈ D1 , (1)

(2)

f ◦ sx = s f (x) ◦ f ,

(2.1)

2 The structure of bounded symmetric domains (1)

109

(2)

where sx , resp. s f (x) , are the symmetries of D1 , resp. D2 , with respect to x, resp. f (x). It is readily checked that this implies that, if Gi = Aut (Di )o , then there is a covering G1 of G1 and a homomorphism

ϕ : G1 −→ G2 for which f is equivariant: f (g · x) = ϕ (g) · f (x) . To see this, let G1 ⊂ G1 × G2 be the connected component of the set of pairs (g1 , g2 ) such that f (g1 · x) = g2 · f (x); via (2.1), check that p1 : G1 −→ G1 is surjective; now let G1 be the product of those simple factors of G1 that map non-trivially to G1 . Note that we may assume G1 ⊂ G1 × G2 , hence G1 = G  (R)o , where G  is an algebraic group over R, and G1 −→ G1 is a finite covering. Proposition 2.2 † Let D1 and D2 be bounded symmetric domains, let oi ∈ Di be basepoints, and let Gi = Aut (Di )o . There are natural bijections between the following sets: (a) holomorphic symmetric maps f : D1 −→ D2 such that f (o1 ) = o2 ; (b) maps f : D1 −→ D2 such that f (o1 ) = o2 and, for all x ∈ D1 , θ ∈ R, f ◦ ux (eiθ ) = u f (x) (eiθ ) ◦ f , where ux (eiθ ), resp. u f (x) (eiθ ), are the automorphisms fixing x, resp. f (x), and multiplying by eiθ in the tangent spaces; (c) connected coverings π : G1 −→ G1 and homomorphisms

ϕ = ϕ1 × ϕ2 : R × G1 −→ G2 , such that π × ϕ2 : G1 −→ G1 × G2 is injective and such that, if we lift the homomorphism θ −→ uo1 (eiθ ) as follows:  m6 G1 m m u π m m m m  m / G1 R uo 1

then uo2 (eiθ ) = ϕ (θ , u (θ )). Proof First of all, note that any map f : D1 −→ D2 is ϕ2 -equivariant for at most one ϕ2 : G1 −→ G2 . Indeed, introduce G1 ⊂ G1 × G2 as above. Then † A slight modification of a result of Satake and Kuga.

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III Compactifications of locally symmetric varieties

 where K  = {g2 ∈ G2 | g2 ◦ f = f }. Now K  ⊂ Stab o , Lie G1 = Lie G1 × Lie K, 2  is compact. Furthermore, since f is ϕ2 -equivariant, the map π × ϕ2 : so K G1 → G1 × G2 factors through G1 . We thus get a map

. Lie G1 = Lie G1 → Lie G1 = Lie G1 × Lie K As G1 is semisimple without compact factors, there are no homomorphisms  so it follows that the image of Lie G in Lie G1 × Lie G2 is from G1 to K, 1 uniquely determined. Since G1 is connected and π × ϕ2 is injective, G1 and ϕ2 are uniquely determined. Now start with f : D1 −→ D2 , holomorphic and symmetric. Construct ϕ2 : G1 → G2 as above. Let σi : Gi −→ Gi be the Cartan involution with respect to Ki = Staboi . Then f is also σ2 ◦ ϕ2 ◦ σ1 -equivariant, so ϕ2 ◦ σ1 = σ2 ◦ ϕ2 . Therefore dϕ2 preserves the Cartan decomposition, in particular dϕ2 (p1 ) ⊂ p2 . Also, since f is holomorphic, dϕ2 : p1 −→ p2 is complex-linear. Now, every element of D1 equals exp(a) · o1 , with a ∈ p1 . So calculate:   f ◦ uo1 (eiθ ) (exp a · o1 ) = f exp(eiθ a) · o1   = ϕ exp(eiθ a) · o2   = exp dϕ2 (eiθ · a) · o2   = exp eiθ · dϕ2 (a) · o2 = uo2 (eiθ ) · ϕ2 (exp a) · o2 = uo2 (eiθ ) · f (exp a · o1 ) . This proves that f has the property in (b) for x = o1 ; the general case follows by conjugating. To go from (b) to (c), let αθ : G1 −→ G1 be conjugation by u (θ ), and let βθ : G2 −→ G2 be conjugation by uo2 (eiθ ). Then (b) shows that f is both ϕ2 equivariant and βθ−1 ◦ ϕ2 ◦ αθ -equivariant. This means that ϕ2 (G1 ) commutes with uo2 (eiθ )· ϕ2 (u (θ ))−1 for all θ , which means precisely that ϕ : R×G1 −→ G2 as in (c) can be constructed. Conversely, given ϕ : R × G1 −→ G2 , let K1 be the centralizer of u , and let K2 be the centralizer of uo2 . Then ϕ (R × K1 ) ⊂ K2 , so ϕ defines f : (R × G1 )/(R × K1 ) −→ G2 /K2 , i.e., f : D1 −→ D2 . Using the fact that ϕ1 (θ ) = uo2 (eiθ ) · ϕ2 (u (θ ))−1 centralizes Im ϕ , we check easily the identity in (b).

Corollary 2.3 There is a bijection between the following two sets: (a) holomorphic symmetric maps f : H −→ D such that f (i) = o; and

2 The structure of bounded symmetric domains

111

(b) homomorphisms ϕ : U 1 × SL(2, R) −→ G such that   ϕ eiθ , hSL (eiθ ) = ho (eiθ ) , where SL



iθ

h (e ) =

cos θ − sin θ

sin θ cos θ

 .

Proof The only new point here is that SL(2) is simply connected as an algebraic group, so, if G1 = Aut (H), we can assume G1 = SL(2, R). We remark that, if f : D1 −→ D2 ,

ϕ2 : G1 −→ G2 have the properties of Proposition 2.2, then dϕ2 : g1 −→ g2 commutes with the Cartan involutions, and hence dϕ2 (k1 ) ⊂ k2 , dϕ2 (p1 ) ⊂ p2 , dϕ2 (p1,± ) ⊂ p2,± ; hence, extending ϕ2 to ϕ2 : G1,C −→ G2,C , we have  ϕ2 (K1,C ) ⊂ K2,C ,  ϕ2 (P1,± ) ⊂ P2,± .

Therefore we can extend f , getting the following commutative diagram: f

D1 ∩

−−−−−→

p1,+ ∩

−−−−−→

Dˇ 1

−−−−−→

f

fˇ

D2 ∩ p2,+ ∩ Dˇ 2

2.3 We now take up the study of the roots of G and GC . Let t ⊂ k be a Cartan subalgebra of k. Then t contains Im (dho ), so that t is also a Cartan subalgebra of g. Let Ψ = tC -root system of gC ,

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III Compactifications of locally symmetric varieties

so that gC = tC +

∑ gϕ .

ϕ ∈Ψ

gϕ

A root ϕ is called compact if ⊂ kC ; it is called non-compact if gϕ ⊂ pC . We denote: ΨK = the compact roots , ϕ Ψ+ p = the non-compact roots with g ⊂ p+ , ϕ Ψ− p = non-compact roots with g ⊂ p− ,

and − Ψ p = Ψ+ p ∪ Ψp .

One can choose a linear ordering on Ψ such that Ψ+ p ⊂ {positive roots} and ⊂ {negative roots}. For example, we may choose as positive roots those corresponding to a Weyl chamber in it whose closure contains (dho )(i). For ϕ ∈ Ψ, define Ψ− p

hϕ ∈ it by 2

ϕ , ψ  = ψ (hϕ ) for all ψ ∈ t∗ , ϕ , ϕ 

where ·, · denotes the Killing form. Choose root vectors eϕ ∈ gϕ such that [eϕ , e−ϕ ] = hϕ , and such that the complex conjugation of gC with respect to g permutes eϕ and e−ϕ , whenever ϕ ∈ Ψ p . Let xϕ = eϕ + e−ϕ , yϕ = i(eϕ −e−ϕ ) for ϕ ∈ Ψ+ p . These elements form a basis over R of p such that Jxϕ = yϕ , Jyϕ = −xϕ . Two roots ϕ and ψ are called strongly orthogonal, denoted by

ϕ ⊥⊥ ψ , if neither of ϕ ± ψ is a root; in this case ϕ and ψ are orthogonal. HarishChandra [5], p. 583, chooses in the following inductive way a maximal set of strongly orthogonal roots:

γ1 , . . . , γr

2 The structure of bounded symmetric domains

113

is the maximal set of roots such that each γi is the smallest element of Ψ+ p  .† We write h , x , e , e , y for h  , x  , e  , strongly orthogonal to γ1 , . . ., γi−1 i i i −i i γi γi γi e−γ i , yγi (where i = 1, . . . , r). The following theorem is basic to the theory of hermitian symmetric domains. Theorem 2.4 (Harish-Chandra) (i) The subspace a = ∑ri=1 Rxi ⊂ p is a maximal commutative subalgebra of p; hence A = exp(a) is the connected component of the group of real points of a maximal split torus A in G , i.e., A = A (R)o . Moreover, the action of K · A is transitive, i.e., K · A · o = D. (ii) There exists a morphism

ϕ : U 1 × SL(2, R)r −→ G such that:   (a) ϕ u, hSL (u), . . . , hSL (u) = ho (u) ; (b) dϕ on the ith factor SL(2, R) is given by   b+c b−c a b · yi + · ihi , = a · xi − dϕ c −a 2 2 and hence dϕ (sl(2, R)r ) = a + Ja + [a, Ja] ; in particular, dϕ induces an isomorphism between the subalgebra of “diagonal matrices” in sl(2, R)r and a; (c) ϕ induces a symmetric holomorphic map

ϕ : Hr −→ D equivariant with respect to ϕ , where SL(2, R) acts on H as usual and U 1 acts trivially, taking (i, i, . . . , i) ∈ Hr to o ∈ D. Proof (i) We refer the reader to Helgason [6], p. 314, for the somewhat delicate, but elementary, verification that a is maximal. Let A = exp(a) ⊂ G. Then, by Cartan, G = K ·A·K , † These roots are called γi to distinguish them from the closely related roots γi defined on a real split torus below. Incidentally, it should be pointed out that {γi } is a very special maximal set of strongly orthogonal roots: Theorem 2.4 below definitely does not hold for every such maximal set.

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III Compactifications of locally symmetric varieties

and hence D = K ·A·K ·o = K ·A·o . (ii) Let gi = Rxi + Ryi + Rihi ; then ∼

gi −→ sl(2, R) by the map indicated in (ii), (b). We thus obtain

ϕ2 : SL(2, R)r −→ G such that dϕ2 is given by (ii), (b). Moreover, noting that Ad ho (u) fixes each hi , and multiplies ei by u2 and e−i by u−2 , one checks easily that    ; ; sin θ cos θ ; ; Ad ho (u); = Ad ϕ2 I, . . . , , . . . , I ; if u = eiθ , − sin θ cos θ gi gi and hence that     ; ; sin θ sin θ cos θ cos θ ; ; Ad ho (u); = Ad ϕ2 ,..., . ; − sin θ cos θ − sin θ cos θ ∑ gi ∑ gi    sin θ cos θ Since ho (u) and ϕ2 , . . . both lie in T = exp t, they − sin θ cos θ commute, and we may write:    sin θ cos θ ho (u) = ϕ1 (u) · ϕ2 ,... , − sin θ cos θ where ϕ1 (u) centralizes Im ϕ2 . Defining ϕ by

ϕ = ϕ1 × ϕ2 : U 1 × SL(2, R)r −→ G , it satisfies both (ii), (a) and (b). Finally, (ii), (c) follows from Proposition 2.2. Definition 2.5 The number r appearing in the above theorem is called the rank of D or the R-rank of G. The map ϕ has the following strong universal property. Proposition 2.6 (Satake) Every symmetric holomorphic map  : H −→ D ψ  (i) = o is of the form such that ψ  (z) = k · ϕ(. . . , z, . . . , i, . . .) ψ for some k ∈ K and some distribution of z’s and i’s among the r variables of ϕ.

2 The structure of bounded symmetric domains

115

 is ψ2 -equivariant for some homomorProof According to Corollary 2.3, ψ phism

ψ = ψ1 × ψ2 : U 1 × SL(2, R) −→ G   such that ho (eiθ ) = ψ eiθ , hSL (eiθ ) . It follows that:    1 0 ⊂p. dψ R · 0 −1 We use the fact that every element of p is in Ad K · a (the polar decomposition  and ψ by kψ k−1 , we can assume  by k · ψ in g). Therefore, replacing ψ    1 0 dψ R · ⊂a. 0 −1 Then





dψ R · and

0 1

1 0

 ⊂ Ja

   0 1 dψ R · ⊂ [a, Ja] . −1 0

This shows that ψ factors through Im ϕ ⊂ G, and hence that ψ2 factors through

ψ2∗ : SL(2, R) −→ SL(2, R)r . Now, pri ◦ ψ2∗ is either trivial or conjugate to the identity by some element of  has the required form. ki ∈ SO(2, R), from which we see that ϕ2 (k1 , . . . , kr ) · ψ Next, we want to decompose g into irreducible pieces under the restriction of the adjoint representation to U 1 × SL(2, R)r . The irreducible real representations of U 1 × SL(2, R)r are easily enumerated as follows. (a) The irreducible real representations of U 1 are: • U0 = R, trivial representation; • Uk = R2 , with action  cos kθ iθ e −→ − sin kθ

sin kθ cos kθ

 ,

for k = 1, 2, . . .. Note that these representations are only irreducible as real representations: they split over C into the direct sum of two one-dimensional representations with characters eiθ −→ (eiθ )±k ,

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III Compactifications of locally symmetric varieties

respectively. (b) The irreducible representations of SL(2, R) are • Wk , the k’th symmetric power of the standard representation, for k = 0, 1, 2, . . .. Note that these are complex-irreducible as well. It follows easily that the irreducible representation of U 1 × SL(2, R)r are of the form Ui ⊗W j1 ⊗ . . . ⊗W jr . The fact that U 1 , acting via ho on gC , has only the characters z2 , 1, z−2 implies easily that, if this representation appears in gC , then r

i + ∑ jk = 0 or 2 , k=1

and we get: Proposition 2.7 The irreducible representations of U 1 × SL(2, R)r which appear in g are at most the following: (ai ) (bi j ) (ci ) (d) (e)

U0 ⊗ (W0 ⊗ · · · ⊗W2 ⊗ · · · ⊗W0 ) (one W2 ) ; U0 ⊗ (W0 ⊗ · · · ⊗W1 ⊗ · · · ⊗W1 ⊗ . . . ⊗W0 ) (two W1 ) ; U1 ⊗ (W0 ⊗ · · · ⊗W1 ⊗ · · · ⊗W0 ) (one W1 ) ; U2 ⊗ trivial ; U0 ⊗ trivial .

Moreover, the image via dϕ of the ith copy of SL(2, R)r defines a representation of type (ai ), and this is the only one of this type. Furthermore, (d) does not occur at all. Proof Let V be a representation of type (ai ) or (d). Then, in both cases, V contains a 2-dimensional subrepresentation W , where h0 acts by the representation U2 , containing some 0 = w ∈ W on which a acts trivially. The first condition implies that w ∈ p, whence by the maximality of a we get w ∈ a. This leads to the given factors of type (ai ). This proposition gives us a bird’s-eye view of the complex root decomposition of gC in which ad tC is diagonalized, and, simultaneously, of the real root decomposition of g in which ad a is diagonalized. To be precise, let r

a = ∑ R · hi ⊂ it , i=1

2 The structure of bounded symmetric domains

117

and define

ϕ ∼ ψ ⇐⇒ ϕ − ψ |a ≡ 0 , for ϕ , ψ ∈ Ψ . Proposition 2.7 tells us which representations of a may occur in g and, in particular, tells us that, for all ϕ ∈ Ψ, one of the following occurs: (I) ϕ ∼ ±γi , for some i, in which case gϕ ⊂ factor of type (ai ), hence in fact ϕ = ±γi ; (II) ϕ ∼ 12 (±γi ± γ j ), for some i = j, in which case gϕ ⊂ factor of type (bi j ) ; (III) ϕ ∼ ± 12 γi , for some i, in which case gϕ ⊂ factor of type (ci ) ; (IV) ϕ ∼ 0, in which case gϕ ⊂ factor of type (e) . But because we also know how ho acts on each of these factors, we can say more: (I) each (ai )-factor is just R(ihi ) + Rxi + Ryi , giving one positive non-compact root γi and one negative non-compact root −γi ; (II) each of the (bi j )-factors is 4-dimensional and each gives one positive noncompact root, ∼ γ  −γ  ∼± i2 j,

γi +γ j 2 ;

two compact roots, one positive and one negative, γ  +γ 

respectively, and one negative non-compact root, ∼ − i 2 j ; (III) each of the (ci )-factors is 4-dimensional and each gives one positive noncompact root, ∼ γi /2, two compact roots, one positive and one negative, ∼ ±γi /2, respectively, and one negative non-compact root, ∼ −γi /2; (IV) each of the (e)-factors is 1-dimensional and gives a compact root, ∼ 0. We can coarsen our complex root decomposition g = tC ⊕

∑ gϕ

ϕ ∈Ψ

by lumping together root spaces with equivalent roots. Let RΨ be the set of non-zero linear maps a −→ R given by restricting roots ϕ ∈ Ψ to a , so that gC = Z(a )C ⊕

∑

ψ  ∈RΨ



gψ ,

(∗ )

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III Compactifications of locally symmetric varieties

where Z(a )C = aC ⊕ ∑ϕ ∼0 gϕ , 

gψ = ∑ϕ ∈Ψ,ϕ ∼ψ  gϕ = eigenspace in gC , where ad a is given by the character ψ  . Then this is essentially the real root decomposition of g with respect to the split torus A = exp a. In fact, if     1 1 i ,... , c = ϕ ..., √ i 1 2 then Ad c(a) = a . Therefore, if RΨ = (Ad c)∗ (RΨ ) is the induced set of linear maps a −→ R, we get an isomorphic decomposition of g via ad a. But now a is real, so it is a real decomposition: g = Z(a) ⊕

∑

gψ ,

(∗)

ψ ∈RΨ

where gψ = Ad c(g(Ad c)

∗−1 (ψ )

)

= eigenspace in g, where ad a is given by the character ψ . What is RΨ, i.e., which combinations of factors actually occur in g in the above proposition? First of all, (ai ) occurs once. We will denote the corresponding roots of a by ±γi : these are simply the roots of a occurring in the image under dϕ of the ith copy of sl(2, R); in fact, in Rxi + Ryi + Rihi . As for the rest, the result is: Proposition 2.8 Let D be simple of rank r. Then either of the following two possibilities occurs: Case Cr :

Case BCr :

RΨ =

6

7 ± 12 (γi + γ j ) for i ≥ j; ± 12 (γi − γ j ) for i > j ,

all (bi j )-factors occur, but no (ci )-factors occur; 7 6 1 1 1 RΨ = ± 2 (γi + γ j ) for i ≥ j; ± 2 (γi − γ j ) for i > j; ± 2 γi , all (bi j )- and all (ci )-factors occur.

In both of these cases, the Weyl group (the automorphisms of a induced by Ad Norm (A)) is the group of all signed permutations γi −→ ±γσ (i) with σ a permutation of {1, . . . , r}. If we order the real roots so that γ1 > · · · > γr , the simple roots R ∆ are:  γr Case Cr αi = (γi − γi+1 )/2, 1 ≤ i ≤ r − 1; and αr = γr /2 Case BCr .

2 The structure of bounded symmetric domains

119

Proof In the image under ϕ of the ith factor of SL(2, R) there exists an element wi which normalizes a, sends γi into −γi , and fixes the γ j for j = i. Let si = Ad (wi ): then si belongs to the Weyl group W . In particular, this implies that the γi are orthogonal to each other. By Proposition 2.7, the other roots are among the following: (bi j ) : 12 (±γi ± γ j ) , (ci ) : ± 12 γi . The Weyl group W is generated by the reflections around the roots, i.e., by the si and, if 12 (±γi ± γ j ) occurs as a root, by the symmetry which interchanges the two roots γi and γ j , leaving fixed the roots γk , for k = i, j. Since D is simple, W acts irreducibly on a; i.e., W /(sign changes) is a transitive group of permutations of {1, . . . , r} generated by transpositions. It follows that W contains all permutations of the γi among themselves. This implies that all 1 1 2 (±γi ± γ j ) are roots and that either all or none of the 2 γi are roots. The first case leads to the type Cr , and the second case leads to the type BCr . The simple roots are now easily written down.

2.4 We now return to the general case, i.e., drop the assumption that D is simple. Theorem 2.9 below determines the position of D inside p+ , with D sitting inside p+ via the Harish-Chandra embedding. For X ∈ p+ , define the linear operator T (X) : p− −→ kC , Y −→ [Y, X] . If

τ : gC −→ gC denotes complex conjugation with respect to gc , we put a positive-definite hermitian form on gC : Bτ (u, v) = −B(u, τ v) , for u, v ∈ gC . Let T ∗ (X) : kC −→ p− be the adjoint of T (X) with respect to Bτ .

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III Compactifications of locally symmetric varieties

Theorem 2.9 (Harish-Chandra, Hermann) Let r

D0 = D ∩ ∑ C · ei . i=1

Then (i) with ϕ defined as in Theorem 2.4,   r

D0 =

∑ ai ei | |ai | < 1

  = Im ϕ : Hr −→ D ;

i=1

(ii) D = Ad K(D0 ) = {X ∈ p+ | T ∗ (X) ◦ T (X) < 2Idp− } . In particular, if a, b ∈ C, with |a| + |b| ≤ 1, then X,Y ∈ D =⇒ aX + bY ∈ D . (Part (ii) is called the Hermann convexity theorem.) Proof We use the fact that D −→ p+ is K-equivariant, where K acts via Ad on p+ . Thus D = K · A · o, so D can be described inside p+ as D = Ad K(A · o). Now, a calculation with SL(2) shows that   A·o =

r

∑ ai ei | ai ∈ R, −1 < ai < 1

i=1

and

 Im ϕ =



r

∑ ai ei | ai ∈ C, |ai | < 1

.

i=1

The isomorphism p+ ∼ = p is also K-equivariant and it is well-known that in p, for every k ∈ K, either (a) Ad k(a) ∩ a ⊂ singular subalgebra of a; or (b) Ad k(a) = a, and hence k lies in the Weyl group. Thus, if D∩ ∑ri=1 Rei were bigger than ∏ri=1 (−1, +1)ei , then one could find x ∈ / ∏ri=1 (−1, +1)ei D ∩ ∑ri=1 Rei such that x lies in no singular subalgebra and x ∈ with r

x = (Ad k)(y) ,

y ∈ ∏(−1, +1)ei . i=1

Then k would be in the Weyl group, and hence Ad k would be a permutation plus sign change, and hence x ∈ ∏ri=1 (−1, +1)ei , a contradiction. Therefore D∩ ∑ri=1 Rei = ∏ri=1 (−1, +1)ei . Since Ad K contains rotations in the planes C· ei (via ϕ (SO(2, R)r )), it follows that D0 is rotation-invariant and (i) is proven.

2 The structure of bounded symmetric domains

121

To prove (ii), note that the sets on both sides are K-invariant, so it suffices to consider elements r

X = ∑ ai ei ,

ai ∈ R ,

i=1

and show that T ∗ (X) ◦ T (X) < 2Id ⇐⇒ |ai | < 1 for i = 1, . . . , r . Expressed in terms of operator norms,  S 2 = sup{Bτ (S(Y ), S(Y )) | Bτ (Y,Y ) = 1} , the condition on the LHS is just  T (X) <

√ 2.

We thus have to show that  T (X) <

√ 2 ⇐⇒ |ai | < 1 for i = 1, . . . , r .

This is done by an explicit computation: write Y ∈ p− as r

r

Y = ∑ bi e−i + ∑ where

∑ bα e−α + ∑ ∑

i=1 α ∈Pi

i=1

i< j α ∈Pi j

bα e−α ,

(2.2)

7 6 Pi = positive non-compact roots ϕ with ϕ ∼ 12 γi , 6 7 Pi j = positive non-compact roots ϕ with ϕ ∼ 12 (γi + γ j ) .

We compute [X,Y ], using the vanishing of various brackets: [X,Y ] = ∑ ai bi [ei , e−i ] + ∑ i

∑ ai bα [ei , e−α ] + ∑ ∑

i α ∈Pi

i = j α ∈Pi j

ai bα [ei , e−α ] . (2.3)

Furthermore, by our normalization of the eα ,

τ (eα ) = −e−α ; hence α = β =⇒ Bτ (eα , eβ ) = 0, and all terms in the sum (2.2) are orthogonal to each other. Using this, together with the Jacobi identity, one computes Bτ ([ei , e−α ], [ei , e−β ]) = α (hi ) · B(e−α , eβ ) , and this expression vanishes unless α = β . Thus the brackets in sum (2.3) are orthogonal to each other.

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III Compactifications of locally symmetric varieties

We may now compare  Y 2 = ∑ |bi |2  e−i 2 + ∑

∑ |bα |2  e−α 2

i α ∈Pi

i

+∑

∑

i< j α ∈Pi j

|bα |2  e−α 2

with  [X,Y ] 2 = ∑ |bi |2 |ai |2 |γi (hi )|  e−i 2 i

+∑

∑ |bα |2 |ai |2 |α (hi )|  e−α 2

+∑

∑

i α ∈Pi

i< j α ∈Pi j



 |ai |2 |bα |2 |α (hi )| + |a j |2 |bα |2 |α (h j )|  e−α 2 .

< 1, for i = 1, . . . , r, it is evident that  T (X) 2 < 2. Conversely, Thus, if |ai |√ if  [X,Y ] < 2  Y , for all Y ∈ p− , then, as one sees by plugging Y = e−i into the above expression, we must have |ai | < 1 for i = 1, . . . , r. Corollary 2.10 (of proof) Let X ∈ D and write X = ∑ ai ei + ∑ bα eα , where α runs through all positive non-compact roots restricting to either 12 γi or 12 (γi + γ j ) (i.e., not restricting to γi ). Then |ai | < 1 for i = 1, . . . , r. This can be seen by calculating [X, e−i ] in the same way as above; cf. Langlands [8], p. 110.

2.5 As a final topic, suppose G is a semi-simple algebraic group defined over Q such that D = G (R)o /K, for K ⊂ G (R)o maximal compact, is a bounded symmetric domain. (This usually means that G (R)o /(center) = Aut (D)o , but it also allows G (R)o to have compact factors, i.e., G (R)o /(compact normal subgroup) = Aut (D)o .) In this case, an important role is played by the maximal Q-split tori A ⊂ G . (If A is non-trivial and G is Q-simple, then, in fact, G (R)o has no compact factors.) Choose B ⊂ G , a maximal R-split torus such that A ⊂ B. Let s = dim A = Q-rank G and r = dim B = R-rank G . Note that compact factors

3 Boundary components

123

do not change the root structure so that the preceding results are still applicable. The roots of G with respect to B are a basis

γi ∈ XQ (B) = Hom (B, Gm ) ⊗ Q for 1 ≤ i ≤ r plus some subset of the further elements, 1 2 (±γi ± γ j ),

1 ≤ i < j ≤ r;

± 12 γi ,

1≤i≤r.

Baily and Borel [1], pp. 467–468, prove by a purely root-theoretic analysis the following basic fact. Proposition 2.11 There is a partition {1, . . . , s} = I0 ∪ I1 ∪ I2 ∪ . . . ∪ Ir such that the subtorus A is defined by 6 7 A = γi = 1 for i ∈ I0 ; γi = γ j for i, j ∈ Ik , where k > 0 ; or, in additive notation, XQ (A ) ∼ = XQ (B)/



subspace spanned by γi for i ∈ I0 and γi − γ j for i, j ∈ Ik , where k > 0

 ;

s

∼ = ∑ Qβi , where βi = image of any γ j , j ∈ Ii . i=1

(Their result is stated differently, but boils down to the above.) Corollary 2.12 The Q-roots are ±β1 , . . . , ±βs plus a subset of the further elements ±βi /2, (±βi ± β j )/2 for i = j. If G is Q-simple, then all (±βi ± β j )/2 occur and either all ±βi /2 occur or none do. The Q-Weyl group is then the full group of permutations and sign changes of the βi .

3 Boundary components In this section we decompose the closure D of D inside p+ into the disjoint union of lower-dimensional bounded domains, the boundary component. We analyze the structure of boundary components and determine their normalizers and centralizers.

3.1 Start with a = ∑ Rxi ⊂ p

124

III Compactifications of locally symmetric varieties

and sl(2, R)r ∼ = ∑ Rxi + ∑ Ryi + ∑ Rihi ⊂ g as in Theorem 2.4. Let S ⊂ {1, . . . , r} be any subset.† Define the subalgebra lS of g:  ϕ  g + [gϕ , g−ϕ ] . lS = ∑ ϕ ∈R Ψ ϕ= ∑ a jγ j j ∈S

In the decomposition of Proposition 2.7, lS may be described as follows: lS = ∑(ai -factor) ⊕ i∈S /

 ⊕

∑ (bi j -factors) ⊕ ∑(ci -factor)

i< j i, j∈S /

i∈S /

the part of the e-factors spanned by [x, y] x, y ∈ some bi j -factor or some ci -factor

 .

Hence lS,C can be written as a sum over the complex roots ϕ ∈ Ψ:  ϕ  lS,C = ∑ g + [gϕ , g−ϕ ] . ϕ ∼ ∑ a j γ j j ∈S

ϕ 0

This shows that lS is stable under Ad ho (eiθ ); hence (a) lS = k ∩ lS ⊕ p ∩ lS and (b) pC ∩ lS,C = p+ ∩ lS,C ⊕ p− ∩ lS,C , which we abbreviate as follows: pS,C = p+,S ⊕ p−,S . By (a), lS is a reductive subalgebra of g, and, since it is generated by nilpotent elements, lS is semi-simple without compact factors. By (b), lS is of hermitian type. Let LS ⊂ G be the corresponding subgroup. Then DS = LS /LS ∩ K is a bounded symmetric domain and we have a symmetric embedding of DS in D. In fact, we have even more, because LS commutes with the subgroup (modulo center)

∏ SL(2, R)i i∈S

† In what follows, we consider S interchangeably as S ⊂ {γ1 , . . . , γr } or S ⊂ {1, . . . , r}.

3 Boundary components

125

arising from the subalgebra

∑(Rxi + Ryi + Rihi ) .

i∈S

This shows that we get a whole equivariant diagram of symmetric holomorphic maps: f

∆s × DS ∩

1 −−−−− −−→

Cs × p+,S ∩ (P1 )s × Dˇ S

D ∩

f

2 −−−−− −−→ p+ ∩ f3 −−−−−−−→ Dˇ

where s = |S|. (Here we identify the domain SL(2, R)/SO(2) with the open unit disc ∆ instead of H; we will only do this for the purpose of the discussion that follows in Subsection 3.1.) Define FS = f2 ((1, . . . , 1) × DS ) ⊂ p+ , FˇS = f3 ((1, . . . , 1) × Dˇ S ) ⊂ Dˇ . Note that FS lies in ∂ D and is a complex submanifold of p+ . In fact, f2 is linear and is just the inclusion map of the following subspace of p+ : 

∑ Cei + ∑ Cei + ∑ ∑ i∈S /

i∈S

%

i< j α ∈Pi j i, j∈S /

Ceα + ∑

∑ Ceα

.

i∈S / α ∈Pi

&'

(

p+,S

Here, as in Subsection 2.4,

6 7  1  Pi j = ϕ ∈ Ψ+ p | ϕ ∼ 2 (γi + γ j ) , 7 6 1  Pi = ϕ ∈ Ψ+ p | ϕ ∼ 2 γi .

Thus by Hermann convexity applied to DS : 7 6 FS = ∑ ei + X ∈ p+,S | TS (X)∗ ◦ TS (X) < 2Id p−,S , i∈S

where TS (X) : p−,S −→ lS,C ∩ kC is given by TS (X)(Y ) = [Y , X] . We use this last description to prove:

126

III Compactifications of locally symmetric varieties

Lemma 3.1 F S = FˇS ∩ D and all holomorphic maps

λ : ∆ −→ p+ such that Im (λ ) ⊂ D ,

Im λ ∩ FS = 0/ ,

map ∆ into FS . Proof Let X = ∑i ai ei + ∑i ∑α ∈Pi aα eα + ∑i j ∑α ∈Pi j aα eα ∈ p+ and assume that X ∈ D and ai = 1, for i ∈ S. Then we claim that aα = 0 for α ∈ Pi , i ∈ S, and aα = 0 for α ∈ Pi j , i or j ∈ S. For this, calculate [X, e−i ] as in the proof of the Hermann convexity theorem: [X, e−i ] = ai [ei , e−i ] +

∑ aα [eα , e−i ] + ∑

α ∈Pi

α ∈Pi j i = j

aα [eα , e−i ] ,

hence  [X, e−i ] 2 =2|ai |2  e−i 2  + ∑ |aα |2  eα 2 + α ∈Pi

∑



α ∈Pi j i = j

|aα |2  eα 2

.

Since X ∈ D, we must have  [X, e−i ] 2 ≤ 2  e−i 2 by the Hermann convexity theorem. Hence, if ai = +1, then  2  e−i 2 + ∑ |aα |2  eα 2 + ∑ |aα |2  eα 2 ≤ 2  e−i 2 , α ∈Pi

α ∈Pi j i = j

which shows the claim. Hence, for such X, we may write X = ∑ ei + X , i∈S

where X ∈ p+,S . But, since, for Y ∈ p−,S , [X,Y ] = [X,Y ], we conclude that TS (X)∗ ◦TS (X) ≤ 2Id p−,S , i.e., X ∈ DS = closure of DS inside p+,S . This proves: D ∩ {X | coeff. ai of ei is 1, for i ∈ S} = F S . We can now prove the lemma. The function fi , which to x = ∑ ai ei + ∑ i∈S

∑ bα eα + ∑ ∑

i α ∈Pi

i< j α ∈Pi j

bα eα ∈ p+

associates ai ∈ C, is a linear function bounded above by 1 on D; if i ∈ S, then fi

3 Boundary components

127

takes on the value 1 on FS . The maximum principle, applied to the holomorphic function fi ◦ λ : ∆ −→ C, implies now that fi ◦ λ ≡ 1 on ∆. The preceding considerations show that λ (∆) ⊂ F S . But, by the Hermann convexity theorem, DS is a convex subset of p+,S , thought of as a real vector space; hence, for every x ∈ ∂ FS ⊂ ∑i∈S ei + p+,S , there exists a linear functional
on p+ and a real number a such that
> a on FS and
(x) = a. Now, again by the maximum principle, this time applied to
◦ λ , we conclude that Im (λ ) ∩ ∂ FS = 0/ =⇒ Im λ ⊂ ∂ FS ,

i.e., Im λ ∩ FS = 0/ ,

which contradicts our assumptions; hence Im λ ⊂ FS . This proves that FS is a boundary component of D in the following sense. Definition 3.2 A boundary component of a bounded symmetric domain D is an equivalence class in D under the equivalence relation generated by x ∼ y if there exists a holomorphic map

λ : ∆ −→ p+ such that Im (λ ) ⊂ D, and x, y ∈ Im λ . We now have: Theorem 3.3 (i) D is the disjoint union of boundary components. (ii) The boundary components of D are just the sets k∈K ,

k · FS ;

S ⊂ {1, . . . , r} ,

with possible repetitions. They are hermitian symmetric domains of rank r − |S|. (iii) Decompose D = D1 × · · · × Dn into simple factors. Then the boundary components of D are the products of boundary components of the simple factors Di . (iv) A boundary component of a boundary component is a boundary component. (v) For every boundary component F, there are holomorphic symmetric maps H ∩ P1

f

F −−−−− −−→

f

F −−−−− −−→

D ∩ Dˇ

128

III Compactifications of locally symmetric varieties such that fF (i) = o, fF (∞) ∈ F, and equivariant with respect to a morphism

ϕF : U 1 × SL(2, R) −→ G , such that ϕF (eiθ , hSL (eiθ )) = ho (eiθ ).† Proof Part (i) is trivial. For part (ii), let X ∈ F = boundary component of D . Since K is maximal compact, D = Ad K · (∏ri=1 [−1, +1] · ei ); hence we can find k ∈ K and a subset S ⊂ {γ1 , . . . , γr } such that r

Ad (k)(X) = ∑ ai ei ,

ai ∈ [−1, +1] .

i=1

But K also contains elements normalizing a and inducing arbitrary sign changes, so we may assume ai ≥ 0. Let S = {i | ai = 1}. Then Ad (k)(X) ∈ FS . Part (iii) is straightforward; check that x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) in D are equivalent if and only if xi ∼ yi in Di for all i. Part (iv) is an immediate consequence of (ii): let D1 be a boundary component of D, and let D2 be a boundary component of D1 . Then ) *

∑ ei + DS

D1 = k · FS = k ·

,

i∈S

and hence

*

) D2 = k ·

∑

ei + D2

,

i∈S

where D2 is a boundary component of DS . But then * ) D2 = k ·

∑ ei + DS∪S

i∈S

for some k ∈ LS ∩ K, S ⊂ {1, . . . , r} \ S, by (ii) applied to D = DS . Thus * ) D2 = k · k ·

∑

i∈S∪S

ei + DS∪S

= k · k · FS∪S is a boundary component of D. † The question of uniqueness of the pair ( fF , ϕF ) is taken up in Theorem 3.7 below.

3 Boundary components

129

Part (v), when F = FS , follows by taking ϕF = ϕS , where   iθ iθ iθ x , . . . , %&'( ϕS (e , x) = ϕ e ; . . . , %&'( e ,... , if i∈S

if i∈S /

where ϕ : U 1 × SL(2, R)r −→ G is Harish-Chandra’s map (see Section 2.3), and taking fF = fS to be the corresponding map of symmetric spaces. For general F = k · FS , let ϕF = kϕS k−1 and fF = k fS . This theorem shows that the FS are good models for studying any one boundary component. They are equally good for studying arbitrary flags of boundary components because of the next result. Proposition 3.4 (i) If S1 ⊂ S2 , then F S1 ⊃ F S2 . (ii) If D ⊃ F1 ⊃ F2 ⊃ · · · ⊃ Ft is any flag of boundary components, then there are subsets S1 ⊂ S2 ⊂ · · · ⊂ St ⊂ {1, . . . , r} and an element k ∈ K such that k · Fi = FSi ,

1 ≤ i ≤ t.

Proof If S1 ⊂ S2 , then LS1 ⊃ LS2 , hence DS1 ⊃ DS2 and     D ⊃ ∑ ei + DS1 ⊃ ∑ ei + DS2 , i∈S1

%

&' F S1

(

i∈S2

%

&'

(

F S2

since ∑i∈S2 \S1 ei ∈ DS1 . The second part is proved just as part (iv) above: first find k1 ∈ K such that k1 · F1 = FS1 ; then find k2 ∈ K ∩ LS1 such that k2 k1 · F2 = FS2 , and so on.

3.2 Our next purpose is to determine the normalizer N(F) of a boundary component F: N(F) = {g ∈ G | gF = F} ,

130

III Compactifications of locally symmetric varieties

and to show that the pair ( fF , ϕF ) in part (v) is unique. It is easiest to attack N(FS ) first. As in the proof of the theorem, let

ϕS : U 1 × SL(2, R) −→ G be given by x , . . .) , ϕS (eiθ , x) = ϕ (eiθ ; . . . , %&'( eiθ , . . . , %&'( if i∈S /

if i∈S

where again ϕ is Harish-Chandra’s map. Let wS : Gm −→ G be    t 0 . wS (t) = ϕS 1, 0 t −1 Consider the associated parabolic subgroup   −1 PS = P(w−1 ) = g ∈ G | lim w (t)gw (t) exists , S S S t−→0

which is the intersection of G with a real parabolic subgroup PS ⊂ G . It is easy to calculate Lie PS using the real root decomposition: Lie PS = Z(a) +

∑

gϕ ,

ϕ ∈RΨ dwS ,ϕ ≥0

where dwS , ϕ  is the inner product of dwS (1) ∈ a and ϕ ∈ Hom (a, R). Since  1 i∈S dwS , γi  = 0 i∈ /S, it follows that {ϕ ∈ RΨ | dwS , ϕ  ≥ 0} =

61

7

1 /S 2 (±γi ± γ j ), ± 2 γi , i, j ∈ 61 1 ∪ 2 (γi ± γ j ), 2 γi , i ∈ S, any

7 j .

Lemma 3.5 PS ⊂ N(FS ). Proof We use the Cayley transformation   1 1 cS = ϕS 1, 1−i 1

i −i

 . f

S Here cS ∈ GC , and one checks, via the equivariant map Hs × DS −→ D and its f S 1 s ˇ ˇ extension (P ) × DS −→ D, that

cS (DS ) = FS

and

where we identify DS with LS · o in D.

cS (Dˇ S ) = FˇS ,

3 Boundary components

131

Now, every g ∈ PS either carries FS to FS or to a boundary component disjoint from FS . Therefore it suffices to show that g · cS (o) ∈ cS (Dˇ S ) for every g ∈ PS,C , because then, for g ∈ PS , g · cS (o) ∈ D ∩ cS (Dˇ S ) = cS (DS ), hence g(FS ) ⊂ F S ; since dim g(FS ) = dim FS , it then follows that gFS = FS . Therefore we only need to show that ˇ c−1 S gcS (o) ∈ DS , or c−1 S gcS ∈ LS,C · KC · P− . But define

     −1 PS,C = c−1 P c = g ∈ G | lim w (t)gw (t) exists , S,C S C S S S t−→0

where wS = c−1 S wS cS : Gm,C −→ GC . Then wS (eiθ ) = ϕS

  cos θ 1, sin θ

− sin θ cos θ

 ,

 is easy to calculate using the complex root decomposition: and Lie PS,C  = tC ⊕ Lie PS,C

= tC ⊕

∑

gϕ

ϕ ∈Ψ dwS ,ϕ ≥0

∑ gϕ

± γ  ±γ  ϕ ∼ i2 j

⊕ ±γ 

or 2 i i, j∈S /

∑ gϕ

−γ  − γ  ϕ ∼ i2 j

⊕ −γ 

or 2 i i∈S, any j

∑ gϕ

− γ  +γ  ϕ ∼ i2 j

⊕

∑ gϕ .

ϕ ∼0

i∈S, any j

Then tC ⊂ kC , the second term generates lS,C , the third term is in p− , and the fourth and fifth terms are in kC . Let b be the subalgebra generated by the first, third, and fourth term, and let c be the subalgebra generated by the first four terms. Then b and c are ideals, and b generates a normal subgroup contained in KC · P− . Therefore, c generates a normal subgroup contained in  ⊂ LS,C · KC · P− , and, since the fifth term is contained in kC , we finally get PS,C LS,C · KC · P− · KC = LS,C · KC · P− , as required. Proposition 3.6 PS = N(FS ).

132

III Compactifications of locally symmetric varieties

Proof It suffices to show this when G is simple because, in the general case, everything decomposes into a product. But when G is simple, applying the Weyl group, we can assume that S = {1, . . . , b}. Then note that all but one of the simple roots αi is zero on Im dwS . Therefore the associated parabolic PS is a maximal real parabolic subgroup of G . Since PS ⊂ N(FS )  G , the connected components of PS and N(FS ) coincide. But then N(FS ) normalizes N(FS )o , and hence normalizes Lie N(FS ), which equals the Lie algebra Lie PS . But PS,C is the full normalizer of Lie PS inside GC , so N(FS ) ⊂ G ∩ PS,C = PS , hence the asserted equality. We now prove: Theorem 3.7 For all boundary components F ⊂ D, the equivariant pair ( fF , ϕF ) fF : H −→ D ,

ϕF : U 1 × SL(2, R) −→ G , where fF is symmetric with fF (i) = o and fF (∞) ∈ F, is unique, and, if wF is defined by    t 0 , wF (t) = ϕF 1, 0 t −1 then

  −1 N(F) = P(w−1 ) = g ∈ G | lim w (t)gw (t) exists . F F F t−→0

Finally, if N(F1 ) = N(F2 ), then F1 = F2 . Proof Because of Proposition 3.6, the middle part follows from the first part. Moreover, because of Theorem 3.3 (iii), to prove the rest we may assume G simple. Now, by Proposition 2.6, all such ϕF arise as kϕS k−1 , for some k ∈ K. The first part therefore amounts to the following statement: k · FS1 = FS2 , k ∈ K =⇒ kϕS1 k−1 = ϕS2 . But k · FS1 = FS2 implies rank(FS1 ) = rank(FS2 ), hence |S1 | = |S2 |. Therefore, by Proposition 2.8, there is an element w of the Weyl group Norm (a)/Cent(a) inducing a permutation of the xi such that wS1 = S2 : we may realize w as an element of K ∩ Norm (a). This reduces us to proving k ∈ K ∩ N(FS ) =⇒ kϕS = ϕS k .

3 Boundary components

133

Let σ be the Cartan involution. Then k ∈ N(F S ) =⇒ lim wS (t)kwS (t)−1 exists t−→0

=⇒ lim σ (wS (t))σ (k)σ (wS (t))−1 exists t−→0

=⇒ lim wS (t)−1 kwS (t) exists. t−→0

Thus t → wS (t)kwS (t)−1 extends to a morphism P1 −→ G . This must be constant, i.e., kwS = wS k. But k ∈ K implies kho = ho k. Thus       sin θ t 0 cos θ iθ and e ϕS 1, ϕ , S − sin θ cos θ 0 t −1 both centralize k. Since these elements generate U 1 × SL(2, R), we also get kϕS = ϕS k. As for the final assertion, when G is simple, we saw in (the proof of) Proposition 3.6 that all the N(F) are maximal parabolic. Let g ∈ G with gF1 = F2 . Then gN(F1 )g−1 = N(F2 ) = N(F1 ), hence g normalizes N(F1 ) so that g ∈ N(F1 ), whence F1 = gF1 = F2 . Corollary 3.8 If F 1 ⊃ F2 are two boundary components of D, then there is a unique symmetric holomorphic map f : H2 −→ D such that f (i, i) = o , f (i, ∞) ∈ F1 , f (∞, ∞) ∈ F2 . In particular, wF1 and wF2 commute with each other. Proof Combine Proposition 3.4 with Theorem 3.7. Let oF = fF (∞): this is the natural basepoint of F determined by the basepoint o ∈ D. Proposition 3.9 When G is simple and hence D is irreducible, the association F −→ N(F) defines a bijection between the set of boundary components of D and the set of maximal real parabolic subgroups of G. In general, if G = G1 × · · · × Gk , D = D1 × · · · × Dk , where Gi = Aut (Di )o , with Gi simple, then F −→ N(F) defines a bijection between the set of boundary components F = F1 × · · · × Fk of D, where here we allow Fi = Di , and the set of real parabolics P = P1 × · · · × Pk of G, with Pi either maximal real parabolic in Gi , or Pi = Gi .

134

III Compactifications of locally symmetric varieties

Proof The general case reduces immediately to the case where G is simple. Theorem 3.7 shows injectivity. Proposition 3.6 shows that N(F) is maximal. Moreover, by the proof of Proposition 3.6, the maximal real parabolic corresponding to any simple root do occur as N(FS ) for suitable S, hence we get surjectivity.

3.3 Pursuing the same ideas, we can determine the centralizer Z(F) of a boundary component F: Z(F) = {g ∈ G | gx = x for all x ∈ F} . The result is (for the connected component Z(F)o ): Theorem 3.10 Let F ⊂ D be a boundary component and let wF : Gm −→ G be as in Theorem 3.7. Then (1) N(F)o is a semi-direct product: N(F)o = Z(wF )o
W (F) , where W (F) = {g ∈ G | lim wF (t)gwF (t)−1 = e} t−→0

= unipotent radical of N(F)o , Z(wF )o = connected component of centralizer of wF = a Levi component of N(F)o . (2) Z(wF )o is generated by two commuting connected subgroups Gh (F) and 
)/A) and 
(F) with finite intersection A (so that Z(wF )o ∼ G = (Gh × G 
(F)
W (F) . Z(F)o = G (3) When F = FS , then Gh (FS ) = LS ; in general, Gh is semi-simple without compact factors and Gh /(center) is isomorphic to Aut (F)o . Proof The first part is a simple decomposition which applies to any parabolic subgroup P(w) for any one-parameter subgroup w, and is proved by decomposing g into 0, +, and − eigenspaces under Ad w and exponentiating. To prove (2) and (3), we may assume F = FS (getting the general case by conjugating by k ∈ K). In the notation above, note that Lie Z(wS ) = Z(a) +

∑

ϕ ∈RΨ dws ,ϕ =0

gϕ .

3 Boundary components

135

Moreover, the ϕ ∈ RΨ with dwS , ϕ  = 0 are of two types:

ϕ = 12 (±γi ± γ j ), ± 12 γi ,

(a)

ϕ=

(b)

1 2 (γi − γ j )

,

i, j ∈ /S,

i, j ∈ S .

Clearly, lS ⊂ Lie Z(wS ). But also, since no roots of type (a) and of type (b) add up to a root, it follows that lS is an ideal in Lie Z(wS ). This proves that we have 
,S )/A as required. By Proposition 2.7, we a decomposition Z(wS )o ∼ = (LS × G know that Z(a) = a ⊕ m(a) , where m(a) = type (e)-factors in the decomposition of g = Z(a) ∩ k . Note that if σ is the Cartan involution of g, then

σ (gϕ ) = g−ϕ ; hence, as has already been noted earlier,

σ (lS ) = lS . Taking into account the fact that 
,S = {x ∈ Lie Z(wS ) | [x, lS ] = 0} , Lie G 
,S ) = Lie G 
,S . Since Lie G 
,S is normalized by a, we it follows that σ (Lie G deduce the following explicit expression: 
,S = Lie G

∑ γ −γ i

ϕ= 2 i, j∈S

j

gϕ + ∑ Rxi + (ideal in m(a)) . i∈S

(The middle factor comes from the fact that

∑ Rxi = {x ∈ a | γi (x) = 0 ,

i∈ / S} = {x ∈ a | [x, lS ] = (0)} .)

i∈S


(FS ) acts identically on FS . Since it commutes with LS , Next, we check that G which acts transitively on FS , it suffices to check that 
,S , gcS (o) = cS (o) for all g ∈ G or that  c−1 S gcS (o) = o for all g ∈ G
,S , or that 
,S ) ⊂ kC + p− . Ad cS (Lie G

136

III Compactifications of locally symmetric varieties

But Ad cS (x) = x, for all x ∈ m(a) , Ad cS (gϕ ) =

∑

gϕ



if ϕ = 12 (γi − γ j ) ,

γ −γ  ϕ∼ i 2 j

Ad cS (xi ) = hi

i, j ∈ S ,

if i ∈ S ,

and hence 
,S ) ⊂ Ad cS (Lie G

∑

ϕ∼

γi −γ j 2



gϕ + tC + m(a) .


,S centralizes FS . All these factors are in kC , so G Next, we check that W (FS ) centralizes FS . Again, since W (FS ) is normalized by LS , it suffices to show that for all g ∈ W (FS ) ,

gcS (o) = cS (o) , or that

Ad cS (LieW (FS )) ⊂ kC + p− . But Ad cS (LieW (FS )) =

∑

gϕ .

ϕ ∈Ψ dwS ,ϕ <0

These roots are: (i) ϕ ∼ − 12 (γi + γ j ) or − 12 (γi ), i ∈ S, any j ; (ii) ϕ ∼ − 12 (γi − γ j ), i ∈ S, j ∈ /S, which are all compact, or negative non-compact. This proves that 
,S
W (FS ) ⊂ Z(FS )o . G Since LS /(center) acts faithfully on FS , we must have equality here, and this finishes the proof. The above proof gives us another useful piece of information: for all F, let H ∩ P1

f

F −−−− −→ D ∩ fF −−−−−→ Dˇ

be as in Theorem 3.3; in particular, fF (i) = o , fF (∞) = oF .

3 Boundary components

137

Define now (recall that so denotes the symmetry of D at o): o0F = fF (0) = so (oF ) , F 0 = so (F) = boundary component containing o0F .

If F = FS , then  In general, since

1 0

−i 1 o0F



o0FS = c−1 S (o) . ∈ SL(2, C) carries i to 0,

   1 −i ·o . = ϕF 1, 0 1


(F) fixes o0 . Proposition 3.11 G F Proof Note that the symmetry so interchanges o0F and the basepoint oF ∈ F, 
(F), hence, for g ∈ G go0F = gso (oF ) = so (σ (g)(oF )) = so (oF ) = o0F , 
⊂ N(F). because σ preserves G

3.4 The next topic we want to discuss is the natural projection of D onto one of its boundary components F:

πF : D −→ F .

138

III Compactifications of locally symmetric varieties

We begin by considering an analog of the linear projection π : C −→ C1 of a cone onto one of its boundary components, discussed in Chapter II, Section 3. Please bear with us though – this will not immediately define πF , as it is the “wrong projection.” Fix a basepoint o ∈ D and let wF : Gm −→ G be the homomorphism associated to F and o, i.e., the equivariant pair f : H −→ D ,

ϕ : U 1 × SL(2, R) −→ G ,    t 0 with f (i) = o, f (∞) ∈ F, and let wF (t) = ϕ 1, . Then Gm acts 0 t −1 ˇ hence, for all x ∈ D, ˇ we may via wF algebraically on the projective variety D; define: pF (x) = lim wF (t) · x . t−→0

For instance, pF (o) = o0F . Now, for all g ∈ N(F)C = P(w−1 F )C , we calculate: pF (go) = lim wF (t) · g · o t−→0

= lim (wF (t)gwF (t)−1 )(wF (t) · o)

t−→0 = g(o0F )

(3.1)

,

where g = lim wF (t)gwF (t)−1 , which exists by assumption. Moreover, note t−→0

that N(F)C = Z(wF )C ·W (F)C , N(F 0 )C = Z(wF )C ·W (F 0 )C , and that g is just the projection of g into its Z(wF )C -component. Therefore ˇ ⊂ Dˇ by g ∈ N(F 0 )C , hence g · o0F ∈ F 0 . Now define the open subset D(F) ˇ = D(F)



g · D = N(F)C · o

g∈N(F)C

(the last equality occurs because N(F) acts transitively on D). By the first ˇ is open; by the second, it is the orbit of an algebraic group description, D(F) ˇ and hence it is a Zariski-open subset of a acting on a projective variety D, ˇ is a Zariski-open subset of D. ˇ Now formula (3.1) subvariety. Therefore D(F) shows that ˇ ⊂ Fˇ 0 and pF (D) ⊂ F 0 , (a) pF (D(F)) ˇ −→ Fˇ 0 is equivariant for N(F)C (b) pF : D(F)

3 Boundary components

139

(not N(F 0 )C !), where N(F)C acts on Fˇ 0 by N(F)C −→ Z(wF )C → N(F 0 )C −→ Aut (DF )C ; ˇ hence pF is continuous on D(F). But now a map defined like pF is automatically a morphism of varieties whenever it is continuous. In fact, embed Dˇ ⊂ PnC such that Dˇ ⊂ hyperplane, and such that wF acts on Pn too, via (x0 , . . . , xn ) −→ (t t0 x0 , . . . ,t rn xn ), where r0 ≥ · · · ≥ rn . Let rm−1 > rm = rn . Let L1 , resp. L2 , be the linear spaces x0 = · · · = xm−1 = 0, resp. xm = · · · = xn = 0. Then pF on Pn \ L2 is the linear projection with center L2 and image L1 . Moreover, L1 ∩ L2 = 0/ and pF (L2 ) ⊂ L2 . Thus, if pF is continuous in the classical topology on any connected S ⊂ Pn , ˇ ⊂ Pn \ L2 . then either S ⊂ L2 or S ⊂ Pn \ L2 . It follows that D(F) ˇ ˇ Now, to define πF : D(F) −→ F, we proceed as follows. Take the symmetries so : Dˇ −→ Dˇ , soF : Fˇ −→ Fˇ . Define

πF (x) = soF (so (pF (x))) (since so (F 0 ) = F, this makes sense). Then, for all g ∈ N(F)C , if g is its projection into Z(wF )C and g is its projection into Aut (DF )C , then

πF (gx) = soF (so (pF (gx))) = soF (so (g(pF x))) = soF (σ (g)(so (pF x))) = soF (σF (g)(so (pF x))) = g(soF (so (pF x))) = gπF (x) . Here σ : GC −→ GC denotes the Cartan involution relative to the basepoint o, and σF : Aut (DF )C −→ Aut (DF )C denotes the Cartan involution relative to the basepoint oF . Therefore ˇ ⊂ Fˇ and πF (D) ⊂ F ; (a ) πF (D(F))  ˇ and on Fˇ in the natural way, (b ) πF is equivariant for N(F)C acting on D(F) ˇ i.e., restricting the action of GC on D ; (c ) πF is a morphism of varieties. As a corollary of (b ), we deduce the following intrinsic definition of πF independent of the choice of basepoint:

140

III Compactifications of locally symmetric varieties

(d ) πF (x) = the point of Fˇ with stabilizer equal to the image in Aut (DF )C of N(F)C ∩ (Stab. of x). Sometimes πF on D is called the geodesic projection of D onto F. The reason for this is as follows: for all x ∈ D, let fF,x be the unique symmetric holomorphic map defined by fF,x : H −→ D , i −→ x , ∞ −→ pt. of F . Then fF,x (iR) is a geodesic in D through x and one checks easily – check it first for x = o, then use (b ) – that

πF (x) = fF,x (∞) = lim fF,x (it). t−→∞

(3.2)

Note that if, at the beginning of Subsection 3.4, we had interchanged the roles of F and F 0 and defined pF 0 : D −→ F by pF 0 (x) = lim wF (t) · x , t−→∞

this would look similar to (3.2). However, wF (R) · x is not a geodesic in general, and pF 0 is definitely not N(F)-equivariant! Incidentally, an alternative to our round-about approach via pF would be to define πF directly using (b ); this would require a careful root analysis via Proposition 2.7 to establish the key lemma: 
,C ·WC , N(F)C ∩ (KC · P− ) ⊂ Kh,C · Ph,− · G where Kh , Ph are the subgroups of Gh analogous to K and P in G.

3.5 The final topic we want to discuss is that of rational boundary components. As in Subsection 2.5, say D = G (R)o /K, where G is an algebraic group defined over Q. Definition 3.12 A boundary component F of D is rational if its normalizer N(F) is defined over Q (i.e., N(F) = N (F)(R) ∩ G (R)o for some algebraic subgroup N (F) defined over Q). To understand the effect of this definition, let A ⊂ G be a maximal Qsplit torus, and let B ⊂ G be a maximal R-split torus with A ⊂ B. As in Subsection 2.5, let γi , 1 ≤ i ≤ r, be the strongly orthogonal real roots and let {1, . . . , r} = I0 ∪ · · · ∪ Is be the partition so that A is defined by γi = 1 for i ∈ I0 and γi = γ j for i, j ∈ Ik , k = 1, . . . , s. For i ≥ 1, let βi be the restriction to A of

3 Boundary components

141

γ j , j ∈ Ii . For all S ⊂ {1, . . . , r}, let FS be the associated boundary component. Then we may state: Lemma 3.13 The boundary component FS is rational if and only if S = Ii1 ∪ · · · ∪ Iit , where 1 ≤ i1 , . . . , it ≤ s. Proof If S has the above property, then wS (Gm ) is a one-parameter subgroup of A . Since A is split, wS is defined over Q, and hence so is PS . By Proposition 3.6, this means FS is rational. Note that if G is Q-simple, then this construction gives all maximal Q-rational parabolics containing A . To prove the converse, we may assume that G is Q-simple since A and N (FS ) are products of their intersections with the Q-simple factors. Then N (FS ) is a maximal Q-rational parabolic (see (2) in the following list), and hence S is of the desired form. In general, rational boundary components behave just like ordinary boundary components. Thus one proves easily the following. (1) If we decompose G into its Q-simple factors as G = G1 × · · · × Gk , and let D = D1 × · · · × Dk be the corresponding decomposition, then a boundary component F = F1 × · · · × Fk is rational if and only if the Fi are rational. (2) If G is Q-simple, then the association F −→ N(F) defines a bijection between the set of rational boundary components and the set of maximal Qparabolics of G . If G = G1 × · · · × Gt over R, then each rational F decomposes as F = F1 × · · · × Ft , with F i  Di ; hence every maximal Q-parabolic P decomposes as P = P1 × · · · × Pt , with Pi ⊂ Gi a maximal R-parabolic. (3) Every rational boundary component equals gFS for some g ∈ G (Q) and some S (for which FS is rational). (4) If s = Q-rank G , there is a symmetric holomorphic map f

Hs ∩

1 −−−− −→

(P1 )s

2 −−−− −→

f

D ∩ Dˇ

associated to

ϕ : U 1 × SL(2, R)s −→ G such that ϕ (1, diagonal matrices) is a maximal Q-split torus and the rational boundary components are the G (Q)-transforms of the boundary components FS containing i , . . . , %&'( ∞ , . . .) . f2 (. . . , %&'( i ∈S

i∈S

142

III Compactifications of locally symmetric varieties

(5) Jumping ahead and using the results of Section 4, we may ask, for F a rational boundary component, which of the factors of N (F) is Q-rational. We will have (see Section 4) that the algebraic connected component of N (F) equals [Gh (F) · G
(F) · M (F)] · V (F) · U (F) . Then (a) wF is in a maximal Q-split torus so it is rational; hence Gh · G
· M , and U and V , as 0, 1, 2-eigenspaces for Ad wF , are Q-rational; (b) Gh · M as the centralizer of U in Gh · G
· M is Q-rational; hence G
as the normal complement to Gh · M in Gh · Gl · M is Q-rational. (6) Because of (5), we may speak of rational boundary components of rational boundary components and we claim: given boundary components F1 < F2 , with F2 rational, then F1 is rational as a boundary component of F2 if and only if it is rational as a boundary component of D. In fact, our assumptions say that N(F1 ) ∩ N(F2 ) is a real parabolic, with N(F2 ) defined over Q. We may assume G is Q-simple. By the uniqueness of the expression of parabolics as an intersection of maximal parabolics, N(F1 ) ∩ N(F2 ) is defined over Q if and only if N(F1 ) is defined over Q. Note that if G = Aut (D)o , then we may have G (R)o = G · M0 , with M0 compact and acting trivially on D. From the point of view of algebraic groups over Q, M0 may be conjugate to simple factors of G, and hence not defined over Q and impossible to throw out. Then M0 appears as an “extra” factor in each N (F)(R) and combines with the M(F) to be introduced in Section 4. In particular, Gh (F) · M (F) is a semi-simple algebraic group over Q such that F = [Gh (F) · M (F)](R)o /(max. compact), i.e., we recover for F the presentation we have for D.

4 Siegel domains of the third kind 4.1 Let D be a bounded symmetric domain, and let F be a boundary component. The purpose of this section is to work out in considerable detail the structure of the group N(F) and from this to realize D as a particular open subset of a rather simple ambient space D(F): this is an abstract version of Piatetskii-Shapiro’s models of D as “Siegel domains of third kind.” We will briefly indicate at the end how to make the link with the more concrete Siegel domains. First we study N(F) more closely. As in Section 3, it is easiest to write

4 Siegel domains of the third kind

143

things out explicitly for N(FS ), and then to observe that the same things happen for every F by conjugating. In this way, we now want to introduce the fundamental 5-factor decomposition of N(F). Special case F = FS We have seen that n(FS ) := Lie N(FS ) = Z(a) ⊕ ϕ=

∑ ±γ ± γ i 2

j

∑ γ ±γ

gϕ ⊕ ±γ

ϕ = i 2 j or i∈S, any j

or 2 i i, j∈S /

gϕ . γi 2

Under the homomorphism wS : Gm −→ N(FS ), we may decompose Lie N(FS ) into the direct sum of three eigenspaces, where Ad (wS (t)) is multiplication by 1, t, t 2 , respectively: n(FS )0 = Z(a) ⊕

∑

±γ ± γ ϕ = i2 j

or i, j∈S /

n(FS )1 =

∑

γ ±γ ϕ= i 2 j

gϕ :

or i∈S, j∈S /

n(FS )2 =

∑ γ +γ

ϕ= i 2 i, j∈S

gϕ ⊕ ± γi 2

∑γ −γ i

ϕ= 2 i, j∈S

gϕ ; j

call this v(FS ) ;

γi 2

gϕ :

call this u(FS ) .

j

These are the root spaces gϕ , where dwS , ϕ  = 0, 1, or 2, respectively. Then u + v is the maximal nilpotent ideal, so, defining as in Theorem 3.10 W (FS ) = unipotent radical of N(FS ) , we have u(FS ) + v(FS ) = LieW (FS ) . Define: U(FS ) = commutative subgroup of W (FS ) with Lie algebra u(FS ) ; V (FS ) = exp v(FS ), a subset of G diffeomorphic to v(FS ) . Then W (FS ) ∼ = V (FS ) ×U(FS ) , where U(FS ) ⊂ center W (FS ) and W (FS )/U(FS ) is an abelian Lie group ∼ = V (FS ) .

144

III Compactifications of locally symmetric varieties

We also want to refine the decomposition of Z(wS ) introduced in Theorem 3.10. If σ is the Cartan involution of g then

σ (gϕ ) = g−ϕ for all ϕ ∈ RΨ ; hence

σ ([gϕ , g−ϕ ]) = [gϕ , g−ϕ ] , and [gϕ , g−ϕ ] = [gϕ , g−ϕ ] ∩ a ⊕ [gϕ , g−ϕ ] ∩ m(a) . % &' ( call this [gϕ ,g−ϕ ](e)

Note that we have lS = gh (FS ) =

∑

±γ ± γ ϕ = i2 j

or i, j∈S /

± γi 2

(gϕ + [gϕ , g−ϕ ](e) ) + ∑ Rxi . i∈S /

Define analogously: g
(FS ) =

∑ γ −γ

ϕ= i 2 i, j∈S

j

(gϕ + [gϕ , g−ϕ ](e) ) + ∑ Rxi . i∈S


(FS ) generated by the gϕ and the xi . It is also This is the subalgebra of Lie G 
(FS ) is the sum of g
(FS ) and a compact ideal 
(FS ), so Lie G an ideal in Lie G m(FS ). Therefore, we have n(FS )0 = gh (FS ) ⊕ g
(FS ) ⊕ m(FS ) . Globally, Z(wS )o = Gh (FS ) · G
(FS ) · M(FS ) % &' ( 
(FS ) this is G

(which stands for the direct product of three groups modulo a finite subgroup), where Gh (FS ) is semi-simple, with no compact factors , G
(FS ) is reductive, with no compact factors , M(FS ) is compact . Thus, finally, N(FS )o = [Gh (FS ) · G
(FS ) · M(FS )] ×V (FS ) ×U(FS ) .

4 Siegel domains of the third kind

145

General case We can use that F = k · FS , for some k ∈ K, S ⊂ {1, . . . , r}, and hence N(F) = kN(FS )k−1 , to get the same decomposition for N(F). However, we may also characterize the decomposition intrinsically using the homomorphism wF associated canonically to F. Then Lie N(F) = sum of 0, 1, 2-eigenspaces for Ad wF (t) = Lie Z(wF ) ⊕ v(F) ⊕ u(F) . Let V (F) = exp v(F), U(F) = exp u(F). Next, Z(wF )o decomposes into Gh (F) 
acts identically on F. Writing G 
as 
(F), as in Theorem 3.10, where G and G a product of its non-compact and compact factors, we obtain N(F)o = [Gh (F) · G
(F) · M(F)] ×V (F) ×U(F) .

4.2 Next, we want to look deeper at the group-theoretic structure ofN(F)o . To t 0 state the first result, note that dϕF maps the k’th eigenspace for Ad 0 t −1 in SL(2, R) into the k’th eigenspace for Ad wF (t) in g. In particular    0 1 ∈ u(F) , ωF := dϕF 0, 0 0 and hence

   1 1 ΩF := ϕF 1, = exp ωF ∈ U(F) . 0 1

With this definition, we have: Theorem 4.1 (1) [Gh (F) · M(F)] ×W (F) centralizes U(F) and u(F). (2) The orbit of ΩF by G
(F), 6 7 C(F) = gΩF g−1 | g ∈ G
(F) , is an open homogeneous cone in U(F), self-adjoint with respect to the positive-definite quadratic form x, y = −B(x, σ (y)) on u(F), hence on U(F). The centralizer of ΩF in G
(F) is the maximal compact subgroup G
(F) ∩ K, hence C(F) ∼ = G
(F)/G
(F) ∩ K .

146

III Compactifications of locally symmetric varieties

Proof We show first that† G
(F) ∩ K = {g ∈ G
(F) | Ad g(ωF ) = ωF } , and that Ad M(F) fixes ωF . In fact, note that    1 −i o0F = ϕF 1, ·o 0 1 = exp(−iωF ) · o ; 
(F) = G
(F) · M(F), hence, for all g ∈ G g · o = g exp(iωF ) · exp(−iωF ) · o = exp(iAd g(ωF )) · go0F . By Proposition 3.11, go0F = o0F . So g · o = exp(iAd g(ωF )) · exp(−iωF ) · o = exp(i(Ad g(ωF ) − ωF )) · o . But nothing in UC fixes o, so g ∈ K if and only if Ad g(ωF ) = ωF . Next, count dimensions taking F = FS . Using the fact that σ (gϕ ) = g−ϕ , we find dim G
(F)/G
(F) ∩ K = dim {−1-eigenspace of σ in g
(F)} = |S| +

∑ dim g(γi −γ j )/2 .

i> j i, j∈S

But, using Proposition 2.7 and the remarks following it, we have dim g(γi −γ j )/2 = # of (bi j )-factors in g = dim g(γi +γ j )/2 , for i > j , and moreover gγi is 1-dimensional and lies entirely in the (ai )-factor, i.e., the i’th copy of sl(2, R). Thus, |S| +

∑ dim g(γi −γ j )/2 = ∑ dim g(γi +γ j )/2 = dim u(F) .

i> j i, j∈S

i≥ j i, j∈S

Therefore, g −→ Ad g(ωF ) defines an isomorphism of G
(F)/G
(F) ∩ K with an open subset C(F) of u(F). It follows from Chapter II, Proposition 1.10, that C(F) is a homogeneous cone, self-adjoint for any inner product ·, · such that t (Ad g)−1 = Ad (σ (g)), or Ad g(x), Ad σ g(y) = x, y for all x, y ∈ u(F) . † This argument is due to P. Deligne.

4 Siegel domains of the third kind

147

The inner product −B(x, σ (y)) has this property. This proves (2). As for (1), by checking roots one sees [gh (F), u(F)] = [v(F) + u(F), u(F)] = (0) , and hence Gh (F) ×W (F) centralizes U(F). Finally, since [m(F), ωF ] = (0) , we have, for all x ∈ g
(F), [m(F), [x, ωF ]] = [[m(F), x], ωF ] = (0) , hence [m(F), u(F)] = (0) , and M(F) centralizes U(F). Corollary 4.2 ϕF (U 1 ) ⊂ Gh (F) × M(F). Proof Since ϕF (U 1 ) and wF commute, ϕF (U 1 ) ⊂ Gh × G
× M and, by Theorem 4.1, it suffices to show that ϕF centralizes u(F). But if F = FS , then u(F) ⊂ factors of type (ai ), (bi j ) ,

i, j ∈ S ,

and

ϕFS (eiθ ) = ϕ (eiθ ; . . . , %&'( 1 , . . . , %&'( eiθ , . . .) . if i∈S

if i∈S /

The next theorem will show that U(F) is the center of the unipotent radical W (F) of the parabolic subgroup N(F). We denote by J the operator on the vector space v(F) J = Ad (ϕF (−i, I)) . Then, clearly, J defines a complex structure on v(F): ϕF (−1, −I) = h0 (−1) = u0 (1) = e; hence J 2 = Ad ϕF (−1, I) = Ad ϕF (1, −I) = wF (−1) , which is −Id on v(F). Theorem 4.3 [v, Jv] ∈ C(F) \ {0}, for all 0 = v ∈ v(F). We have the consequence: Corollary 4.4 U(F) is the center of W (F).

148

III Compactifications of locally symmetric varieties

Proof of Theorem 4.3 Since C is self-adjoint with respect to ·, ·, it is sufficient to show that v = 0 =⇒ [v, Jv], y > 0 for all y ∈ C . But, writing y = Ad g−1 (ωF ) with g ∈ G
, and noting that Ad (σ (g))u, u  = u, Ad (g)−1 u  for all u, u ∈ U(F) , we obtain [v, Jv], y = [v, Jv], Ad g−1 (ωF ) = Ad σ (g)[v, Jv], ωF  = [Ad σ (g)v, Ad σ (g)Jv], ωF  = [Ad σ (g)v, JAd σ (g)v], ωF  , where, for the last equality, we used ϕF (U 1 ) ⊂ Gh × M, so ϕF (U 1 ) and G
commute. Thus it suffices to show v = 0 =⇒ [v, Jv], ωF  > 0 . Now, [v, Jv], ωF  = −B([v, Jv], σ (ωF )) = B(v, [σ (ωF ), Jv]) = B(v, σ ([ωF , σ Jv])) . So it suffices to prove [ωF , σ Jv] = −v, for all 0 = v ∈ v(F) , and use the fact that B(x, σ (y)) is negative-definite. But    0 1 ◦ Ad ϕF (−i, I) σ ◦ J = Ad ϕF i, −1 0    0 1 , = Ad ϕF 1, −1 0 and hence

     0 1 [ωF , σ Jv] = (Ad ΩF − I) Ad ϕF 1, v −1 0       −1 1 0 1 = Ad ϕF 1, v − Ad ϕF 1, v. −1 0 −1 0

Now suppose we put together the +1 and −1-eigenspaces for Ad wF (t): the +1-space is v(F) and the whole space is given by h = {x ∈ g | Ad wF (−1)x = −x} ,

4 Siegel domains of the third kind 149   t 0 which is a subspace invariant under SL(2, R). Since has only ±10 t −1 eigenspaces, h is the sum of copies of the usual two-dimensional   representation 1 R2 . Therefore we need only calculate in R2 , with v = : 0         −1 1 1 0 1 1 1 − =− . −1 0 0 −1 0 0 0

4.3 Next, we want to interpret the group-theoretic structure of N(F)o geometrically as a decomposition of D. First, by Proposition 3.6, N(F), and hence N(F)o , acts transitively on D, so that, using our more refined decomposition of N(F)o , we can write D∼ = ([Gh (F) · G
(F) · M(F)] ×W (F)) /[Kh (F) · K
(F) · M(F)] , where Kh (F) = Gh (F) ∩ K = a maximal compact subgroup of Gh (F) , K
(F) = G
(F) ∩ K = a maximal compact subgroup of G
(F) . Therefore, firstly, there is an N(F)o -equivariant mapping ΦF : D −→ C(F) with ΦF (o) = ΩF , defined by the map of homogeneous spaces ([Gh · G
· M] ×W ) /[Kh · K
· M] −→ ([Gh · G
· M] ×W ) / ([Gh · K
· M] ×W ) ∼ = G
/K
. Secondly, the whole domain D can be decomposed as a real manifold: D∼ = F ×C(F) ×W (F) by x −→ (πF (x), ΦF (x), w(x)) , where w(x) is defined as follows. Let

πF (x) = gh (πF (o)) with gh ∈ Gh , ΦF (x) = g
(ΩF ) with g
∈ G
, then x = w(x) · gh · g
· o with w(x) ∈ W (F) .

(4.1)

150

III Compactifications of locally symmetric varieties

ˇ To understand the situation better, we introduce a new open subset D(F) ⊂ D. Definition 4.5 Let D(F) = U(F)C · D =




g∈U(F)C g · D.

Note that, since U(F) is a normal subgroup of N(F)o , it follows that N(F)o · U(F)C is a subgroup of GC , which clearly acts transitively on D(F). Recall the point: o0F = exp(−iωF ) · o ∈ D(F) . Lemma 4.6 Kh (F) · G
(F) · M(F) = {g ∈ N(F)o ·U(F)C | go0F = o0F }. Proof We saw in Proposition 3.11 that G
· M fixes o0F . Moreover, Kh fixes o and Kh commutes with exp(−iωF ), so Kh fixes o0F . This shows ”⊂.” But, since KC · P− is the stabilizer of o in GC , the RHS equals N(F)o ·U(F)C ∩ exp(−iω F ) · KC P− · exp(iωF ) (intersection inside GC ). A simple root calculation shows that the Lie algebra of this group equals the Lie algebra of the LHS. But considering GC as the real points of RC/R GC , both N(F)o ·U(F)C and KC · P− are the connected components of the real points of algebraic subgroups. So this intersection has finitely many components. Since Kh ⊂ Gh is maximal compact, and W (F) ·U(F)C is torsion-free, there is no group H with N(F)o ·U(F)C ⊃ H



(finite index)

Kh × G
× M.

Therefore, if we take o0F as basepoint of D(F), we get an isomorphism: ∼

(Gh (F)/Kh (F)) ×W (F) ·U(F)C −→ D(F) , (g, w) −→ w · g(o0F ) ,

(4.2)

and hence F ×V (F) ×U(F)C ∼ = D(F) . The projection πF to F is again just given by

D(F)

Gh /Kh ∼ =

∼ =

−→ N(F)o ·U(F)C /Kh · G
· M ·W ·UC ,

∼ =

N(F)o ·U(F)C /Kh · G
· M

F

4 Siegel domains of the third kind

151

which is N(F)o ·U(F)C -equivariant, and hence is the πF defined in Section 3.4. Also N(F)o is a subgroup of N(F)o ·U(F)C , so projection onto the imaginary part of the U-coordinate is given by N(F)o ·U(F)C /N(F)o , ∼

←

−→

∼ =

N(F)o ·U(F)C /Kh · G
· M D(F)

U(F)

which is also N(F)o ·U(F)C -equivariant, where N(F)o ·U(F)C acts on U(F) via the vertical isomorphism in the preceding diagram, which amounts to letting N(F)o act on U(F) by conjugation and letting iU(F) act on U(F) by translation. We call this ΦF because we claim that we get a commutative diagram involving this projection: Φ

F −−−− −→

D ∩

Φ

C(F) ∩

F D(F) −−−− −→ U(F) .

To see this, let x = w · gh · g
· o ∈ D, with w ∈ W , gh ∈ Gh , g
∈ G
. Then ΦF (x) = g
(ΩF ), and, since G
acts on C(F) by conjugation, we obtain ΦF (x) = g
exp(ωF )g−1
. But x = w · gh · g
· exp(iωF ) · exp(−iωF ) · o 0 = w · gh · g
· exp(iωF ) · g−1
· oF

(since g
o0F = o0F ).

Since w · gh ∈ N(F), and g
· exp(iωF ) · g−1
∈ iU(F) ⊂ U(F)C , the imaginary part of the U(F)C -coordinate of x is just g
· exp(ωF ) · g−1
. In fact: Lemma 4.7 D = {x ∈ D(F) | ΦF (x) ∈ C(F)}. Proof Let x = g · exp(iu) · o0F be any element of D(F), where g ∈ N(F)o and u ∈ u(F). If u ∈ C(F), then u = Ad h(ωF ) with h ∈ G
, and hence   x = g · h exp(iωF )h−1 · o0F = g · h · exp(iωF ) · o0F

(since h · o0F = o0F )

= g·h·o ∈ D .

The idea of Siegel domains is that D(F) is a much simpler complex manifold than D, and that D is defined inside D(F) by the tube-domain-like requirement ΦF (x) ∈ C(F) .

152

III Compactifications of locally symmetric varieties

To see what D(F) is, note that because of the isomorphism (4.2) above, U(F)C acts freely on D(F), making it into a principal homogeneous space over D(F) := D(F)/U(F)C ∼ N(F)o ·U(F)C /Kh · G
· M ·U(F)C = = N(F)o /Kh · G
· M ·U(F) . In turn, V (F) acts freely on D(F) , making it into a principal homogeneous space over F: D(F) πF fibers U(F)C

πF

 D(F)

pF fibers V (F)

 F

Since U(F)C is a complex subgroup of GC , we have that U(F)C acts holomorphically on D(F), and D(F) has a complex structure making all the maps holomorphic. Note, however, that V (F) has no natural complex structure and that D(F) −→ F is only real-analytically a principal homogeneous space with group V (F). This is as far as we need to carry this analysis of D and D(F). The full picture, however, is the following. (i) There is a holomorphic section of D(F) over D(F) , such that D(F) ∼ = D(F) ×U(F)C . (ii) For each x ∈ F, there is a complex structure Jx : V (F) −→ V (F) (with JoF being the J defined above) such that (V (F), Jx ) acts complex-analytically on p−1 F (x). (iii) Altogether, D(F) is a complex vector bundle over F, which can be trivialized as D(F) ∼ = Ck × F , such that each v ∈ V (F) acts as (x, a) −→ (x + λv (a), a) where λv (a) is holomorphic in a and linear in v. Via (i) and (iii), we get a holomorphic isomorphism: D(F) ∼ = U(F)C × Ck × F

4 Siegel domains of the third kind

153

(NB not the same as the more elementary group-theoretic isomorphism D(F) ∼ = U(F)C ×V (F) × F in (4.2) above).

(iv) In this product representation ΦF (x, y, z) = Im x − hz (y, y) , where hz is a real-bilinear quadratic form Ck × Ck −→ U(F) depending real-analytically on z. Thus 9 : D∼ = (x, y, z) ∈ U(F)C × Ck × F | Im x ∈ C(F) + hz (y, y) . For proofs, see Koranyi–Wolf [7] and Satake [11]. We summarize the essential maps that we will use in Section 5 below: C(F) O ΦF

⊂

U(F) O ΦF

⊂ D(F) ⊂ Dˇ D+ 5 55 + 5  π  : p.h.s. for U(F) + 55  C + 5  F πF +D(F) + + pF : p.h.s. for V (F) +  F This whole diagram is N(F)o -equivariant; all but D and C(F) is N(F)o ·U(F)C equivariant. An interesting topic that we have not seen explored is to investigate the third ˇ open subset of D: ˇ = N(F)C · D = D(F)

g∈N(F)C

g·D .

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III Compactifications of locally symmetric varieties

ˇ is a Zariski-open subset of Dˇ (see Subsection 3.4) and appears to Here, D(F) sit in a double fibration: ˇ D(F) p.h.s. for U(F)C

 ˇ  D(F)  Fˇ

cx. vector bundle and p.h.s. for V (F)

ˇ in which D(F) and D(F) are just the inverse images of F ⊂ F. 4.4 The final point of this section is to compare N(F) and N(F  ) and their decompositions when F ⊂ F  . We formulate the situation in the following theorem. Theorem 4.8 Let F ⊂ F  be two boundary components of D. Then (i) U(F) ⊃ U(F  ), G
(F) ⊃ G
(F  ), Gh (F) ⊂ Gh (F  ). (ii) C(F  ) is a boundary component of C(F). Moreover, fixing F, then the map F  −→ C(F  ) is an order-reversing bijection between the set of boundary components F  with F ⊂ F  of D and the set of boundary components of C(F). (iii) Now assume F = FS , F  = FS , where S ⊃ S . Then AS = (A ∩ G
(FS ))o is a maximal split torus of G
(FS ). Let UD = ∏i∈S exp(gγ j ) ⊂ U(FS ) and define coordinates x1 , . . . , xd on UD by xi (exp gγ j ) = 0 if i = j and xi (ΩFS ) = 1. Then AS · ΩFS = {(x1 , . . . , xd ) ∈ UD | xi > 0 for all i = 1, . . . , d} , 0 , . . . , %&'( 1 , . . .) ∈ UD , ΩFS = (. . . , %&'( i∈S\S

i∈S

and C(FS ) is the boundary component of C(FS ) whose intersection with UD is given by {(x1 , . . . , xn ) | xi = 0 if i ∈ S \ S , xi > 0 if i ∈ S } . (iv) Finally, if D = G (R)o /K, with G defined over Q, and F is a rational boundary component, then the subgroups U(F), G
(F) ⊂ N(F) are defined over Q; in particular, U(F) comes from a Q-vector space and for all F  with F ⊂ F  ,

4 Siegel domains of the third kind 

F  is a rational boundary component of D



 ⇐⇒

C(F  ) is a rational boundary component of C(F)

155 .

Proof By Proposition 3.4, any pair F ⊂ F  is conjugate by some k ∈ K to a pair FS ⊂ F S , where S ⊃ S . For such pairs, (i) is immediate by the explicit formulae at the beginning of this section. Now, A ⊂ G
(FS ) · Gh (FS ) and is a maximal split torus in G. So A = A
· Ah , where A
= (A ∩ G
(FS ))o and Ah = (A ∩ Gh (FS ))o are maximal split tori in G
(FS ) and Gh (FS ). In fact, A
= exp (∑i∈S Rxi ). To investigate about the situation in U(F), let us identify U(F) and u(F) via exp, and calculate instead in u(F). First of all, ωFS is the natural basepoint in

∑ gγ i .

i∈S

(In the notation of Theorem 2.4, it is ∑i∈S −yi2+ihi .) Since A
acts on gγi via the character eγi , it is a simple calculation to describe AS · ωFS as in part (iii) of the theorem. And if S ⊃ S , then, by definition, ωFS is just the projection of ωFS into the subspace

∑ gγi .

i∈S

Now, by definition, C(FS ) = {Ad g(ωFS ) | g ∈ N(FS )} . But as N(FS ) = G
(FS ) · (centralizer of U(FS )), in fact for any group H with G
(FS ) ⊂ H ⊂ N(FS ), we have C(FS ) = {Ad g(ωFS ) | g ∈ H} . For instance, since G
(FS ) ⊃ G
(FS ), it follows that H = N(FS ) ∩ G
(FS ) will do. But let wS : Gm −→ A be the homomorphism defined above, relative to S . Then N(FS ) = P(w−1 S ), so −1 ) of G (F ) defined by w , i.e., H is the parabolic subgroup P(w−1   S
G (F ) S S
S : 9 −1 H = g ∈ G
(FS ) | lim wS (t)gwS (t) exists . t−→0

But wS is a one-parameter subgroup of G
(FS )/(center) = Aut (C(FS ))o (via Ad ). Acting on the basepoint ωFS , lim Ad wS\S (t)ωFS = ωFS .

t−→0

156

III Compactifications of locally symmetric varieties

Note that the parabolic subgroups P(w−1 S )G
(FS ) and P(wS\S )G
(FS ) of G
(FS ) agree, since wS is central in G
. Therefore, by Chapter II, Proposition 3.4, the boundary component of C(FS ) through ωFS is the orbit of ωFS under  P(wS\S )G
(FS ) = P(w−1 S )G
(FS ) in u(FS ), which is just C(FS ), namely the locus of points Ad g(ωFS ), with g ∈ H. To prove the second part of (ii), we no longer normalize F and F  , but instead we may assume G is simple because, in the general case, everything breaks up into a product. The first step is to check that, for any two boundary components F, F  , F ⊂ F  or F  ⊂ F ⇐⇒ N(F) ∩ N(F  ) is parabolic. Ordering the roots ∑ ui γi lexicographically, the corresponding minimal real parabolic P ⊃ A is given by Lie P = Z(a) +

∑

g

all i, j

γi + γ j 2

+∑g i< j

γi − γ j 2

γi

+∑g 2 ; i

and the maximal real parabolics containing P are PS for S = {1}, {1, 2}, . . . , {1, 2, . . . , r} . To prove implication =⇒, if for instance F ⊂ F  , we may assume F = FS , F  = FS , with S ⊃ S ; applying the Weyl group, we may even assume S = {1, . . . , s}, S = {1, . . . , s }, with s ≥ s . Thus N(FS ) ∩ N(FS ) ⊃ P. Conversely, if N(F)∩N(F  ) contains a parabolic, by conjugating we can assume it contains P. Then N(F) = P{1,...,s} and N(F  ) = P{1,...,s } for some s, s . Thus F = F{1,...,s} and F  = F{1,...,s } and, depending on whether s > s or s < s , we get F ⊂ F  or F  ⊂ F. Therefore, fixing F, and using the unique representation of parabolics as intersections of maximal parabolics, we find a bijection: 6  7 F | F ⊂ F  or F  ⊂ F ∼ = {P ⊂ N(F) maximal real parabolic} . But now N(F) ∼ = G
(F) · Gh (F) · M(F) ·W (F), so the maximal real parabolics P ⊂ N(F) are of the form P = P
· Gh (F) · M(F) ·W (F) ,

P
maximal parabolic in G
(F) ,

P = G
(F) · Ph · M(F) ·W (F) ,

Ph maximal parabolic in Gh (F) .

or

Using our standard models again, it is easy to see that F ⊂ F  corresponds to the first type and F  ⊂ F to the second; i.e., we get bijections: 6  7 F | F ⊂ F ∼ = {P ⊂ G
(F) maximal real parabolic} , 6   7 F |F ⊂F ∼ = {P ⊂ Gh (F) maximal real parabolic} .

4 Siegel domains of the third kind

157

But if G is simple then so is G
(F) (and Gh (F)): for our purposes, it is enough to check this for G
(FS ). Then, for every permutation σ of {1, . . . , r} which preserves S, there is an element wσ ∈ Norm (A), such that wσ γi w−1 σ = γσ (i) ; hence wσ wS = wS wσ and wσ ∈ N(FS ). Hence wσ projects to give an element of the Weyl group of the root system of G
(FS ), and it follows that the Weyl group of the root system of G
(FS ) is the full permutation group of S, and hence G
(FS ) is simple. Therefore there is a bijection between maximal real parabolics in G
(F) and boundary components of C(F). Altogether we have a bijection from the F  with F ⊂ F  to the boundary components C of C(F) defined by NG (F  ) ∩ G
(F) = NG
(F) (C ) . In fact, this C is just C(F  ) because NG (F  ) ∩ G
(F) ⊂ NG
(F) (C(F  )) and NG
(F) (C ) is maximal. This proves (ii). Finally, if we have a Q-structure and F is Q-rational, then U(F) is the center of the unipotent radical of F, and hence is defined over Q, and G
(F) is the factor of Z(wF ) which acts non-trivially on U(F), and hence is defined over Q. Then, for all F  ⊃ F, F  is Q-rational ⇐⇒ N(F  ) is defined over Q ⇐⇒ N(F  ) ∩ G
(F) is defined over Q ⇐⇒ NG
(F) (C(F  )) is defined over Q ⇐⇒ C(F  ) is a Q-rational boundary comp. of C(F) .

Appendix: Connected components We need to know in Section 5 that the key diagram of Section 4, namely C(F) O ΦF

⊂

U(F) O ΦF

⊂ D(F) ⊂ Dˇ D+ 5 55  + 5   + 55  π  : p.h.s. for U(F)C + 5  F πF +D(F) + + pF : p.h.s. for V (F) +  F

158

III Compactifications of locally symmetric varieties

is, in fact, N(F)-equivariant, in addition to being N(F)o -equivariant. To check this, note that U(F), being the center of the unipotent radical of N(F)o , is invariant under all outer automorphisms of N(F)o , and hence is normal in N(F). Therefore conjugation by N(F) carries U(F)C into itself, and hence N(F) maps D(F) into itself. Since D(F) ∼ = D(F)/U(F)C , we may define the action of N(F) on D(F) to make πF equivariant. But πF was seen to be N(F)equivariant in Subsection 3.4, and hence pF is also N(F)-equivariant. Next define the action of N(F) on U(F) to be conjugation. Then the equivariance of ΦF follows from: Lemma 4.9 Let G ⊂ G be groups, G normal in G . Let f : X −→ Y be a G-equivariant map of G -spaces. If (i) G is transitive on X, and (ii) there is a point o ∈ X such that f (o) is the only point of Y fixed by Stab G (o), then f is G -equivariant. Proof Easy. Apply the lemma with G = N(F)o , G = N(F), X = D(F), Y = U(F), f = ΦF , o = ωF , noting that o0F ∈ U(F) is the only point fixed by K
. Finally, in view of the fact that the action of N(F) on D(F) maps D into itself, it follows that the action of N(F) on U(F) maps C(F) into itself. A few remarks on connected components may be in order. (1) By definition, G is the connected component both of Aut (D) and of G (R), (2) N(F) is the full subgroup of G fixing F, but may be of finite index in the corresponding subgroup of Aut (D) or in the real points of the algebraic subgroup N (F) ⊂ G (F). (3) By definition, we made Gh (F), G
(F), M(F), W (F) connected, getting N(F)o = [Gh · G
· M] ×W (F) . Then Gh (F) is the connected component of the real points of the algebraic subgroup Gh (F) of G , and G
(F) is the connected component of the real points of the algebraic subgroup G
(F) of G . There are projections Gh (F) → Aut (F), resp. G
(F) → Aut (C(F)), which identify the connected components Aut (F)o , resp. Aut (C(F))o , with Gh (F)/(center), resp. G
(F)/(center). However, W (F), being nilpotent, equals W (F)(R).

5 Statement of the Main Theorem

159

5 Statement of the Main Theorem Let G be a connected semi-simple linear algebraic group defined over Q such that G = G (R)o is the connected component of the group of automorphisms of a hermitian symmetric domain D. For a boundary component F of D, we get the associated groups N(F),W (F),U(F)

(see Section 4) ;

if F is rational, these are algebraic groups defined over Q. Let Γ ⊂ G be an arithmetic group, i.e., Γ ⊂ G (Q), and, for any faithful rational representation ρ : G −→ GL(n), the subgroup ρ (Γ) of GL(n, Q) is commensurable with ρ (G) ∩ GL(n, Z). If F is a rational boundary component of D, we let ΓF = Γ ∩ N(F) , ΓF = subgroup of elements of ΓF acting trivially by conjugation on u(F) , ΓF = group of automorphisms of u(F) induced by ΓF ; these map C(F) into itself , U(F)Z = Γ ∩U(F) . Note that we have an exact sequence of groups: 1 → ΓF → ΓF → ΓF → 1 . Let {σα } be a ΓF -admissible polyhedral decomposition of C(F) ⊂ U(F) (see Chapter II, Definition 4.10). We now construct a “partial compactification in the direction F” of D/U(F)Z , (D/U(F)Z ){σα } . Reconsider the fiber bundle from Section 4: π

1 D(F) (= D(F)/U(F)C ) , D(F)(= U(F)C · D)) −→

where π1 makes D(F) into a principal fiber bundle over D(F) with structure group U(F)C . π

1 From this, we get a quotient fiber bundle D(F)/U(F)Z −→ D(F) , where π 1 is a principal fiber bundle under the algebraic torus (over C)

T (F) = U(F)C /U(F)Z . The chosen collection {σα } defines an equivariant embedding (see Chapter I, Section 1) T (F) ⊂ T (F){σα } .

160

III Compactifications of locally symmetric varieties

We form (D(F)/U(F)Z ) ×T (F) T (F){σα } ; this is the fiber bundle over D(F) associated to π 1 with fiber T (F){σα } . Finally, define: (D/U(F)Z ){σα } = interior of closure of D/U(F)Z in (D(F)/U(F)Z ) ×T (F) T (F){σα } . Note that, since {σα } is a ΓF -invariant collection, the group ΓF is still acting on (D/U(F)Z ){σα } ; we will see later that, in fact, ΓF /U(F)Z acts properly discontinuously on (D/U(F)Z ){σα } . Moreover, recall from Section 4 that we have a map Φ : D(F) −→ U(F) such that (a) Φ is equivariant for N(F) · U(F)C , where N(F) acts on U(F) by inner automorphisms and ia ∈ iU(F) acts on U(F) by translation by a ; (b) Φ−1 (C(F)) = D ; (c) Φ induces a trivialization of the U(F)C -bundle π1 in the imaginary direction: ∼

D(F)/U(F) −→ U(F) × D(F) . (Φ,π1 )

Therefore, denoting as in Chapter I, Section 1 by T (F)c the maximal compact torus in T (F), i.e., U(F)/U(F)Z , we see that Φ induces a trivialization in the “absolute value direction” of the T (F)-bundle π 1 ; ∼ (d) [D(F)/U(F)Z ]/T (F)c −→ U(F) × D(F) . Moreover, Φ extends continuously to maps (which we also call Φ): Φ

(D(F)/U(F)Z ) ×T (F) T (F){σα } ∪





D(F)/U(F)Z

−→

Φ

U(F){σα } ∪ U(F)

by the definition Φ(x, y) = Φ(x) · ord(y) . Here “·” denotes the action of U(F) on the partial compactification U(F){σα } , and this definition is justified by the second equivariance assertion in (a). As in Section 1, define C(F) = interior of closure of C(F) in U(F){σα } .

5 Statement of the Main Theorem The quotient



(D(F)/U(F)Z ) ×T (F) T (F){σα }

161

< T (F)c

is a fiber bundle over D(F) with fiber U(F){σα } . It is, again via Φ, just the product U(F){σα } × D(F) . It therefore follows easily from (b) that (D/U(F)Z ){σα } can also be described as Φ−1 (C(F) ) as follows: Φ

(D/U(F)Z ){σα } = Φ−1 (C(F) ) −→ ∩ (D(F)/U(F)Z ) ×T (F) T (F){σα }

Φ

C(F) ∩

−→ U(F){σα } .

These maps will be useful later in the study of (D/U(F)Z ){σα } . The following definition describes the simplicial data on which our compactification will depend. Definition 5.1 A Γ-admissible collection of polyhedra {σα } is a collection of polyhedra 6 F7 σα ⊂ C(F) , one for every rational boundary component F of D, which are ΓF -admissible and which satisfy the following two compatibility conditions: (1) if F1 = γ · F2 with γ ∈ Γ, then : 9 : 9 σαF1 = γ · σαF2 via the natural isomorphism ∼

γ : C(F2 ) −→ C(F1 ) ; (2) if F1 ⊂ F 2 , then 9 : σαF2 is exactly the set of cones σαF1 ∩C(F2 ) (recall that C(F2 ) = C(F1 ) ∩U(F1 ) (see Subsection 4.4)). We can now formulate our main theorem. Theorem 5.2 (Main Theorem I) Let G be a semi-simple algebraic group defined over Q such that G = G (R)o is the connected component of the automorphism group of a hermitian symmetric domain D. Let Γ ⊂ G be an arithmetic group. Let {σα } be a Γ-admissible collection of polyhedra. Then there exists

162

III Compactifications of locally symmetric varieties

a unique Hausdorff analytic variety D/Γ containing D/Γ as an open dense subset and such that, for every rational boundary component F of D, there are open analytic morphisms πF making the following diagram commutative: D/U(F)Z  D/Γ



/ (D/U(F)Z ){σ F } α   πF  / D/Γ



and such that every point of D/Γ is in the image of one of the maps πF . Furthermore, D/Γ is a compact algebraic space. = = > (D/U(F)Z ) F . Then D/Γ is the quotient Proof of uniqueness Set D/Γ F { σα } = of D/Γ by an equivalence relation which is described by a closed graph (closed because D/Γ is a Hausdorff space): = × D/Γ = . Λ ⊂ D/Γ But because the πF are open, Λ is just the closure of the equivalence relation > defined on F D/U(F)Z by the action of Γ; this shows that D/Γ is unique. The fact that D/Γ is an algebraic space comes from the following by-product of the proof of the Main Theorem. Proposition 5.3 There exists a natural morphism from D/Γ to the Baily–Borel “minimal” compactification (D/Γ)∗ , inducing the identity morphism on D/Γ. Now, (D/Γ)∗ is a projective algebraic variety; hence this proposition implies that D/Γ is a Moishezon space, i.e., an algebraic space over C. Note also that, to prove the proposition, it suffices to construct a continuous map from D/Γ to (D/Γ)∗ : the fact that this map is holomorphic will then follow from the Riemann extension theorem. To construct D/Γ, we shall construct the equivalence relation Λ explicitly and show in Section 6 that Λ is closed. We use the following straightforward lemma. Lemma 5.4 Let F and F  be two rational boundary components such that F  ⊂ F. Then (1) U(F  ) ⊂ U(F) ⊂ N(F  ) and T (F  ) ⊂ T (F);  (2) T (F){σ F  } ∼ = T (F) ×T (F ) T (F  ){σ F  } , an open subset of T (F){σαF } ; α

α

5 Statement of the Main Theorem

163

(3) the quotient by the action of U(F)Z on the left factor of 

(D(F  )/U(F  )Z ) ×T (F ) T (F  ){σ F  } α

is canonically isomorphic to an open subset of (D(F)/U(F)Z ) ×T (F) T (F){σ F  } , α

which is an open subset of (D(F)/U(F)Z ) ×T (F) T (F){σαF } ; (4) this induces an e´ tale map (D/U(F  )Z ){σ F  } −→ (D/U(F)Z ){σαF } . α

= Let Now introduce the following equivalence relation on D/Γ. x1 ∈ (D/U(F1 )Z ){σ F1 } and x2 ∈ (D/U(F2 )Z ){σ F2 } . α

α

Then x1 ∼ x2 if and only if (1) there is a rational boundary component F and an element γ ∈ Γ, such that F1 ⊂ F

and

γ · F2 ⊂ F ;

(2) there is a point x ∈ (D/U(F)Z ){σαF } that (a) projects to x1 via the canonical map defined by Lemma 5.4, (D/U(F)Z ){σαF } −→ (D/U(F1 )Z ){σ F1 } , α

(b) projects to γ · x2 via the canonical map (D/U(F)Z ){σαF } −→ (D/U(γ · F2 )Z ){σ γ ·F2 } . α

The transitivity condition of this relation is an easy consequence of Lemma 5.4 plus the following result. Lemma 5.5 Let x ∈ (D/U(F)Z ){σαF } . Among all rational boundary components F  such that there is some x ∈ (D/U(F  )Z ){σ F  } projecting to x ∈ (D/U(F)Z ){σαF } via the canonical map α

(D/U(F  )Z ){σ F  } −→ (D/U(F)Z ){σαF } , α

there is a maximal one Fx , i.e., F  ⊂ F x .

164

III Compactifications of locally symmetric varieties

We will call this boundary component Fx the boundary component associated with x or say that x belongs to the Fx -stratum. Proof To the point x there is associated, in a canonical way, a polyhedron σα ⊂ U(F): it is the unique polyhedron such that, via the map Φ defined above, Φ(x) = z + ∞ · σα (in the notation of Chapter I). Let Cx ⊂ C(F) be the smallest boundary component such that σα ⊂ Cx . This defines a rational boundary component Fx such that F ⊂ F x , Cx = C(Fx ) ; see Subsection 4.4. It is immediate that there exists a point in (D/U(Fx )Z ){σ Fx } α projecting to x via the canonical map (D/U(Fx )Z ){σ Fx } −→ (D/U(F)Z ){σαF } . α

This argument also shows that Fx is the maximal element with the required properties.

6 Proof of the Main Theorem 6.1 Let G be a connected semi-simple algebraic group defined over Q such that D = G (R)o /K is a bounded symmetric domain. Let D∗ be the union of D and its rational boundary components. In Chapter II, Subsection 4.1, we defined Siegel sets Sω ⊂ D, associated to a minimal Q-parabolic subgroup P ⊂ G and a relatively compact subset ω ⊂ P(R)o . Our first goal is to analyze the “Satake topology” on D∗ , introduced by Satake (see Baily and Borel [1], §4.8), using Siegel sets. The Satake topology is a topology on D∗ which is different from the topology induced via the Harish-Chandra embedding from the vector space topology on p+ . To define it, take any arithmetic subgroup Γ ⊂ G (Q) and choose any fundamental set for Γ: Ω = C · Sω , where C ⊂ G (Q) is a finite subset and where Sω is a Siegel set with respect to a minimal Q-parabolic. A fundamental system of neighborhoods of x ∈ D∗ is, by definition, given by all subsets U ⊂ D∗ such that Γx ·U = U , where Γx = {γ ∈ Γ | γ · x = x} ,

6 Proof of the Main Theorem

165

and such that γ · U ∩ Ω is a neighborhood of γ · x in Ω (for the topology on Ω induced from D ⊂ p+ ), whenever γ · x ∈ Ω. The topology is characterized by the following theorem. Theorem 6.1 The Satake topology is the unique topology on D∗ having the following properties: (i) it induces the natural topology on D and on the closure Sω of any Siegel set Sω ; (ii) the group G (Q) acts continuously on D∗ ; (iii) if x, x ∈ D∗ are not equivalent with respect to the action of an arithmetic group Γ ⊂ G (Q), then there exist neighborhoods U of x and U  of x such / that Γ ·U ∩U  = 0; (iv) let Γ ⊂ G (Q) be an arithmetic group. For every x ∈ D∗ , there exists a fundamental set of neighborhoods {U} of x such that

γ ·U = U if γ · x = x , γ ·U ∩U = 0/ if γ · x = x . The Baily–Borel compactification (D/Γ)∗ of D/Γ is the quotient (D/Γ)∗ = D∗ /Γ , equipped with the quotient topology. Baily and Borel [1] proved: Theorem 6.2 The compactification (D/Γ)∗ is a compact Hausdorff space containing D/Γ as an open dense subset. The compactification (D/Γ)∗ is a finite union of subspaces of the form F/ΓF , where F is a rational boundary component. The closure of F/ΓF in (D/Γ)∗ is the union of F/ΓF and of subspaces  F /ΓF  , of strictly smaller dimension. We wish to describe explicitly the Satake topology in terms of the coordinates on D given by its presentation as a Siegel domain, and, in particular, we want to show that a fundamental system of open sets can be given using the concept of rational core of a cone (see Chapter II, Subsection 5.1). The result is as follows. Theorem 6.3 Let F be a rational boundary component of D and let x ∈ F. Let

166

III Compactifications of locally symmetric varieties

C0 (F) ⊂ C(F) be a rational core. ⎡ there exists a ⎢ neighborhood V of x ∈ D∗ ⎢ ⎣ in the Satake topology such that U ⊃ V ∩ D

Let U ⊂ D be an open set. Then ⎤ ⎡ there exists a ⎥ ⎢ neighborhood E of x ∈ F ⎥ ⇐⇒ ⎢ ⎦ ⎣ and 1 ≤ t < ∞ , such that U ⊃ πF−1 (E) ∩ Φ−1 F (tC0 )

⎤ ⎥ ⎥. ⎦

Proof Fix an arithmetic group Γ ⊂ G (Q), and let Γx = {γ ∈ Γ | γ x = x}. On the LHS, we can assume V is Γx -invariant; on the RHS, we can assume C0 is ΓF -invariant and E is Γx -invariant. Thus we are comparing Γx -invariant open subsets of D, namely V ∩ D and πF−1 (E) ∩ Φ−1 F (C0 ), so we may as well assume U is Γx -invariant. Similarly, if G = G1 × G2 over Q, and this corresponds to D = D1 × D2 , then the Satake topology on D is the product of the Satake topology on D1 and on D2 ; hence, both the sets V ∩ D and the sets

πF−1 (E) ∩ Φ−1 F (C0 ) can be assumed to be products. Therefore we may as well assume U is a product too, and thus the result for D1 and D2 implies it for D. So we may assume G is Q-simple. We start on the left: U ⊃ V ∩ D ⇐⇒ for all Siegel sets Sω ⊂ D with x ∈ Sω , U ∩ Sω is a neighborhood of x in Sω . Here we mean any Siegel set Sω with respect to any minimal Q-parabolic P. This is straightforward. Next we need a lemma about when x ∈ Sω . Lemma 6.4 If Sω is a Siegel set with respect to P, and F is a boundary component of D, then Sω ∩ F = 0/ ⇐⇒ P ⊂ N (F) and F is rational . Proof Let A ⊂ P be a maximal Q-split torus, A = A (R)o and A+ the positive piece, see Chapter II, Definition 4.1. Let K be a maximal compact subgroup such that Lie K ⊥ Lie A, and let p be fixed by K. Then it suffices to show A+ p ∩ F = 0/ ⇐⇒ P ⊂ N (F) , F rational , because, if ω ⊂ P(R) is compact with Sω = ω · A+ p, then ω leaves fixed every F such that P ⊂ N (F). Let s = dim A, and let 1 1 1 2 (β1 − β2 ), 2 (β2 − β3 ), . . . , 2 (βs−1 − βs ), βs ,

or 12 βs

be the simple positive roots (ordering defined by P) as usual. We constructed

6 Proof of the Main Theorem

167

in Subsection 3.5 symmetric holomorphic maps f

1 −→ D ∩ f2 −→ Dˇ

Hs ∩ (P1 )s associated to

ϕ : U 1 × SL(2, R)s → G with ϕ (1, diag. matrices) = A and with     t1 0 ts x = ϕ 1, , . . . , 0 t1−1 0

0

 ∈A,

ts−1

such that βi (x) = ti2 . Then A+ is the image of elements     ts 0 t1 0 ,..., , 0 t1−1 0 ts−1 where t1 ≥ t2 ≥ · · · ≥ ts ≥ 1. Then A+ p = f2 ({(ix1 , . . . , ixs ) | ∞ ≥ x1 ≥ · · · ≥ xs ≥ 1}) , i.e., A+ p meets the boundary components F1 , . . . , Fs , where f2 (∞, i, . . . , i) ∈ F1 , f2 (∞, ∞, i, . . . , i) ∈ F2 , . . . , f2 (∞, ∞, . . . , ∞) ∈ Fs . Now, if

 wi (t) = ϕ (1, %

t 0 0 t −1



 ,..., &'

t 0 0 t −1

 , I, . . . , I) , (

i factors

then all but the i’th simple root vanishes on wi ; hence, P = P(w−1 1 ) ∩ ··· ∩ ) expresses P as an intersection of maximal Q-parabolics, and it is P(w−1 r immediate that Pi = N (Fi ). Next, write out the algebraic connected component of N (F) as Gh (F) · G
(F) · W (F) , where all these factors are defined over Q, Gh (F)(R)o = Gh (F) · M(F) in the notation of Section 4, and W is the unipotent radical. Then a minimal Qparabolic P ⊂ N (F) is equal to h · P
· W , P =P h ⊂ Gh and P
⊂ G
are minimal Q-parabolics. Then, for any basewhere P point o ∈ D, a cofinal collection of Siegel sets for P is given as Sω = (ωW · ωh · ω
) · A+ · o ,

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III Compactifications of locally symmetric varieties

where h (R)o , ω
⊂ P
= P
(R)o , ωW ⊂ W = W (R) ωh ⊂ Ph = P are compact, and where A ⊂ P is a maximal Q-split torus conjugate to the original maximal split torus in P, with Lie A ⊥ Stab (o), and where   β (g) ≥ 1 for all positive roots β of A + o . A = g ∈ A = A (R) | i.e., roots occurring in Lie P Note that A = Ah · A
, where Ah ⊂ Ph and A
⊂ P
are conjugates of the maximal h and P
. Moreover, the root system of A (written multiplicaQ-split tori in P tively, i.e., as a subset Φ ⊂ Hom (A, R>0 )) is given by ±1/2 ±1/2

±1/2

βi±1 , βi βj (plus possibly βi 1≤i≤s 1≤i< j≤s 1≤i≤s

),

with simple positive roots 1/2 −1/2

β1 β2

1/2 −1/2

, β2 β3

1/2 −1/2

, . . . , βs−1 βs

1/2

, and βs or βs

,

and with Ah = {g ∈ A | β1 (g) = · · · = βu (g) = 1} A
= {g ∈ A | βu+1 (g) = · · · = βs (g) = 1} . Our next step is the following lemma. Lemma 6.5 Assume ω
is large enough so that x ∈ IntF (ω
· πF (o)). Let Un ⊂ Gh (F) be sets such that Un · πF (o) is a fundamental system of neighborhoods of x ∈ F, for n = 1, 2, . . .. Then the intersection with Sω of a fundamental system of neighborhoods of x ∈ Sω is given by Sn,t = ωW ·Un · ω
· A
,t · o , where A
,t = {x ∈ A
| β1 (x) ≥ β2 (x) ≥ · · · ≥ βu (x) ≥ t} . Proof Recall  from Section 3that  we can map SL(2, R)u −→ G, taking diagonal t1 0 , . . . to elements x ∈ A
with βi (x) = ti2 , and from matrices δ = 0 t1−1 this obtain a homomorphism

ϕ : Gh (F) × SL(2, R)u −→ G

6 Proof of the Main Theorem

169

plus a symmetric holomorphic map for ϕ : F × Hu ∩ ˇ F × (P1 )u

f

1 −→

f

2 −→

D ∩ ˇ D.

Consider the map g : ωW × ω
× F × A
,1 −→ D , where A
,t = {t = (t1 , . . . ,tu ) | ∞ ≥ t1 ≥ t2 ≥ · · · ≥ tu ≥ t 1/2 } , given by g(u, v, a, it) = u · v · f2 (a, it). Note that Im g is compact, hence closed, and that Sω ⊂ Im g. Therefore, Sω ⊂ Im g, which is why we have introduced g. On the other hand, f2−1 (x) = {(x, ∞, . . . , ∞)}, and hence g−1 (x) = ωW × ω
× {x} × {(∞, . . . , ∞)} . Since ωW and ω
are compact, any open subset of ωW × ω
× F × A
,1 containing g−1 (x) contains

ωW × ω
× {Un · πF (o)} × A
,t for some n and t; and, since g is a proper map, the images Sn,t by g of these sets are a fundamental system of neighborhoods in Im g of x. But, for n large, these are subsets of Sω , hence the lemma follows. Corollary 6.6 Let U ⊂ D be an open subset. Then there exists an open neighborhood V of x ∈ D∗ in the Satake topology with U ⊃ V ∩ D if and only if, for all minimal Q-parabolics P
⊂ G
and all compact subsets ωW ⊂ W and ω
⊂ P
, U ⊃ Sn,t for some n ≥ 1, t ≥ 1, where Sn,t are defined as in Lemma 6.5 above. We now use the product representation D∼ = F ×C(F) ×W (F) introduced in Subsection 4.3. In this representation Sn,t = (Un · πF (o)) × (ω
· A
,t · ΩF ) × ωW ; here ΩF is the basepoint of C(F). If En = Un · πF (o), then, by definition, the

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III Compactifications of locally symmetric varieties

En are a fundamental set of neighborhoods of x in F. Now U is Γx -invariant, and hence U ⊃ Sn,t implies U ⊃ En × (ΓF · ω
· A
,t · ΩF ) × (Γ ∩W ) · ωW . If ωW is large enough, (Γ ∩W ) · ωW = W , so there is no condition on the third factor. Moreover, if we take a finite union of the sets in the middle factor, one for each ΓF -conjugacy class of minimal Q-parabolics P
⊂ G
, while taking the ω
large enough, then

i

while

i

(i)

(i)

ΓF · ω
· A
· ΩF = C(F) ,

(i)

(i)

ΓF · ω
· A
,t · ΩF = t ·C0 (F) ,

where C0 (F) lies between two rational cores of C(F). Thus the condition on U is equivalent to U ⊃ πF−1 (En ) ∩ Φ−1 F (tC0 (F)) for some n ≥ 1,t ≥ 1 . This concludes the proof of Theorem 6.3. In fact, if one adjoins to πF−1 (E) ∩ Φ−1 F (tC0 (F)) a suitable set of elements of the rational boundary components F  with F  ⊃ F, then one can prove that these extended sets, as F, E, and t vary (E is open and relatively compact in F), form a basis of open sets for the Satake topology on D∗ . However, we do not need this additional fact. Combining the above proof with the results of Chapter II, Subsection 4.3, we find: Corollary 6.7 Every Siegel set Sω ⊂ D is covered by a finite number of sets of the form E × (C0 ∩ σαF ) × ωW ⊂ F ×C(F) ×W (F) ∼ =D, where E ⊂ F and ωW ⊂ W (F) are compact, C0 ⊂ C(F) is a rational core, and σαF is one of the polyhedra in our decomposition of C(F).

6.2 The next step is to define a map = = f : D/Γ

 F

(D/U(F)Z ){σαF } −→ (D/Γ)∗ .

6 Proof of the Main Theorem

171

Let x ∈ (D/U(F)Z ){σαF } and let Fx be its associated rational boundary component; hence there exists a point x ∈ (D/U(Fx )Z ){σ Fx } projecting to x via the α canonical map (D/U(Fx )Z ){σ Fx } −→ (D/U(F)Z ){σαF } . α

We then define the image via f of x as the image of x via the succession of maps πF

x (D/U(Fx )Z ){σα } −→ Fx ⊂ D∗ −→ (D/Γ)∗ .

The following facts are immediate. (1) The definition of f (x) is independent of the choice of x . (2) The restriction f |D is just the natural projection from D to D/Γ ⊂ (D/Γ)∗ . = that are equivalent under the equivalence relation intro(3) Two points in D/Γ duced in Section 5 have the same image in (D/Γ)∗ . Indeed, Fγ ·x = γ · Fx , for all γ ∈ Γ. (4) f |(D/U(F)Z ){σα } factors as f

F D∗ /U(F)Z −→ (D/Γ)∗ . (D/U(F)Z ){σα } −→

Proposition 6.8 If (D/Γ)∗ and D∗ /U(F)Z are given the Satake topology, the map = −→ (D/Γ)∗ f : D/Γ and the maps fF through which it factors are continuous. Proof It is clearly sufficient to show that fF is continuous. We use the following elementary lemma. Lemma 6.9 Let X and Y be two topological spaces which are second countable, Y also being a T3 -space, and let X ⊂ X, resp. Y ⊂ Y , be an open and dense, resp. open, subset. Let f : X −→ Y be a map such that f (X) ⊂ Y . Assume that, for every x ∈ X and y = f (x) ∈ Y , and for any neighborhood V of y, f −1 (V ∩Y ) contains a set of the form U ∩ X, where U is a neighborhood of x. Then f is continuous. Proof We have to show that if x1 , x2 , . . . −→ x ,

172

III Compactifications of locally symmetric varieties

then y1 = f (x1 ), y2 = f (x2 ), . . . −→ y = f (x) . Suppose first that xi ∈ X ⊂ X. Let V be any neighborhood of y and choose a neighborhood U of x as in the hypotheses of the lemma. Then, for large enough i, we have xi ∈ U ∩ X . Hence we get yi ∈ f (U ∩ X) ⊂ V for i  0, i.e., yi −→ y. Now consider the general case. Since X ⊂ X is dense, we may find convergent sequences xi1 , xi2 , . . . −→ xi with xi j ∈ X . Let V be an arbitrary open neighborhood of y and let V1 be a neighborhood of y such that V1 ⊂ V 1 ⊂ V . (We use the fact that Y is a T3 -space.) Denote by U, resp. U1 , the neighborhoods of x whose existence is guaranteed by hypothesis. Then, for i  0, we have xi ∈ U1 . Since U1 is a neighborhood of xi for i  0, we have xi j ∈ U1 for i  0, j > J(i) . This shows that yi j ∈ V1 for i  0, j > J(i). Since, by the first part of the proof, the sequence yi j converges to yi we conclude that yi ∈ V 1 ⊂ V for i  0. This finishes the proof. To verify the assumptions of the lemma for fF , we use induction on codim F. Therefore we may assume fF  continuous for all F  with F  ⊃ F. Then to check fF itself is continuous, we need only verify the assumption for x in the F-stratum of (D/U(F)Z ){σαF } . Let y = fF (x) ∈ F. Then a fundamental system of neighborhoods of y in the Satake topology, intersected with D, is given, according to Theorem 6.3, by the sets πF−1 (E) ∩ Φ−1 F (C0 ), where E ⊂ F is a neighborhood of x and where C0 is a rational core of C(F). But πF extends to a continuous map

π F : (D/U(F)Z ){σαF } → F , and ΦF extends to a continuous map ΦF : (D/U(F)Z ){σαF } −→ C(F) . (Recall that C(F) is the interior of the closure of C(F) in U(F){σαF } .) Moreover, C0 contains some cylindrical set a +C(F) ⊂ C(F), and (a +C(F)) := interior of closure of a +C(F) in U(F){σαF }

6 Proof of the Main Theorem

173

is an open subset of C(F) containing all points of U(F){σαF } in the F-stratum, i.e., points x + ∞ · σαF , where σαF meets C(F) (or σαF ⊂ ∂ C(F)). Therefore πF−1 (E) ∩ Φ−1 F (C0 ) (inverse image in D/U(F)Z ) contains  −1   D/U(F)Z ∩ π −1 , F (E) ∩ ΦF (a +C(F)) % &' ( inverse image in (D/U(F)Z ){σ F } α

and the latter terms are neighborhoods of x in (D/U(F)Z ){σαF } .

6.3 We now show that the equivalence relation Λ is closed. Since any pair (y, z) of = is a limit of equivalent points (yi , zi ), where equivalent points in D/Γ = , yi ∈ (D/U(F1 )Z ) ⊂ D/Γ = , zi ∈ (D/U(F2 )Z ) ⊂ D/Γ it suffices to show that, for any two sequences yi , zi ∈ D, where yi = γi zi with γi ∈ Γ, such that yi mod U(F1 )Z −→ y ∈ (D/U(F1 )Z ){σ F1 } , α

zi mod U(F2 )Z −→ z ∈ (D/U(F2 )Z ){σ F2 } , α

it follows that (y, z) ∈ Λ. But, by Proposition 6.8, the images of y and z in (D/Γ)∗ are equal, and hence lie in the same stratum F/ΓF of (D/Γ)∗ . By the definition of f , it follows that δ1 F ⊃ F1 , δ2 F ⊃ F2 , for some δ1 , δ2 ∈ Γ, and where y lies in the δ1 F-stratum, and z lies in the δ2 F-stratum. Replacing yi , y by δ1−1 yi , δ1−1 y and zi , z by δ2−1 zi , δ2−1 z, we can assume F ⊃ F1 and F ⊃ F2 . Next, lift y, z to points y∗ , z∗ ∈ (D/U(F)Z ){σαF } . Since (D/U(F)Z ){σαF } −→ (D/U(Fi )Z ){σ Fi } α

is obtained by dividing by U(Fi )Z plus an open immersion, it follows that y∗ = lim λi · yi , λi ∈ U(F1 )Z , i−→∞

∗

z = lim µi · zi , µi ∈ U(F2 )Z . i−→∞

So, replacing yi by λi · yi , and zi by µi · zi , and y by y∗ , and z by z∗ , it suffices to prove the assertion in the special case: F1 = F2 : call this just F ; y, z ∈ F-stratum of (D/U(F)Z ){σαF } .

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III Compactifications of locally symmetric varieties

Now consider the following continuous maps: (D/U(F)Z ){σαF }

f

F −→

D∗ /U(F)Z ∪ F

−→ (D/Γ)∗ ∪ −→ F/ΓF .

Since y, z have the same image in (D/Γ)∗ , their images in D∗ /U(F)Z differ by γ ∈ ΓF . Again, replacing zi , z by γ −1 zi , γ −1 z, we may assume that y and z have the same image p ∈ F ⊂ D∗ /U(F)Z . Let U1 ⊂ D∗ be a neighborhood of p in / Γ p . Then the Satake topology such that Γ p ·U1 = U1 and γ U1 ∩U1 = 0/ if γ ∈ U2 = fF−1 (U1 /U(F)Z ) is an open subset of (D/U(F)Z ){σαF } containing y and z. Therefore yi mod U(F)Z and zi mod U(F)Z lie in U2 if i  0. In other words, the points yi , zi in D lie in U1 if i  0. Now yi = γi zi , for γi ∈ Γ, so, by the assumption on U1 , we have γi ∈ Γ p , hence γi ∈ ΓF . This reduces us to the following assertion. Proposition 6.10 The action of ΓF /U(F)Z on (D/U(F)Z ){σαF } is properly discontinuous. Consequently, ΓF -equivalence is a closed equivalence relation in (D/U(F)Z ){σαF } × (D/U(F)Z ){σαF } . Proof We use the double fibration: π

p

F F (D/U(F)Z ){σαF } −→ D(F) −→ F.

Recall that U(F) ⊂ W (F) ⊂ N(F) are all defined over Q; hence, the quotient group by which N(F) acts on U(F), namely the image of N(F) −→ Aut (U(F)) , is also defined over Q. Let N(F) = Ker (N(F) −→ Aut (U(F))); this is defined over Q. Finally, N(F) /W (F) is defined over Q and equals Gh up to compact factors. Therefore all the following are arithmetic, hence discrete subgroups: ΓF ⊂ N(F) , ΓF ⊂ N(F) , Γ ∩W (F) = W (F)Z ⊂ W (F) , Γ ∩U(F) = U(F)Z ⊂ U(F) , ΓF  ΓF /W (F)Z

⊂ Aut (U(F)) , ⊂ Gh (F) · (comp. factors) .

We use the following elementary lemma.

6 Proof of the Main Theorem

175

Lemma 6.11 Let X,Y be two Hausdorff topological spaces acted on by groups G, resp. H, and let

φ : G −→ H , f : X −→ Y be a homomorphism and an equivariant continuous map. If H acts properly discontinuously on Y and Ker φ acts properly discontinuously on X, then G acts properly discontinuously on X. We deduce: (1) ΓF /W (F)Z acts properly discontinuously on F; hence, since pF is a principal fiber bundle with structure group V (F), (2) ΓF /U(F)Z acts properly discontinuously on D(F) by the lemma; hence, (3) ΓF /U(F)Z acts properly discontinuously on (D/U(F)Z ){σα } . Next we make use of the ΓF -equivariant map ΦF : (D/U(F)Z ){σαF } −→ C(F) . Since, by the method of proof of Theorem 1.4, ΓF acts properly discontinuously on C(F) , the lemma plus (3) above give the proposition. = by the equivaThis proves that Λ is closed. Let D/Γ be the quotient of D/Γ lence relation Λ. The above proof shows that D/Γ is locally isomorphic at all points of the F-stratum to ? (D/U(F)Z ){σαF } (ΓF /U(F)Z ) . Since this is an analytic space modulo a properly discontinuous group action, D/Γ has an analytic structure, and the maps

πF : (D/U(F)Z ){σαF } −→ D/Γ are open. This, together with the fact that the maps fF are continuous, also shows that the natural map from D/Γ to (D/Γ)∗ is continuous. It remains to check that D/Γ is compact. Now, D/Γ it is covered by the images of finitely many Siegel sets hence, by Corollary 6.7, is covered by the images of finitely many sets of the form S = E × (C0 ∩ σα ) × ωW ⊂ F ×C(F) ×W (F) ∼ =D. We claim that the closures of the images of these sets in (D/U(F)Z ){σαF } are

176

III Compactifications of locally symmetric varieties

already compact, and hence so are the closures of their images in D/Γ. To see this, recall that, dividing by the compact group T (F)c , ? (D/U(F)Z ){σα } T (F)c ∼ = C(F) × D(F) ∼ = F ×C(F) ×V (F) . Now, C0 ∩ σα has compact closure in C(F) , and this decomposition is related to our previous one by D ↓ D/U(F) ∩ (D/U(F)Z ){σαF } /T (F)c

∼ = F ×C(F) ×V (F) ×U(F) ↓ ∼ F ×C(F) ×V (F) = ∩ ∼ F ×C(F) ×V (F) , =

so the closure of the image of our set S in (D/U(F)Z ){σαF } /T (F)c is compact, hence the same happens in (D/U(F)Z ){σαF } .

7 An intrinsic form of the Main Theorem Recall from Borel [2], §17.1, the concept of a neat subgroup Γ ⊂ G (C): this is a subgroup consisting of elements g such that, for one faithful representation,

ρ : G −→ GL(n) (and hence for all representations ρ ), the group inside C∗ generated by the eigenvalues of ρ (g) is torsion-free. Such a Γ is itself torsion-free. But, even more, for every pair of subgroups H1 ⊂ H2 ⊂ G , with H1 normal in H2 , the group ? Γ ∩ H2 (C) Γ ∩ H1 (C) is also torsion-free. We now return to the set-up of Sections 5 and 6: G is semi-simple, defined over Q, and G = G (R)o is the connected group of automorphisms of a bounded symmetric domain D; Γ ⊂ G (Q) ⊂ G is an arithmetic subgroup. We will assume throughout this section that Γ is also neat. Since by [2], Proposition 17.6, any Γ has a neat subgroup Γ1 ⊂ Γ of finite index, this is not too restrictive. It has the effect that not only does Γ act freely on D, ∼ hence Γ −→ π1 (D/Γ), but that also for all rational boundary components F, the group Im (ΓF −→ Aut (F)) acts freely on F, and ΓF acts freely on C(F),

7 An intrinsic form of the Main Theorem

177

and ΓF /U(F)Z acts freely on (D/U(F)Z ){σαF } ; hence (D/U(F)Z ){σαF } −→ D/Γ is e´ tale. Borel [3] showed that the Baily–Borel compactification (D/Γ)∗ enjoys the following property: Every holomorphic map ˚ k × ∆n−k −→ D/Γ f : (∆) extends to a holomorphic map f ∗ : ∆n −→ (D/Γ)∗ . Here, as usual, ∆ is the unit disc, and ∆˚ = ∆ \ {0}. Letting exp : H −→ ∆˚ be the universal covering as usual, such an f lifts to an equivariant map f˜ and a homomorphism φ :

φ : Zk −→ Γ , f˜ : Hk × ∆n−k −→ D . Let

γi = φ (. . . , 0, %&'( 1 , 0, . . .) ,

(7.1)

place i

i.e., γi is the image under φ of the covering map z −→ z + 1 in the i’th factor H. Now, f will not in general extend to f : ∆n −→ D/Γ , and we will analyze here when it does. Enlarge H to H∗ = H ∪ {i∞} , where a fundamental system of neighborhoods of i∞ is given by Wc = {z ∈ H | Im z > c} ∪ {i∞} . Extend the map exp : H → ∆˚ to exp : H∗ → ∆ by exp(i∞) = 0. As a preliminary remark, we have Proposition 7.1 The map f ∗ above lifts to a continuous map f˜∗ : (H∗ )k × ∆n−k −→ D∗ , extending the map f˜ : Hk × ∆n−k −→ D, where here we put on D∗ the Satake topology (see Subsection 6.1).

178

III Compactifications of locally symmetric varieties

Proof Let x ∈ (H∗ )k × ∆n−k , let exp(x) be its image in ∆k × ∆n−k , and let P = f ∗ (exp x). Let Q ∈ D∗ lie over P, and let U1 be a neighborhood of Q in the Satake topology such that, for all γ ∈ Γ, we have γ U1 = U1 if γ Q = Q, but γ U1 ∩ U1 = 0/ if γ Q = Q. Let U2 be the image of U1 in (D/Γ)∗ . Then exp(x) has a neighborhood V2 such that f ∗ (V2 ) ⊂ U2 . Let V1 be the component of exp−1 (V2 ) containing x. Then 

 γ U1 . f˜ V1 ∩ (Hk × ∆n−k ) ⊂ γ ∈Γ

We may assume V1 is a product of sets of the type Wc and discs in C, and hence that V1 ∩ (Hk × ∆n−k ) is still connected. Therefore   f˜ V1 ∩ (Hk × ∆n−k ) ⊂ γ ·U1 for some γ ∈ Γ. We then define f˜∗ (x) = γ · Q . It is easy to check that this f˜∗ is continuous. Consider the point P = f˜∗ (i∞, . . . , i∞, 0, . . . , 0) in D∗ . Let F be the unique boundary component containing it. Then Im f˜∗ ⊂ D ∪



F F 1 ⊃F 1

.

Note that, by continuity, γi P = P, for all i = 1, . . . , k, hence γi ∈ ΓF . The next result was suggested to us by P. Deligne; it generalizes a result of Y. Namikawa [9]. Theorem 7.2 In the notation introduced above, γi ∈ C(F) ∩U(F)Z for all i = 1, . . . , k; hence, f˜ induces a holomorphic map ˚ k × ∆n−k −→ D/U(F)Z , f 0 : (∆) and the following statements are equivalent: (i) all γi are in one and the same σαF ⊂ C(F) for some α ; (ii) f 0 extends to a holomorphic map 0

f : ∆n −→ (D/U(F)Z ){σαF } ; (iii) f extends to a holomorphic map f : ∆n −→ D/Γ . ˚ Proof We prove first that γi ∈ U(F)Z : in fact, restricting f to the i’th factor ∆, holding all other variables constant, we get a map fi : ∆˚ −→ D/Γ

7 An intrinsic form of the Main Theorem

179

that extends to ∆ −→ (D/Γ)∗ , and hence has no essential singularity at 0 ∈ ∆. Since dim ∆ = 1 and D/Γ is compact, fi also extends to a map f i : ∆ −→ D/Γ . Also f i (0) maps to a point of (D/Γ)∗ which is in the image by D∗ −→ (D/Γ)∗ of a stratum F1 , where F 1 ⊃ F. Therefore f i (0) is in the image of the map (D/U(F)Z ){σαF } −→ D/Γ . Since this map is e´ tale, it follows that f i lifts, in a small neighborhood ∆ ⊂ ∆ of 0, to f i : ∆ −→ (D/U(F)Z ){σαF } , 0

and hence fi lifts to fi0 : ∆˚  = ∆ \ {0} −→ D/U(F)Z .   0 : π (∆ ˚  ) −→ π1 (D/U(F)Z ) , so γi ∈ U(F)Z . But γi ∈ Im fi,∗ 1 In particular, we know now that f 0 is defined. Next, we show that f 0 always 0 extends to what we will call here a semi-proper meromorphic map f from ∆n to (D/U(F)Z ){σαF } ; i.e., a many-valued map whose graph 0

Gr ( f ) ⊂ ∆n × (D/U(F)Z ){σαF } is a closed analytic subset, mapping properly to ∆n , and restricting to f 0 on ˚ k × ∆n−k . To see this, consider the following diagram: (∆)

∆n × D/Γ proper, birational

 Gr ( f )

 ⊃ Gr ( f ∗ ) ⊃

 Gr ( f )

∼

 ∆n × (D/Γ)∗

 ⊃ Gr ( f ) ⊃

 ∆n

∼

e´ tale

∼



Gr ( f 0 ) ∼

∆n × (D/U(F)Z ){σαF } ⊃ Gr ( f 0 ) ⊃

 ˚ k × ∆n−k ⊃ (∆)

Since f ∗ exists, it follows that Gr ( f ∗ ) is a closed analytic set isomorphic to ∆n . Since D/Γ −→ (D/Γ)∗ is proper and birational, the closure Gr ( f ) of Gr ( f ) is also analytic, and maps properly to Gr ( f ∗ ), hence to ∆n . Since (D/U(F)Z ){σαF } −→ D/Γ is e´ tale, the closure Gr ( f 0 ) is e´ tale over Gr ( f ) and

180

III Compactifications of locally symmetric varieties

in particular is analytic. Since Gr ( f 0 ) = Gr ( f ) is open dense in both Gr ( f 0 ) and Gr ( f ), in fact Gr ( f 0 ) is just an open subset of Gr ( f ). Finally, Gr ( f 0 ) = Gr ( f ). To see this, let xi ∈ Hk × ∆n−k be a sequence with xi −→ x ∈ (H∗ )k × ∆n−k , such that f (exp xi ) −→ y ∈ D/Γ. But f˜(xi ) −→ f˜∗ (x) and f˜∗ (x) ∈ F1 for some rational boundary component F1 with F 1 ⊃ F. Then the image of y in (D/Γ)∗ is in the image of F1 , so y lifts to a point z in the F1 -stratum of (D/U(F1 )Z ){σ F1 } ; we may choose z so that z α maps to f˜∗ (x)modU(F1 )Z via (D/U(F1 )Z ) F1 −→ D∗ /U(F1 )Z . Now, since { σα }

(D/U(F1 )Z ){σ F1 } −→ D/Γ is e´ tale, there is a unique sequence zi ∈ D/U(F1 )Z α

converging to z, with zi over f (exp xi ). In fact, {zi } must be equal to γ · f˜(xi ) modU(F1 )Z , for some γ ∈ Γ. But then, in D∗ /U(F1 )Z , i→∞

Im (z) = f˜∗ (x) modU(F1 )Z

=

Im (γ · f (xi )) −→ i→∞ γ · Im ( f˜(xi )) −→

γ · f˜∗ (x) modU(F1 )Z ,

i.e., γ · f˜∗ (x) = f˜∗ (x) modU(F1 )Z . Thus γ ∈ ΓF1 . Replace z by γ −1 z; then f 0 (exp xi ) = f˜(xi ) modU(F1 )Z converges to z, so (exp(x), z) ∈ Gr ( f 0 ), as required. ∼ The equivalence of (ii) and (iii) is now clear: since Gr ( f 0 ) −→ Gr ( f ), we see that f is single-valued if and only if f 0 is single-valued. To bring in (i), use the following maps: (D/U(F)Z ){σαF } ⊂ (D(F)/U(F)Z ){σαF } π 1 T (F)−bundle

 D(F)  F

For all x ∈ ∆n , f (x) is a connected compact analytic set. But F is a bounded domain and D(F) is an affine bundle over F; so D(F) contains no positivedimensional compact analytic sets. Thus π 1 ◦ f is single-valued. We can now apply: Lemma 7.3 Let (1) g : ∆n −→ Y be an analytic map; (2) π : E −→ Y be a topologically trivial principal analytic fiber bundle with fiber an algebraic torus T ; let T ⊂ X{σα } be a torus embedding and let π : E{σα } −→ Y be the associated fiber bundle;

7 An intrinsic form of the Main Theorem

181

˚ k × ∆n−k −→ E be a lifting of g, which extends to a semi-proper (3) h : (∆) meromorphic map ∆n −→ E{σα } . Then 

h extends to h : ∆n −→ E{σα }



⎡

⎤ the monodromy elements ⇐⇒ ⎣ γi ∈ π1 (T ) = Ker (π1 (E) −→ π1 (Y )) ⎦ all lie in one σα

and, in this case, h extends to h : ∆n → Eσα , where α is as in the assertion on ˚ k ). the right. Here the γi are the images of the generators in π1 ((∆) Proof Firstly, the result is local on ∆n , so it is also local on Y . So we may assume E ∼ = T ×Y . Following h by projection onto T , it suffices to prove the lemma when Y is a point and E = T . Since h is semi-proper meromorphic, for all β ∈ M(T ) = Hom (T, Gm ), we have that β ◦ h is a meromorphic function on ∆n and h extends to h : ∆n → Eσα ⇐⇒ for all β ∈ σˇ α ∩ M(T ),

β ◦ h is holomorphic on ∆n . But for all β ∈ M(T ), β ◦ h has zeroes and poles only on the coordinate hyperplanes Hi ⊂ ∆n , so we may write k

(β ◦ h) = ∑
i (β ) · Hi , i=1

where
i is a linear function from M(T ) to Z. But then arg (β ◦ h) changes by 2π
i (β ) when going in a loop around Hi ; i.e., arg β changes by 2π
i (β ) when traversing the loop γi in T . But, identifying π1 (T ) with N(T ), hence with Hom (M(T ), Z), arg Xβ in fact changes by 2π γ , β  in a loop homologous to γ . Thus
i (β ) = γi , β  . Therefore

β ◦ h is holomorphic on ∆n ⇐⇒ γi , β  ≥ 0 for all i , hence h extends to h : ∆n −→ Eσα ⇐⇒ γi ∈ σˇˇ α = σα , for all i . Now, if h extends to h : ∆n −→ E{σα } , then h(0, . . . , 0) ∈ Eσα , for some σα , and hence for all β ∈ σˇ α , the function β ◦ h is holomorphic on ∆n , i.e., by what precedes, γi , β  ≥ 0 for all i, and all β ∈ σˇ α ; hence γi ∈ σˇˇ α = σα for all i.

182

III Compactifications of locally symmetric varieties

The only question now is why γi ∈ C(F) in all cases. But this follows because fi = restriction of f to i’th ∆˚ always extends, so, by (iii) ⇒ (i), we have that γi ∈ some σα , hence γi ∈ C(F). The theorem easily extends to the more general setting in which the inclusion ˚ k × ∆n−k ⊂ ∆n (∆) is replaced by U ∩ T ⊂ U, where U is a nice neighborhood of a point in the closed orbit of Xσ , with T ⊂ Xσ a torus embedding. The condition that U ∩ T −→ D/Γ extends to U −→ D/Γ is that the image of a certain natural map

σ −→ C(F) should lie in some σα . The above theorem in the special case k = n = 1 says: starting with any f : ∆˚ −→ D/Γ, we lift it to an equivariant f˜ : H → D. Then the monodromy γ is in C(F) ∩ U(F)Z . If we alter the lifting f˜ of f to δ · f˜, then γ changes to δ γδ −1 . Thus f alone determines canonically an element < 

Γ. µ ( f ) ∈ ΣZ = C(F) ∩U(F)Z F

(Here the union is over all rational boundary components F, and the C(F) ∩ U(F)Z are considered as subsets of G; the action of Γ is through conjugation; µ is short for monodromy.) Let us clarify the meaning of ΣZ . First of all, inside G itself, let  = |Σ|

F

C(F) =

Z = Γ ∩ |Σ|  = Σ



 F

F

C(F) ,

C(F) ∩U(F)Z .

If {σα } is a Γ-admissible collection of polyhedra as in Section 5, and Vα =  = (|Σ|,  {σα }, {Vα }) is a (infinite) res Int σα (linear functions on U(F)), then Σ Z defines conical polyhedral complex in the sense of Chapter I, Section 3, and Σ   an integral structure on Σ; namely, on each σα , let Lα be the linear functions Z . which are integral on Σ  And, for every Γ-admissible collection Since Γ is neat, Γ acts freely on |Σ|. {σα }, this action permutes the strata Int σα and preserves the spaces Vα of

7 An intrinsic form of the Main Theorem

183

linear functions, so we may form a quotient conical polyhedral complex:  |Σ| = |Σ|/Γ , Sα = Im (Int σα ) , Vα = Vα . As a topological space with piecewise-linear structure, |Σ| is independent of {σα }. And ΣZ is a subset of |Σ| and defines an integral structure on the conical polyhedral complex Σ = (|Σ|, {Sα }, {Vα }), by taking Lα to be the linear functions which are integral on ΣZ . Note that, whereas there are infinitely many σα , Σ has only a finite number of strata Sα , and, in fact, if R>0 acts by homotheties on the cones of Σ, then |Σ|/R>0 is compact. Heuristically, |Σ|/R>0 should be thought of as the set of all directions of approach to ∞ in D/Γ, as described by ratios of exponents. Definition 7.4 A structure {Sα ,Vα } of conical polyhedral complexes on |Σ| is admissible if it is induced by a Γ-admissible collection {σα }. Note that an admissible structure {Sα ,Vα } on |Σ| comes from only one collection {σα }: in fact, we can recover the σα by looking at  −→ |Σ| C(F) → |Σ| and (i) taking inverse images of the Sα , (ii) taking their connected components, and (iii) closing them up in C(F). Next, define, in analogy with the definition for a toroidal embedding in Chapter I, Section 3, : 9 ˚ ⊂ D/Γ . R.S.(D/Γ) = ϕ : ∆ −→ (D/Γ)∗ analytic | ϕ (∆) Then, as above, we get a map

µ : R.S.(D/Γ) −→ ΣZ by lifting and considering the monodromy. We can now formulate the Main Theorem I (Theorem 5.2) in a more intrinsic way. Theorem 7.5 (Main Theorem II) Let G , G, D, and Γ be as in Theorem 5.2, where Γ is neat, and define the piecewise-linear topological space |Σ| as above. Then there is a map ⎫ ⎧ ⎫ ⎧ ⎨ toroidal embeddings D/Γ ⊂ D/Γ ⎬ ⎨ admissible conical ⎬ −→ polyhedral stuctures without monodromy, where D/Γ ⎭ ⎩ ⎭ ⎩ Σ = {|Σ|, Sα ,Vα }} is a compact algebraic space such that, if Σ(D/Γ) is the conical polyhedral complex with integral structure

184

III Compactifications of locally symmetric varieties

associated by the theory of Chapter I, Section 3, to the toroidal embedding on the right, there is a unique isomorphism ϕ making the diagram 5Σ µ kkkkk k k k k R.S.(D/Γ) RRRR RRR ) ord

ϕ

 Σ(D/Γ)

commute. In particular, there is a bijection between the set of strata of D/Γ (1) and the set of strata {Sα } of Σ. Furthermore, if {Sα } is a subdivision of (2)

(1)

{Sα }, then D/Γ

(2)

dominates D/Γ .

Proof Since D/Γ ⊂ D/Γ is locally like D/U(F)Z ⊂ (D/U(F)Z ){σαF } , it is certainly a toroidal embedding. Also (D/U(F)Z ){σαF } /(ΓF /U(F)Z ) −→ D/Γ is bijective on the strata of type F. Morever, denoting, as in the proof of Proposition 6.10, by ΓF the intersection Γ ∩ N(F) and by ΓF the image of Γ in Aut (U(F)), ΓF /U(F)Z fixes every stratum of (D/U(F)Z ){σα } , while ΓF permutes without fixed points the strata of type F in (D/U(F)Z ){σα } . It follows that the strata of D/Γ are all of the form Yα /(ΓF /U(F)Z ) , where Yα ⊂ (D/U(F)Z ){σαF } is a stratum corresponding to a σα meeting C(F). The main point here is that ΓF /U(F)Z acts on Y α so as to fix each stratum Yβ ⊂ Y α . Since D/U(F)Z ⊂ (D/U(F)Z ){σαF } is a toroidal embedding without self-intersection, it follows

that, even though D/Γ ⊂ D/Γ may have self-intersection, it is without monodromy (in the sense of Chapter I, Section 3). Moreover, the polyhedral complex C(F)∗ associated to D/U(F)Z ⊂ (D/U(F)Z ){σαF } is given by 



 F σαF = C(F) ∪ .  C ; {σα }; restr. to σα of linear functions on U(F) α

C

Here C

runs through the rational boundary components of C. The equivalence relation Λ is generated by (i) the isomorphisms γ

(D/U(F)Z ){σαF } −→ (D/U(γ F)Z ){σ γ F } , for γ ∈ Γ , α

7 An intrinsic form of the Main Theorem

185

(ii) the e´ tale maps (D/U(F1 )Z ){σ F1 } −→ (D/U(F)Z ){σαF } , α

whenever F 1 ⊃ F. So the polyhedral complex Σ(D/Γ) is obtained by dividing

>

∗ F C(F)

by

(i) the isomorphisms γ : C(F)∗ −→ C(γ F)∗ (ii) the inclusions, whenever F 1 ⊃ F, C(F1 )∗ → C(F)∗ .  and modulo the former we get Σ; this yields Modulo the latter we get Σ, ∼

ϕ : Σ −→ Σ(D/Γ) . The fact that the diagram in the theorem commutes is implied by the commutativity of the following analogous diagram: U(F)Z ∩C(F) _ k5 kkk k k monodromy k kkk kkk k   maps φ : ∆ −→ (D/U(F)Z ){σαF } ˚ ⊂ D/U(F)Z where φ (∆) SSSS SSS ord SSSSS SSSS  ) C(F)∗ The commutativity here is a consequence of Proposition 3.3 in Chapter I. The uniqueness assertion about ϕ follows from the fact that the image of ord, mod(1) (2) ulo homotheties, is dense. Finally, if {Sα } is a subdivision of {Sα }, then (1) (2) {σα } is a subdivision of {σα }; hence there are vertical maps in the diagram %  fffff2 D/U(F)Z  Xy XX XX,

(D/U(F)Z ) (1),F  { σα }   (D/U(F)Z ) (2),F { σα

which induce a map in the vertical direction: (1)

j5 D/Γ '  jjjjj w  D/Γ TTT  TTT* (2) D/Γ

}

186

III Compactifications of locally symmetric varieties

Finally, let us show that there are plenty of non-singular compactifications among those constructed. Recall that we assume throughout this section that Γ is neat. Corollary 7.6 For every Γ-admissible collection {σα } of polyhedra, there ex ists exists a subdivision {σα } such that the morphism π : D/Γ −→ D/Γ is  projective and D/Γ is smooth. Proof Because Γ is neat, the morphism πF in the diagram D/U(F)Z  D/Γ



/ (D/U(F)Z ){σ F } α πF

 /

 D/Γ

is e´ tale. Consequently, it is enough to find, compatibly for all F, a subdivision σαF of σαF such that (D/U(F)Z ){σαF } is smooth and (D/U(F)Z ){σαF } −→ (D/U(F)Z ){σαF } is projective. Now Theorem 4 of TE I , Ch.I, §1, tells us that (D/U(F)Z ){σαF } is smooth if all cones σαF are generated by part of a basis of U(F)Z , and Theorem 11 of TE I , Ch.I, §1, constructs such a subdivision by an inductive procedure. It is also shown there that the resulting morphism will be a normalized blow-up and is, in particular, projective.

References [1] W. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Annals of Math. 84 (1966), 442–528. [2] A. Borel, Introduction aux Groupes Arithm´etiques. Paris: Hermann, 1969. [3] A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Diff. Geom. 6 (1972), 543–560. [4] R. Gunning and H. Rossi, Analytic Functions. Englewood Cliffs, NJ: Prentice Hall, 1965. [5] Harish-Chandra, Representations of semi-simple Lie groups VI, Am. J. Math. 78 (1956), 564–628. [6] S. Helgason, Differential Geometry and Symmetric Spaces. New York: Academic Press, 1962.

7 An intrinsic form of the Main Theorem

187

[7] A. Koranyi and J. Wolf, Generalized Cayley transformations of bounded symmetric domains, Am. J. Math. 87 (1965), 899–939. [8] R. P. Langlands, The dimension of spaces of automorphic forms, Am. J. Math. 85 (1963), 99–125. [9] Y. Namikawa, On the canonical holomorphic map from the moduli space of stable curves to the Igusa monoidal transform, Nagoya Math. J. 52 (1973), 197–259. [10] I. Satake, On the arithmetic of tube domains (blowing up of the point at infinity), Bull. Am. Math. Soc. 79 (1973), 1076–1094. [11] I. Satake, Realization of Symmetric Domains as Siegel Domains of the Third Kind. Lecture notes, Berkeley, 1972. [12] J. Wolf, Fine structure of hermitian symmetric spaces, in Symmetric Spaces, eds. W. Boothby and G. Weiss. New York: Marcel Dekker, 1972, pp. 271–357. [13] J. Wolf, On the classification of hermitian symmetric spaces, J. Math. Mechanics, 13 (1964), 489–495.

IV Further developments

This chapter is divided into two sections. The first section is an immediate application of the construction of D/Γ in the previous chapter. With explicit local coordinates, we are able to give a criterion for the holomorphic extension of higher-order differential forms to cusps. We have also computed the local codimension of the space of “extendable forms” in the space of cusp forms in the case of Hilbert modular surfaces, from which one can compute the plurigenera of Hilbert modular surfaces. The second section concerns a criterion for the projectivity of D/Γ. We shall show that certain D/Γ’s are obtained by blowing up Baily–Borel’s compactification. Our method follows essentially that of Igusa [5], but the situation is much clearer now, since D/Γ has already been constructed.

1 Extension of differential forms to the cusps 1.1 Following the notations of the previous chapter, let G = a connected semi-simple linear algebraic group of hermitian type defined over Q, G = G (R)o , K = its maximal compact subgroup, Γ = an arithmetic subgroup of G, D = G/K, a bounded symmetric domain in CN , D/Γ = the compactification of D/Γ corresponding to a Γ-admissible collection of polyhedra {σα }. Our purpose is to investigate when a top differential form extends to all “cusps” D/Γ \ (D/Γ). 189

190

IV Further developments

Let f be an automorphic form of weight
with respect to Γ: f (γ z) j(γ , z)
= f (z) for all γ ∈ Γ , z ∈ D , where j(γ , z) is the jacobian of the map γ : D −→ D at z, i.e., if

ω = dz1 ∧ · · · ∧ dzN , then γ ∗ ω = j(γ , z)ω . The differential form f ω ⊗
is invariant under Γ, and hence can be considered as a form† in ΩN (D/Γ)⊗
, and every element in ΩN (D/Γ)⊗
is of this type. Our problem is: when does f ω ⊗
extend to D/Γ? Let us first recall how D/Γ was constructed. By Chapter III, Theorem 5.2 (Main Theorem I), we have the following diagram:  / (D/U(F)Z ){σ F } D/U(F)Z α πF

 D/Γ



 D/Γ /

for every rational boundary component F. Furthermore, if Γ is neat [1] (comp. also Chapter III, Section 7), and if each F σα can be generated by a part of a basis of U(F)Z , then πF is unramified and D/Γ is non-singular (see Chapter III, Corollary 7.6). Assume this is the case. Hence, to see if f ω ⊗
extends, it is enough to check if it extends for D/U(F)Z → (D/U(F)Z ){σαF } for every F. In fact, we only need to check this for D/U(F)Z → (D/U(F)Z )σαF for each top-dimensional simplicial cone σαF . Let us fix a rational boundary component F, and embed D into D(F): D(F) = U(F)C × Cm × F , D = {(z, u,t) ∈ D(F) | Im z − ht (u, u) ∈ C(F)} . Introduce local coordinates (zi ), (u j ), (tk ) for each component; we may take ω as

ω=

@

dzi ∧

i

@ j

du j ∧

@

dtk .

k

Now, f has an expansion as a Fourier–Jacobi series: f=

∑

ρ ∈L∗

ϕρ (u,t) exp(2π iρ , z) ,

† Assume there are no singularities in D/Γ.

1 Extension of differential forms to the cusps

191

where  ,  is a positive-definite inner product on U(F) for which C(F) is self-adjoint, as defined in Chapter II, Section 1, and L∗ = {ρ ∈ U(F) | ρ , x ∈ Z, for all x ∈ U(F)Z } . This series is convergent in some cylindrical set S(K, r) = {(z, u,t) ∈ D(F) | t ∈ K, Im z − ht (u, u) − r ∈ C(F)} , where K is a compact set in F. If none of the Q-simple components of G is isomorphic to SL2 /Q, then we have “Koecher’s theorem”:

ϕρ = 0 only for ρ ∈ L∗ ∩C(F) , (see Baily [1], p. 299). We now express f ω ⊗
in terms of the local coordinates in (D/U(F)Z )σαF . Recall that (D/U(F)Z )σαF is the interior of the closure of (D/U(F)Z ) in (D/U(F)Z ) ×T (F) T (F)σαF , and T (F) = U(F)C /U(F)Z . And, by our assumption,

σαF = ∑ R≥0 Pi , where {P1 , . . . , Pn } is a basis for U(F)Z . Let {Q1 , . . . , Qn } ⊂ L∗ be the dual basis of {P1 , . . . , Pn }, σˇ α = ∑ R≥0 Qi . Then T (F)σαF is isomorphic to Cn with coordinates {wr }: wr = exp(2π iQr , z) . In terms of (wr ), (u j ), (tk ), which are coordinates on (D/U(F)Z )σαF , we may write f = ∑ ϕρ (u,t) exp(2π iρ , z) ρ

= ∑ ϕρ (u,t) exp(2π i ∑ ρr Qr , z) (where ρr = ρ , Pr ) ρ

= ∑ ϕρ (u,t) ∏ wρr r , ρ

ω=

@

dzi ∧

i

@

r

du j ∧

@

j

= constant ·

dtk

k

@

dwr ∧

r

@

du j ∧

j

@

dtk

k

1 . ∏r wr

Hence, fω

⊗


= const. ∑ ϕρ ∏ ρ

r

wρr r

1 (∏r wr )


)

@ r

dwr ∧

@ j

du j ∧

@ k

*⊗
dtk

.

192

IV Further developments

Now, (D/U(F)Z )σαF \ (D/U(F)Z ) is given by extends to (D/U(F)Z ){σαF } if and only if




r {wr

= 0}, and hence f ω ⊗


ϕρ = 0 implies ρ , Pr  ≥
for all r . Since {Pr } form a basis for U(F)Z and {σαF } is admissible, there exists, for every P ∈ U(F)Z ∩C(F), an α such that P can be expressed as follows: P = ∑ ar Pr with ar ∈ Z≥0 , where σαF = ∑ R≥0 · Pr . If P = 0, then ρ , P = ∑ ar ρ , Pr  ≥ (∑ ar )
≥
. This proves Theorem 1.1 Let D be a bounded symmetric domain in CN , let Γ be a neat arithmetic subgroup of Aut (D)o , let f be an automorphic form of weight
with respect to Γ, i.e., f ω ⊗
∈ ΩN (D/Γ)⊗
, and let D/Γ be the compactification of D/Γ corresponding to a Γ-admissible collection of polyhedra {σαF }, where each σαF can be generated by a part of a basis of U(F)Z . Then f ω ⊗
extends to D/Γ if and only if, for every rational boundary component F, in the Fourier expansion of f at F, namely f=

∑

ρ ∈U(F)∗Z

ϕρF (u,t) exp(2π iρ , z) ,

ϕρF satisfies ϕρF = 0 =⇒ ρ , P ≥
for all non-zero P ∈ U(F)Z ∩C(F). Now, on any complex manifold U of dimension N, let us call sections of (ΩUN )⊗

-fold top differentials. Recall the general fact that, if V1 ,V2 are bimeromorphic compact analytic manifolds, and ω1 , ω2 are meromorphic
fold top differentials on V1 , V2 which correspond to each other, then ω1 is regular, i.e., without poles, if and only if ω2 is regular. Therefore, for any noncompact analytic manifold U of the form U \ Z, with U a compact manifold, and Z a closed analytic subset, and for any holomorphic
-fold top differential η on U, the condition that η extend to U is independent of the choice of U. Let us call such η simply extendable forms. We see indeed from the above theorem that the criterion for f ω ⊗
to extend to D/Γ is independent of the choice of {σαF }. Thus we may rephrase the conclusion as: Corollary 1.2 The
-fold top differential f ω ⊗
on D/Γ is extendable if and

1 Extension of differential forms to the cusps

193

only if, for every F, we have that ϕρF = 0 implies ρ , P ≥
, for all 0 = P ∈ U(F)Z ∩C(F).

1.2 Let Γ ⊂ Γ be an arithmetic subgroup of G, and define U(F)Z = U(F) ∩ Γ . Apply the previous theorem to Γ , but keep f as an automorphic form with respect to Γ. Then f ω ⊗
extends to D/Γ if and only if we have

ϕρF = 0 =⇒ ρ , P ≥
for all F and all 0 = P ∈ U(F)Z ∩C(F) .

(1.1)

Assume that, for some positive integer q, U(F)Z ⊆ qU(F)Z for all F. Then (1.1) is a consequence of the following condition:

ϕρF = 0 =⇒ ρ , P ≥


q

for all F and all 0 = P ∈ U(F)Z ∩C(F) .

(1.2)

Now start with a cusp form of weight
with respect to Γ and choose q ≥
; then f will satisfy (1.2) via the following proposition. Proposition 1.3 If f is a cusp form on D of weight
with respect to Γ, then, for every rational boundary component F of D,

ϕρF = 0 =⇒ ρ , P ≥ 1, for all non-zero P ∈ U(F)Z ∩C(F) . Proof Since f is a cusp form, ϕ0F = 0 for every boundary component F. Assume ϕτF = 0 for some τ ∈ L∗ ∩C(F), with τ , P = 0 for all 0 = P ∈ U(F)Z ∩C(F) . Define Hτ = {x ∈ U(F) | τ , x = 0} . Then Hτ ∩C(F) defines a rational boundary component of C(F). By Chapter III, Theorem 4.8, there is a rational boundary component F  , with 

F ⊂ F and C(F  ) = Hτ ∩C(F) . Consider the Fourier expansion of f at F  : f=

∑

ρ  ∈L∗ ∩C(F  )



ϕρF (u ,t  ) exp(2π iρ  , z ) .

Decompose U(F)C into U(F)C = U(F  )C ⊕ Ck z = z + z .

194

IV Further developments

Then 

ϕ0F =

∑

ρ ∈L∗ ∩C(F  ) ρ ,C(F  )=0

ϕρF (u,t) exp(2π iρ , z ) ,

which would not be identically zero because ϕτF = 0. But this contradicts the fact that f is a cusp form. We recall that, for any compact analytic manifold V of dimension N, the Kodaira dimension of V is defined to be * )

κ (V ) = tr.degC

∞ $

Γ(V, (ΩVN )⊗
) − 1 .


=0

Then −1 ≤ κ (V ) ≤ N, and V is said to be of general type if κ (V ) = N. Equivalently, this means that, for some
, there are N + 1
-fold top differentials ω0 , . . . , ωN such that ω1 /ω0 , . . . , ωN /ω0 are algebraically independent meromorphic functions on V . If U is a non-compact manifold of type U \ X, with U a compact analytic manifold and X a closed analytic subset, then we make the same definitions but using extendable
-fold top differentials on U. Note that, in both cases, the Kodaira dimension is a biholomorphic invariant. Theorem 1.4 There exists Γ ⊂ Γ such that D/Γ is of general type. Proof Let f0 , . . . , fN be modular forms of weight
such that f1 / f0 , . . . , fN / f0 are algebraically independent, and let f be a cusp form of weight m with respect to Γ. Then f · f0 , . . . , f · fN are cusp forms of weight
+ m. Let Γ ⊂ Γ be an arithmetic subgroup of G such that, for all rational boundary components F, we have (
+ m + 1) ·U(F)Z ⊃ U(F)Z . Then f · fi all extend to sections of (ΩN )⊗(
+m) over D/Γ . Hence the Kodaira dimension of D/Γ is equal to N.

1.3 Let K = a real quadratic number field, O = its ring of integers, D = H × H = the product of upper half-planes Γ = SL(2, O)/{±I} .

 Then γ =

1 Extension of differential forms to the cusps  a b ∈ Γ acts on z = (z1 , z2 ) ∈ D by c d   az1 + b a z2 + b , , γz = cz1 + d c z2 + d 

195

where x → x is the conjugation automorphism of K. Let f be an automorphic form of weight
: f (γ z) = (cz1 + d)2
(c z2 + d  )2
f (z) . Then f has a Fourier expansion:

∑

f (z) =

α ∈L∗ ∩V

aα exp(2π iα , z) ,

where V = the positive quadrant in R2 , α , z = α1 z1 + α2 z2 , for α = (α1 , α2 ), z = (z1 , z2 ), L = {a ∈ R2 | z −→ z + a lies in Γ}, L∗ = {α ∈ R2 | α , a ∈ Z for all a ∈ L}, and this series is convergent for Im z1 Im z2  0. Theorem 1.1 implies that f (dz1 ∧ dz2 )⊗
extends to D/Γ if and only if aα = 0 for all α such that α , a <
, for some 0 = a ∈ L ∩V . We are going to study the number of those terms modulo the unit action. More precisely, if we identify L with O, and define (x, y) = xy + x y for x, y ∈ K , then L∗ = {ρ ∈ K | (ρ , x) ∈ Z, for all x ∈ O} . It is known as the complementary module of L. If we fix a base for L: L = Z · 1 + Z ·W , then L∗ = Z ·

1 W +Z· . W −W  W −W 

Let C be the cone of totally positive numbers in K. Also fix some positive integer
. Define

ϕ (α ) = min (α , x) = min (α , x) , 0 =x∈L∩C

x∈L∩C

S = {α ∈ L∗ ∩C | ϕ (α ) <
} .

196

IV Further developments

The group of totally positive units U+ in O acts on S via ε : α → εα .† The following theorem was conjectured by Hirzebruch. Theorem 1.5 The set S /U+ has 1 2
(
− 1)

r

∑ (bk − 2)

k=1

elements, where (b1 , . . . , br ) is the cycle associated to the continued fraction expansion of W . Before proving the theorem, we first construct a decomposition of C∗ such that ϕ is linear on each sector. By taking the convex hull of L ∩C, we obtain an admissible decomposition of C. Write each vertex Vk as Vk = pk − qkW with pk , qk ∈ Z. Assume W > 1 > W  . Then we have pk−1 qk − qk−1 pk = 1 ,

(1.3)

pk−1 qk+1 − qk−1 pk+1 = bk ≥ 2 ,

(1.4)

 > Vk , Vk > Vk+1 ,Vk+1

(1.5)

Vk+1 +Vk−1 = bkVk ,

(1.6)

pk pk = W, lim = W . k−→−∞ qk qk

(1.7)

lim

k−→∞

The union of r consecutive sectors forms a fundamental domain with respect to the action of totally positive units (see Chapter I, Section 5, or Hirzebruch [4]). For each k, define Wk =

pk+1 − pk qk+1 − qk  − W ∈ L∗ . W −W  W −W 

By (1.5), Wk > 0 and Wk > 0, hence Wk ∈ L∗ ∩C. Furthermore,   pk+1 − pk qk+1 − qk  (Wk ,Vk ) = − W (pk − qkW ) W −W  W −W    pk+1 − pk qk+1 − qk − W (pk − qkW  ) + − W −W  W −W  = pk (qk+1 − qk ) − qk (pk+1 − pk ) = 1, by (1.3) . † The action induced by Γ is actually α → ε 2 α , which will change our number by a factor of 1 or 2.

1 Extension of differential forms to the cusps

197

Similarly, (Wk−1 ,Vk ) = 1 . Define Ck∗ = R>0Wk + R≥0Wk−1 . Note from (1.3) and (1.4) that pk − pk−1 pk+1 − pk ≥ , qk+1 − qk qk − qk−1 with equality holding only when bk = 2 , in which case Wk = Wk−1 and Ck∗ is only a ray. The collection of Ck∗ with bk ≥ 3 covers C∗ because bW a a − ∈ C∗ ⇐⇒ W > > W  ,   W −W W −W b and the fact that lim

k−→∞

pk − pk−1 =W , qk − qk−1

lim

k−→−∞

pk − pk−1 = W  , by (1.7) . qk − qk−1

Lemma 1.6 Ck∗ = {α ∈ C∗ | ϕ (α ) = min (α , x) = (α ,Vk )}. x∈L∩C

Proof Start with an element on the left. Now,

α ∈ Ck∗ =⇒ α = aWk−1 + bWk , for some a, b ≥ 0 =⇒ (α ,Vk ) = a + b ≤ (α , x) , for all x ∈ L ∩C . Conversely, suppose ϕ (α ) = (α ,Vk ) = 0. Since {R>0Wi + R>0Wi−1 } covers C∗ , we have α ∈ R>0Wi + R>0Wi−1 for some i. Assume that i = k. Without loss of generality, let us consider the case i < k. Then α = aWi−1 + bWi for some a, b > 0; hence a(Wi−1 ,Vk ) + b(Wi ,Vk ) = (α ,Vk ) ≤ (α ,Vi ) = a + b , and (Wi−1 ,Vk ) = 1 . But we know that (Wi−1 ,Vi−1 ) = (Wi−1 ,Vi ) = 1 . Hence Vk lies on the straight line joining Vi−1 and Vi . By convexity, all the vertices Vi−1 ,Vi , . . . ,Vk lie on the same line, and we get ∗ = · · · = C∗ ∗ Ci∗ = Ci+1 k−1 ⊂ Ck ,

198

IV Further developments

hence α ∈ Ck∗ . The collection of r consecutive Ck∗ forms a fundamental domain in C∗ with respect to the action of totally positive units. Define Nk = #{α ∈ L∗ ∩Ck∗ | ϕ (α ) <
}, so that #(S /U + ) =

∑ Nk .

bk ≥3

Proof of Theorem 1.5 It suffices to prove Nk = 12
(
− 1)(bk − 2) . Introduce the following elements of L∗ : qk−1 −pk−1 + W ,  W −W W −W  pk+1 qk+1 Q= + W . W +W  W −W  P=

Define Pi =

i bk − i P+ Q for 1 ≤ i ≤ bk . bk bk

We have (1) P1 = Wk , Pbk −1 = Wk−1 (from (1.6)); (2) Pi ∈ L∗ , because

−pk 1 bk (P − Q) = W −W 

qk  + W −W  W (from (1.6));

(3) (Pi ,Vk ) = 1 (from (1)); (4) {Pi , Pi+1 } form a basis for L∗ .

: 9 1 W To see the last point, calculate the determinant with respect to the basis W −W  , W −W  of L∗ : ; ; ; ; i+1 bk −i−1 ; ;; 1 ; bk ; ; −pk−1 qk−1 ;; bk det{Pi , Pi+1 } = ; i = · bk = 1 . bk −i ; ; ; ; b ; p −q b k+1 k+1 k b k

k

∗ = R P +R P Define Ck,i >0 i ≥0 i+1 for i = 1, . . . , bk − 2, and let ∗ | ϕ (α ) <
} , Nk,i = #{α ∈ L∗ ∩Ck,i

so that Nk = ∑ Nk,i . Since {Pi , Pi+1 } forms a base for L∗ , and ϕ (Pi ) = ϕ (Pi+1 ) = 1, we get Nk,i = 1 + 2 + · · · + (
− 1) = 12
(
− 1) , and hence Nk = (bk − 2) · 12
(
− 1) .

2 Projectivity of D/Γ

199

2 Projectivity of D/Γ 2.1 Let D be a bounded symmetric domain, let Γ be an arithmetic subgroup of Aut (D)o , and let {σαF } be a Γ-admissible decomposition. Then we have constructed the associated compactification D/Γ of D/Γ. In this section, we are going to study the relation between D/Γ and Baily–Borel’s compactification (D/Γ)∗ = D∗ /Γ. The main result is that D/Γ, for certain {σαF }, is the blowingup of D∗ /Γ at a certain sheaf of ideals. Consequently D/Γ is projective in these cases. For simplicity, we shall assume in this section that Γ is neat. (Things can be worked out without this assumption, but things will be much cleaner if we assume Γ to be neat.) The relevant decompositions are similar to the projective subdivisions in TE I,† Ch. III, §1. We shall describe them first. As in Chapter III, Section 7, define  = |Σ|

 F

 C(F) , |Σ| = |Σ|/Γ and ΣZ .

Definition 2.1 A Γ-admissible decomposition {σαF } is projective if there exists a continuous convex piecewise-linear function ϕ : |Σ| −→ R such that (1) ϕ (x) > 0 for x = 0 ; (2) ϕ is linear on the image of σαF , and σαF are the maximal polyhedral cones in C(F) on which ϕ is linear ; (3) ϕ is integral on ΣZ . The existence of such decompositions follows from the theory of co-cores (see Chapter II, Section 5). Indeed, let {∆F } be a system of ΓF -polyhedral co-cores, one for each rational boundary component F, such that 

(1) for F ⊂ F , ∆F ∩C(F  ) = ∆F  , (2) for F  = γ F with γ ∈ Γ, we have ∆F  = γ ∆F . From Chapter II, Section 5, we know that the cones over the faces of ∆F define a Γ-admissible decomposition. Fix a sufficiently divisible integer N. For each F, let ϕF be the unique convex piecewise-linear function on C(F) such that it has value N at each face of ∆F and is linear on the cone over each face. Then {ϕF } defines a function ϕ on |Σ| with the required properties. Let us start with a projective Γ-admissible decomposition {σαF } and ϕ : |Σ| −→ R satisfying the required properties. Then ϕ defines a collection of continuous piecewise-linear functions {ϕF } on {C(F)} such that ϕF is linear † Recall this reference from p. x.

200

IV Further developments

on each σαF . We define a dual piecewise-linear function ϕF∗ on C(F) such that, in the terminology of Chapter II, Subsection 5.2, {ϕF∗ (λ ) ≥ 1} = the core dual to the co-core {ϕF (x) ≥ 1} . More explicitly, let {Pα ,i } = vertices of σαF ∩ {ϕF = 1} , and define

ϕF∗ (λ ) = min λ , Pα ,i  , for λ ∈ C(F) . α ,i

Moreover, for all top-dimensional σαF , let

λα ∈ U(F)∗Z = Hom (U(F)Z , Z) be the linear function such that

λα |σαF = ϕF |σαF (this exists by assumptions (2) and (3) on ϕ ). Note that ϕF∗ (λα ) = 1. Now we can define the ideal of the blowing-up as follows. Let x ∈ F/Γ(F) ⊂ D∗ /Γ. Then the holomorphic functions around x are described by the Fourier–Jacobi series (for notation, see Subsection 2.2 below) of the following form: f=

∑

θρ (u,t) exp(2π iρ , z) .

ρ ∈C(F)∩U(F)∗Z

Define Im,x = { f ∈ Ox | θρ = 0 only for ρ ∈ U(F)∗Z ∩C(F) and ϕF∗ (ρ ) ≥ m} . Since ϕF∗ is convex, Im,x is an ideal. We shall see later that {Im,x } form a coherent sheaf of ideals Im concentrated at the boundary. (Note that Im depends on the choice of ϕ as well as on m.) We shall often denote Im simply by I . Our aim is to prove the following theorem. Theorem 2.2 Let {σαF } be a projective Γ-admissible decomposition, let Im be the sheaf of ideals on D∗ /Γ constructed as above (with a suitable m fixed in the  be the normalization of the blowing-up course of the proof), and let (D/Γ) I

 is isomorphic to the compactification D/Γ associated to at I . Then (D/Γ) I F {σα }. Corollary 2.3 If {σαF } is projective, then the associated D/Γ is projective.

2 Projectivity of D/Γ

201

Corollary 2.4 There are Γ-admissible collections {σα } of polyhedra such that the associated compactification D/Γ is non-singular and projective. Proof Indeed, start with some projective {σαF } and then apply the refining procedure of Chapter III, Corollary 7.6.

2.2 Let F be a rational boundary component of D, let x ∈ F ⊂ D∗ , let x be the image of x in D∗ /Γ, and let U be a neighborhood of x in D∗ such that U ∩ D = πF−1 (E) ∩ Φ−1 F (
C0 ) , where E is a relatively compact open neighborhood of x in F such that • γ E ∩ E = 0/ for id = γ ∈ Γ(F) , •C0 is the interior of a core in C(F) , •
> 0 , • γ U = U for γ ∈ Γx . By Chapter III, Section 6, such U actually form a fundamental system of neighborhoods of x . Let f be a holomorphic function on U ∩ D; such an f defines an element in Ox if, for all y ∈ U ∩ D, f (γ y) = f (y) for all γ ∈ Γx . To study this invariance condition we need explicit forms of the N(F)action. As in Chapter III, Section 4, and in Koranyi–Wolf [7], D can be realized as a Siegel domain of the third kind: D∼ = {(t, u, z) ∈ F × Ck ×U(F)C | Im z − Re Lt (u, u) ∈ C(F)} , where Lt is a quasi-hermitian form (a sum of a hermitian form Ht and a symmetric form St ) depending analytically on t, where 2k = dimV (F), and where Ht (u, u) ∈ C(F), for all u. Recall that, for each t0 ∈ F, the action of V (F) on the points (t0 , u, ·) ∈ D makes Ck into a principal homogeneous space over V (F), and hence identifies V (F) with Ck via mapping x to the second coordinate of x(t0 , 0, ·). But this identification varies with t0 . Proposition 2.5 (1) In the coordinates (t, u, z), the action of N(F) consists of quasi-linear

202

IV Further developments transformations, i.e., every γ ∈ N(F) acts by: z −→ Az + a(u,t) , u −→ Bt u + bt , t −→ g(t) ,

where A ∈ Aut (C(F)), Bt is linear in u, and Bt , bt , g(t), and a(u,t) are analytic in t and u. (2) W (F) consists of the following transformations: ⎧ ⎨ z −→ z + a + 2iLt (u, bt ) + iLt (bt , bt ) , (b, a) = u −→ u + bt , ⎩ t −→ t , where a ∈ U(F), b ∈ V (F), and bt ∈ Ck is given by the identification of V (F) and Ck , depending analytically on t. (3) Let Z  (F) ⊂ Z(F) be the subgroup consisting of transformations of the form: ⎧ ⎨ z −→ Az + iLt (Bt u, Bt u) − iALt (u, u) , (B, A) = u −→ Bt u , ⎩ t −→ t , with A, Bt satisfying AHt (u, u) = Ht (Bt u, Bt u), A ∈ Aut (C(F)) . Then we have the semi-direct product decomposition: Z(F) = Z  (F)
W (F) . For the proof of (1) and (2), see [7] and [9]; for (3), see [6]. We make some remarks about (2) and (3). (i) By simple computations, one can show (b, a)(b , a ) = (b + b , a + a + 2[b, b ]) , where [b, b ] = Im Ht (bt , bt ). Hence (b, a)(b , a )(b, a)−1 (b , a )−1 = (0, 4[b, b ]) . (ii) The form that (B, A) is given shows that (B, A) transforms D to D (in the given realization). The condition AHt (u, u) = Ht (Bt u, Bt u) guarantees that Z  (F) normalizes W (F). By computation, we have that, if γ = (b, a)(B, A), then

γ (b , a )γ −1 = (Bb , Aa + 4[b, Bb ]) .

2 Projectivity of D/Γ

203

(iii) For every β ∈ V (F), there is a unique function bt as in (2) such that (b, 0) represents the action of β , and all the bt correspond to some β . (iv) In the notation of Chapter III, Section 4, G
(F) · M(F) is the identity component of Z  (F). Let V (F)Z be the image of W (F) ∩ Γ in V (F) considered as the quotient W (F)/U(F). Let ΓF be the image of Z(F) ∩ Γ in Aut (C(F)). Note that this is a subgroup of finite index in the ΓF defined in Chapter III, Section 5, which was the image of N(F) ∩ Γ in Aut (C(F)). Lemma 2.6 The following sequences are exact: 1

−→

U(F)Z

−→ W (F) ∩ Γ −→ V (F)Z

1 −→ W (F) ∩ Γ −→

Z(F) ∩ Γ

−→

ΓF

−→

1,

−→

1.

Proof The exactness of the first sequence follows from the definition of V (F)Z . To prove that the second sequence is exact, we need to prove ker (Z(F) ∩ Γ −→ Aut (C(F))) = W (F) ∩ Γ . Recall from Chapter III, Section 4, that we have Z(F)o = [G
(F) · M(F)]
W (F) . Therefore ker (Z(F) ∩ Γ −→ Aut (C(F))) /W (F) ∩ Γ is contained in the compact factors, and, since it is discrete, it is finite. But ker (Z(F) ∩ Γ → Aut (C(F))) /W (F) ∩ Γ ⊂ (N (F) ∩ Γ)/W (F) ∩ Γ , which is torsion-free, since Γ is neat and N (F) and W (F) are both defined over Q. Lemma 2.7 [G
(F) ∩ Γ]
[W (F) ∩ Γ] is of finite index in Z(F) ∩ Γ. Proof Since G
, W , and G

W are defined over Q, the following are arithmetic subgroups: G
(F) ∩ Γ ⊂ G
(F) , W (F) ∩ Γ ⊂ W (F) , (G
(F)
W (F)) ∩ Γ ⊂ G
(F)
W (F) . From the general theory of arithmetic subgroups, (G
(F) ∩ Γ)
(W (F) ∩ Γ) is also an arithmetic subgroup of G
(F)
W (F), and hence (G
(F) ∩ Γ)
(W (F) ∩ Γ) is of finite index in (G
(F)
W (F)) ∩ Γ. But Z(F)/(G
(F)
W (F)) is compact, so Z(F) ∩ Γ/(G
(F)
W (F)) ∩ Γ is also finite.

204

IV Further developments

Going back to our invariance condition, if we start with f in Ox , then f (γ y) = f (y), for all γ ∈ Γx , y ∈ U ∩ D. The invariance condition by the elements in U(F)Z implies that f admits the Fourier–Jacobi series expansion: f (t, u, z) =

∑

ρ ∈U(F)∗Z

θρ (u,t) exp(2π iρ , z) .

If γ ∈ V (F)Z and lifts to (b, a) ∈ W (F) ∩ Γ, the invariance condition shows

θρ (u + bt ,t) = θρ (u,t) exp(2π iρ , −2iLt (u, bt ) − iLt (bt , bt ) − a) .

(2.1)

For simplicity, call ebt (u) the exponential factor appearing here. (Note that a is uniquely determined modulo U(F)Z .) Define the line bundle Lρ on the π family of complex tori (Ck × F)/V (F)Z −→ F as C × Ck × F modulo the following action of V (F)Z : (α , u,t) −→ (evt (u)α , vt + u,t) , v ∈ V (F)Z . Equation (2.1) just means θρ ∈ Γ(π −1 (E), Lρ ). It is shown in [9] and [1] that the convergence of the series for f requires that ρ ∈ C(F) whenever θρ = 0. Define g(u) = exp(−2π ρ , St (u, u)) , evt (u) = evt (u)g(vt + u)g(u)−1 . By simple computations, evt (u) = exp(2π ρ , 2Ht (u, v) + Ht (v, v)) · exp(−2π iρ , a) . In the notation of [8], this shows that the line bundle Lρ |π −1 (t) is algebraically equivalent to L (4ρ , Ht ). Furthermore, for ρ ∈ L∗ ∩ C(F)∗ , since Ht (u, u) ∈ C(F) for all u, we have that 4ρ , Ht  is positive-definite. By the remarks following Proposition 2.5, Im 4ρ , Ht (u, v) is integral on V (F)Z × V (F)Z . In particular, for our λα defined in Subsection 2.1, we know Lmλα |π −1 (t) is generated by its sections if m ≥ 2, and hence so is Lmλα |π −1 (E) . Now consider A ∈ ΓF , and assume it lifts to (d, c)(B, A) in Z(F) ∩ Γ. By Lemma 2.7, for all A ∈ ΓF , we can get such liftings while choosing (d, c) in a finite set. For γ = (d, c)(B, A), the invariance condition gives:

θA∗ ρ (u,t) = θρ (Bt u + dt ,t) exp(2π iρ , a(u,t)) , where a(u,t) = iLt (Bt u, Bt u) − iALt (u, u) + 2iLt (Bt u, dt ) + iLt (dt , dt ) + c .

(2.2)

2 Projectivity of D/Γ

205

Note that by Lemma 2.6, if A ∈ ΓF , then (d, c) is uniquely determined modulo W (F) ∩ Γ, and Bt is uniquely determined. Conversely, start with θλ ∈ Γ(π −1 (E), Lλ ) for some neighborhood E of x in F and define θA∗ λ by (2.2) (A ∈ ΓF ). Then form the sum f θλ =

∑

θA∗ λ exp(2π iA∗ λ , z) .

A∈ΓF

By Lemma 2.6, and the discussion above, the local ring Ox is generated by such fθλ if we know their convergence. For this, we prove the following proposition. Proposition 2.8 For all A ∈ ΓF , a ∈ C(F), and θλ as above, there is an M > 0 such that | fθλ | ≤ M in πF−1 (E) ∩ Φ−1 F (a +C(F)) . Proof Since F is Q-rational, V (F)/V (F)Z is compact. Let S ⊂ V (F) be a compact fundamental set for V (F)Z . Using the action of V (F) on Ck × F, let S1 ⊂ Ck be a compact set such that S · [(0) × E] ⊂ S1 × E. By (2.2), we have |θA∗ λ (u,t)| = |θλ (Bt u + dt ,t)| exp(−2π λ , Im a(u,t)) . Writing ht = Re Lt , we find Im a(u,t) = ht (Bt u, Bt u) − Aht (u, u) + 2ht (Bt u, dt ) + ht (dt , dt ) . Decompose Bt u as ut + bt , with ut ∈ S1 and b ∈ V (F)Z . Then |θλ (Bt u + dt ,t)| = |θλ (ut + dt + bt ,t)| = |θλ (ut + dt ,t)| exp(−2π λ , a (u,t)) , where, by (2.1), a (u,t) = −2ht (ut + dt , bt ) − ht (bt , bt ) = −2ht (dt , bt ) + ht (ut , ut ) − ht (ut + bt , ut + bt ) = −2ht (dt , Bt u) + 2ht (dt , ut ) + ht (ut , ut ) − ht (Bt u, Bt u) . Since t is in a compact set and the (d, c) are in a finite set, the dt are in a compact set. Further, ut ∈ S1 , which is compact. Hence |θA∗ λ (u,t) exp(2π iA∗ λ , z)| ≤ C · exp(−2π A∗ λ , Im z) exp(2π λ , Aht (u, u)) = C · exp(−2π A∗ λ , Im z − ht (u, u)) ≤ C · exp(−2π A∗ λ , a) , for suitable constants C,C ,C . Let an = #{A ∈ ΓF | n ≤ A∗ λ , a ≤ n + 1} ;

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IV Further developments

then (because the action of ΓF is fixed-point free) an ≤ # {x ∈ U(F)∗Z ∩C(F) | n ≤ (x, a) ≤ n + 1} , which grows like the volume of C(F) ∩ {(x, a) = n}. Hence an ≤ const. · nK for some constant K. Therefore,

∑ exp(−2π A∗ λ , a) ≤ ∑ an exp(−2π n) ≤ const. · ∑ nK exp(−2π n) < ∞ . n≥0

n≥0

Proposition 2.9 Ox is generated by fθλ with θλ ∈ Γ(π −1 (E), Lλ ) and λ ∈ C(F) ∩U(F)∗Z . Proof This follows from Proposition 2.8 and the fact that
C0 is, modulo ΓF , contained in a finite union of cylindrical sets a +C(F). Proposition 2.10 Assume Int σαF ⊂ C(F), with σαF top-dimensional. Fix a ∈ C(F), K > 0, and θλ ∈ Γ(π −1 (E), Lλ ), where ϕF∗ (λ ) ≥ m. Then there exists M > 0 such that | fθλ exp( − mλα , z)| < M F in πF−1 (E) ∩ Φ−1 F (σα + a) ∩ {|u| ≤ K}.

Proof Recall the definition of ϕF∗ : we have points {Pi,α } such that

σαF = ∑ R≥0 Pi,α and λα , Pi,α  = 1 ,

ϕF∗ (λ ) = min λ , Pi,α  . i,α

Since ϕF∗ (A∗ λ ) = ϕF∗ (λ ) ≥ m, we have A∗ λ − mλα ≥ 0 in σα , and hence A∗ λ − mλα , a ≥ 0 for all A ∈ ΓF . The rest of the proof runs parallel to that of Proposition 2.8, where we proved the similar statement for fθλ . We now prove that the ideals Im,x defined in Subsection 2.1 form a coherent sheaf of ideals. We first define a sheaf of principal ideals J on D/Γ in the following steps. (1) Define JF on (D/U(F)Z ){σαF } : Let Xα = (D/U(F)Z )σαF , and, for all topdimensional σαF , let JF,α = OXα Xλα , where Xλα = exp(2π iλα , z) .

2 Projectivity of D/Γ

207

The {JF,α } define an ideal JF on (D/U(F)Z ){σαF } since, if σα ∩ σβ = σγ ,

then on Xγ we have JF,γ = OXγ Xλα = OXγ Xλβ .

(2) If F ⊂ F  , then C(F) ⊃ C(F  ), and we have the glueing map:

αF  ,F : (D/U(F  )Z ){σ F  } −→ (D/U(F)Z ){σαF } . α

F

The {σα } are the cones σαF ∩ C(F  ). We get αF  ,F by dividing out U(F)Z ,   plus an open immersion. If σαF = σαF ∩C(F  ), then λα = λα on σαF , hence

αF∗  ,F (JF ) = JF  . (3) Similarly, if F  = γ F for some γ ∈ Γ, then we have

αF  ,F : (D/U(F  )Z ){σ F  } −→ (D/U(F)Z ){σαF } , α

with

 {σαF } = {γσαF }.

Now,

{λα }

is just {γ ∗ λα }, hence

αF∗  ,F (JF ) = JF  . (4) As in Chapter III, Section 6, define = = D/Γ

 F

(D/U(F)Z ){σαF } .

= and define J = as  on D/Γ Let ιF be the injection (D/U(F)Z ){σαF } −→ D/Γ 3 F ιF,∗ JF . = −→ D/Γ, and define J by (5) Let p be the map D/Γ   αF∗  ,F ιF∗ (s) = ιF∗  (s) if F ⊂ F  −1  Γ(U, J ) = s ∈ Γ(p (U), J ) | . or F  = γ F for some γ ∈ Γ It is easy to check that in each step we get a principal sheaf of ideals and that J is locally generated by Xλα . Proposition 2.11 Let f : D/Γ −→ D∗ /Γ be the map defined in Chapter III, Section 5, and let Im = f∗ J m . Then (1) Im is a coherent sheaf of ideals. (2) Let x ∈ F/Γ(F) ⊂ D∗ /Γ. Then Im,x is the ideal of holomorphic functions around x such that in their Fourier–Jacobi series expansion,

∑ θρ exp(2π iρ , z) , the coefficient θρ = 0 only for ρ ∈ U(F)∗Z ∩C(F) and ϕF∗ (ρ ) ≥ m.

208

IV Further developments

Proof (1) follows from Grauert’s coherency theorem, since f is obviously proper. To prove (2), let V be an open neighborhood of x in D∗ /Γ; let U be a connected component of the inverse image of V in D∗ . We may choose U as at the beginning of this section: U ∩ D = πF−1 (E) ∩ Φ−1 F (
C0 ) . F Let Uα = U ∩ Φ−1 F (σα ). Then

Γ(V, Im ) = { f ∈ OV | f · X−mλα is holomorphic in Uα for all α } . If f ∈ Γ(V, Im ), consider the Fourier expansion of f : f = ∑ θρ exp(2π iρ , z) . As above, let Pi,α be the vertices of σαF , and let Di,α be the corresponding codimension-one strata of Uα \Uα ∩ D. Then a term

θρ exp(2π iρ − mλα , z) of f · X−mλα vanishes on Di,α to order ρ − mλα , Pi,α  (or has a pole on Di,α if ρ − mλα , Pi,α  < 0). Thus f ∈ Γ(V, Im ) implies that θρ = 0 only when ρ − mλα , Pi,α  ≥ 0 for all i, α . But this means that

ϕF∗ (ρ ) = min ρ , Pi,α  ≥ min mλα , Pi,α  = m . Conversely, if ϕF∗ (ρ ) ≥ m whenever θρ = 0, then, by Proposition 2.10, the function f · X−mλα is bounded on U ∩ D ∩ Φ−1 F (σα + a), for all α and a, hence extends holomorphically to Uα .

2.3 We will now give the proof of Theorem 2.2. We shall in fact prove slightly more: namely, in the notation of Proposition 2.11, we shall prove that, for suitable m, D/Γ = normalization of blow-up of D∗ /Γ along Im , J m = f ∗ Im .

2 Projectivity of D/Γ

209

Step I We first remark that it suffices to prove the theorem for some normal subgroup Γ of finite index in Γ. Write X = D/Γ and H = Γ/Γ , and let X  = D/Γ be the compactification corresponding to the same {σαF }. By uniqueness of D/Γ, we have X∼ = X  /H . Write S = D∗ /Γ and S = D∗ /Γ . The group H acts on S and defines a map h : S −→ S which induces an isomorphism S /H ∼ = S. Lemma 2.12 (h∗ Im )H = Im . Proof Consider the following commutative diagram: X  S

g

f h

/X  /S

f

We have J and J  on X and X  , both are generated locally by Xλα , and f∗ J m = Im and f∗ J m = Im . Moreover J and J  are related by J = (g∗ J  )H ; hence (h∗ Im )H = (h∗ f∗ J m )H = ( f∗ g∗ J m )H = f∗ (g∗ J m )H = f∗ J m = Im .

A  is the integral closure I k k Lemma 2.13 If f ∗ Im = J m , then Imk m of Im , for all k ≥ 1. Proof Since f  is proper, whenever f ∗ K is a sheaf of principal ideals, then f∗ f ∗ K = KB. Therefore  A k Imk = f∗ (J mk ) = f∗ f ∗ (Imk ) = I m .

Introduce the notation XI for the normalization of the blow-up of a variety X along a coherent sheaf of ideals I . It is well known that ∼ XI n for all n ≥ 1 , (a) XI =  (b) XI ∼ = XI# with I# the integral closure of I . We will apply the general fact:

210

IV Further developments

Lemma 2.14 Let X be a normal quasi-projective variety, let H be a finite group acting on X, and let I be an H-invariant coherent sheaf of ideals on X. If n denotes the order of H, and π : X −→ X/H the canonical map and J = π∗ (I n )H , then = J . XI /H ∼ = (X/H) Moreover, J induces the ideal I n · OX on XI . I

Proof It is easy to check that  , Xπ ∗ J /H ∼ = (X/H) J ∗ J . To do this, it Bn = π and hence the lemma follows if we prove that I n ∗ suffices to prove that I and π J generate the same sheaf of ideals on X#I . The difficult point here is to check that J generates the full sheaf (I · OX )n . I Take any point x ∈ XI , let x be its image in X, and let U ⊂ X be an H-invariant

affine neighborhood of x. It is easy to see that there is an f ∈ Γ(U, I ) that generates the principal ideal sheaf I · OX at each of the points σ x, σ ∈ H. I Then f  = ∏σ ∈H (σ f ) is a section of J on U/H which generates (I · OX )n I at x. Bn )H . Corollary 2.15 The same thing holds if we let J equal π∗ (I Proof The same proof works in fact. Now, assuming the theorem for Γ , we have an m such that  ∗  m X ∼ . = SI  , f Im = J m

Therefore, by Lemmas 2.12 and 2.13,  H   k H (h∗ (I m ) ) = (h∗ Imk ) = Imk ,

and, by Lemma 2.14,   ∼  X∼ = SI = X  /H ∼  /H = (S /H)Imk = SImk , m

and Imk induces the ideal J mk on X  , hence Imk induces the ideal J mk on X. Step II Since, by Step I, it suffices to prove the theorem for some normal subgroup of finite index of Γ, we can make the following assumption about Γ. αF = σαF \ σαF ∩ ∂ C(F). (When there is no confusion, we For each σαF , let σ shall drop the superscript or subscript F.)

2 Projectivity of D/Γ

211

Assumption λα , x < A∗ λα , x for all x ∈ σ˜ α , id = A ∈ ΓF . This assumption is justified for the following reason. There are only finitely many A ∈ ΓF such that Aσα ∩ σα ∩ C(F) = 0; / since there are only finitely many σα modulo ΓF , we may take a subgroup Γ of finite index (e.g., a suitable  congruence subgroup) such that all such A are not in ΓF for all F. With the above assumption, we have: Proposition 2.16 (1) Let xn = (tn , un , zn ) be a sequence in D, with lim(tn , un ) = (t, u), lim Re zn = x and lim Im zn = y + ∞ · σγ , where σγ is a face of the top-dimensional σα and Int σγ ⊂ C(F). Then lim fθmλα (xn )X−mλα (zn ) = θmλα (t, u) .

n−→∞

(2) Assume further that σγ is a face of the top-dimensional σα and σβ , with θmλα (t, u) = 0, and let z = x + iy. Then lim

fθmλ (xn ) β

n−→∞ f θ (xn ) mλα

= Xm(λβ −λα ) (z) ·

θmλβ (t, u) θmλα (t, u)

.

Proof For the proof of (1), by the same argument as in Proposition 2.8, |θmA∗ λα Xm(A

∗λ

α −λα )

(zn )| ≤ const. · exp(−2π mA∗ λα − λα , Im zn ) .

By the assumption on Γ, for A = id , we have that A∗ λα − λα is positive on σγ , and A∗ λα − λα , Im zn  −→ ∞ uniformly as n −→ ∞; therefore, except for A = id , every term of fθmλα · X−mλα (zn ) tends to zero uniformly. (2) follows from (1), since Im zn = y + εn + wn with εn −→ 0, wn ∈ σγ ; hence lim Xλβ −λα (zn ) = lim exp(−2π iλβ − λα , z + εn + wn ) = Xλβ −λα (z) .

#α = For later use, we may put (2) in a slightly more general form. Define σ {λ ∈ U(F)∗ | λ ≥ 0 on σα }. Since {σα } is the biggest decomposition such that λα is linear on each σα , #α is generated by the λβ − λα , where β runs through the topit follows that σ dimensional simplices such that σβ ∩ σα has codimension one. #α ∩U(F)∗Z | λ ≡ 0 on σγ }. Then Iγ is generated by the λβi − Let Iγ = {λ ∈ σ λα with λβi ≡ λα on σγ , i.e., every λ ∈ Iγ is of the form λ = ∑ ai (λβi − λα ), where the ai are positive rational numbers.

212

IV Further developments

There exists k ∈ Z≥0 such that λ + kλα ∈ C(F), and is a positive linear combination of the λβi and λα ; hence λ + kλα , x < λ + kλα , Ax , for all id = A ∈ ΓF , x ∈ Int σγ . Proposition 2.17 With the same notation as in Proposition 2.16, and for λ , k as above, if θkλα (t, u) = 0, then lim

n−→∞

fθλ +kλα (xn ) fθkλα (xn )

=

θλ +kλα (t, u) λ X (z) . θkλα (t, u)

Step III Recall from Chapter III, Section 5 the following commutative diagram:  / (D/U(F)Z ){σ F } D/U(F)Z α   πF     / D/Γ D/Γ >

The map πF is e´ tale, πF is surjective, and D/Γ is the unique compact analytic space with these properties. Consider the map f : D/Γ −→ D∗ /Γ . By Proposition 2.11, f ∗ Im is a subsheaf of J m . By Proposition 2.16, (1), if m ≥ 2, then Im has a section at each point which, locally on D/Γ, is a unit times Xmλα : this is because, if m ≥ 2, then Lmλα |π −1 (E) is generated by its sections, so we can find θmλα with θmλα (t, u) = 0. Therefore, if m ≥ 2, we have f ∗ Im = J m , hence f ∗ Im is principal. Since D/Γ is normal, by the = I , there is an analytic morphism ψ such that universal property of Y = (D/Γ) the following diagram is commutative: D/Γ

ψ

/Y EE }} EE } EE }} }} f EE } " ~} D∗ /Γ

We shall prove that ψ is a local isomorphism, i.e., ∼

ψ ∗ : Oy −→ Ox , for all x ∈ D/Γ , where y = ψ (x) . Then Y will have the property characterizing D/Γ, and hence ψ will be an isomorphism.

2 Projectivity of D/Γ

213

Step IV Since ψ induces the identity map on D/Γ, it is clear that ψ ∗ is injective, so it suffices to prove ψ ∗ is surjective, and again we only need to check this for each stratum of (D/U(F)Z ){σα } . We first recall the notion of strata of (D/U(F)Z )σα . For each face σγ of σα , we have the orbit Oγ of T (F) in T (F)σα , cf. TE I, Ch. I; see also Chapter I, Section 1. Let Sγ be the subset of (D/U(F)Z )σα given by Sγ = (D(F)/U(F)Z ) ×T (F) Oγ . >

Then (D/U(F)Z )σα can be broken up into Sγ . For x ∈ Sγ , OSγ ,x is generated by t, u, and by Xλ with λ ∈ Iγ = {λ ∈ U(F)∗Z | λ ≡ 0 on σγ }. We have a sequence of maps: ιγ

π

ψ

F D/Γ −→ Y . Sγ → (D/U(F)Z ){σα } −→

Claim To prove ψ ∗ is surjective, it is sufficient to prove that (ψ ◦ πF ◦ ιγ )∗ is surjective for all F, α , γ . Proof of the claim Indeed, assume ψ ∗ is not surjective at y ∈ Y . Then there is a curve in D/Γ mapping to y under ψ . (By Zariski’s Main Theorem, if ψ −1 (y) is finite, then ψ is a local isomorphism at each point of ψ −1 (y).) Hence there is a curve in (D/U(F)Z ){σα } mapping to y under ψ ◦ πF for some F. It follows that, for some stratum Sγ in (D/U(F)Z )σα , there is a curve in Sγ mapping to y under ψ ◦ πF ◦ ιγ . If we can prove that (ψ ◦ πF ◦ ιγ )∗ is surjective at y for all F, α , γ , then such a curve cannot occur in any stratum Sγ , and hence ψ ∗ is surjective. By this reduction step, we only need to consider the situation

ψ : Sγ −→ Y , with x ∈ Sγ , ψ (x) = y ∈ Y , and to prove that ψ ∗ OY,y = OSγ ,x . Note that, if Sγ is a stratum in (D/U(F)Z ){σα } , we may choose F to be the associated boundary component of x, i.e., Int σγ ⊂ C(F). The local ring OSγ ,x is generated by t, u, and Xλ with λ ∈ Iγ . Let t(x) = t, u(x) = u. We now assume m ≥ 3, so that Lmλα |π −1 (t) is very ample and we can choose

θmλα ∈ Γ(π −1 (t), Lmλα ) with 0 ≤ i ≤ dimV (F) so that (i)

(i)

u −→

θmλα (t, u) (0)

θmλα (t, u)

214

IV Further developments (i)

is a local isomorphism at u. Therefore, OSγ ,x is generated by t, for λ ∈ Iγ . Applying Proposition 2.16, (2) for f

(i)

θmλ

α

θmλ

α (0) α

θmλ

and Xλ

, we know that all the θ ’s are in

ψ ∗ OY,y . Then applying Proposition 2.17, all Xλ with λ ∈ Iγ are in ψ ∗ OY,y . Hence ψ ∗ is surjective, and the proof of Theorem 2.2 is complete.

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Index

Γ-admissible collection of polyhedra, 161 Γ-admissible polyhedral decomposition, 37, 75
-fold top differentials, 192 k-gon of rational curves, 20 admissible structure of polyhedral complexes, 183 associated boundary component, 164 Baily–Borel compactification, 165 best rational approximations, 35 boundary component rational of a bounded domain, 140 of a homogeneous self-adjoint cone, 52 boundary component of a bounded domain, 127 central co-core, 92 centralizer of a boundary component, 134 characteristic function of a convex non-degenerate cone, 38 co-core, 82 compact root, 112 compact torus, 2 complementary module, 195 cone, 38 k-irreducible, 56 central, 93 non-degenerate, 38 perfect, 93 self-adjoint, 38 self-dual, 38 conical polyhedral complex, 11 core, 77 cusp, 15, 26 cylindrical set, 100 decomposition into central cones, 90 exposed point, 85

extendable form, 192 extreme point, 82 flag of k-boundary components, 64 standard, 64 fundamental 5-factor decomposition, 143 geodesic projection, 140 Hermann convexity theorem, 120 hermitian symmetric space, 105 Hilbert modular surface, 26 Hilbert modular variety, 103 horocycle, 16 idempotents mutually orthogonal, 46 complete set of, 47 maximal set of, 47 integral structure on a conical polyhedral complex, 12 Jordan algebra, 43 formally real, 47 kernel, 77 rationally locally polyhedral, 84 Kodaira dimension, 194 Koecher’s theorem, 191 Krein–Milman theorem, 82 level-k structure, 20 Levi-pseudoconvex, 104 manifold of general type, 194 mutation, 51 natural basepoint of a boundary component, 133 neat subgroup, 176 non-compact root, 112 normalizer of a boundary component, 129

229

230 ord, 2 Peirce decomposition, 46 perfect co-core, 92 polyhedral cone, 71 Q-rational, 72 polyhedral kernel, 84 projective Γ-admissible decomposition, 199 quasi-hermitian form, 201 rank of a hermitian symmetric domain, 114 rational boundary component, 140 rational partial polyhedral decomposition, 6 real root decomposition, 118 Riemannian symmetric space, 105 Satake topology, 164 semi-conical convex set, 81 semi-proper meromorphic map, 179 Siegel set, 37, 67 strict homomorphism (of semi-groups), 7 strictly or strongly commuting elements, 44 strongly convex subset (of euclidean space), 33 strongly orthogonal roots, 112 strongly pseudoconvex subset (of a complex manifold), 33 symmetric domain, 106 symmetric map, 108 symmetric space non-euclidean, 106 of compact type, 106 of non-compact type, 106 simple, 106 toroidal embedding, 9 of analytic spaces, 9 without monodromy, 10 without self-intersection, 10 torus embedding, 2 tube domain, 98 universal elliptic curve, 15 Voronoi decomposition of the first type, 90 Weil pairing, 20

Index