Automorphic Functions and the Geometry of Classical Domains (Mathematics and Its Applications)

utomorphic Functions and the Geometry of Classical omalns •

I. I. PYATETSKII-SHAPIRO Academy of Sciences, Moscow

GORDON AND BREACH

Science Publishers NEW YORK

LONDON

PARIS

Copyright © 1969 by GORDON AND BREACH, 150 Fifth Avenue, New York, N.Y. 10011

SCIENCE PUBLISHERS, INC.

Library of Congress catalog card number: 78-92622 Editorial Office for the United Kingdom: Gordon and Breach Science Publishers Ltd. 12 Bloomsbury Way London W.C.1. Editorial Office for France: Gordon & Breach 7-9 rue Emile Dubois Paris 14e Distributed ill Canada by: The Ryerson Press 299 Queen Street West Toronto 2B, Ontario

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Publisher's Note This monograph is devoted to the theory of automorphic functions of several complex variables. This is the first Soviet publication devoted to this subject (if we neglect the translation of Siegel's Automorphic Functions of Several Complex Variables). The book contains a detailed discussion of so-called classical domains and Siegel domains. The book is aimed at scientists and post-graduate students studying the theory of functions of a complex variable, as well as students specializing in this field.

Contents v

Publishers Note Introduction . Chapter J-Siegel Domains 1. Siegel Domains of Genus 1 2. Siegel Domains of Genus 2 3. Siegel Domains of Genus 3 4. Bounded Holomorphic Hulls

15 16 21 30 41

Chapter 2-The Geometry of Homogeneous Domains 1. Statement of Fundamental Results 2. j-algebras 3. Normalj-algebras 4. j-ideals 5. Homogeneous Siegel Domains of Genus 2 6. Universal j-algebras . 7. Canonical Models of Bounded Homogeneous Domains 8. Canonical Models of Symmetric Domains 9. The Geometry of Classical Domains . 10. Classical Domains of the First Type . 11. Classical Domains of the Second and Third Types

45 45 46 51 64

66 73 76 80 83 91

114

Chapter 3-Discrete Groups of Analytic Automorphisms of Bounded Domains 1. Introduction 2. Construction of the Extension of the Factor Space!!} / r . 3. Analytic Normal Spaces 4. Poincare Series . 5. Lemmas 6. Arithmetic Groups in Symmetric Domains 7. The Andreotti-Grauert Method

131 131 133 136

Chapter 4-Automorphic Forms Introduction . 1. Fourier-Jacobi Series. 2. Automorphic Forms . 3. The Theorem on Algebraic Relations

163 163 163 173 177

vii

140 146 153 159

viii

CONTENTS

Chapter 5-Abelian Modular Functions 1. Statement of Fundamental Results 2. The Domains K (U, R) 3. The Modular Groups Q5(~{, R)

179 179 184 194

Chapter 6-Classijication of Bounded Homogeneolls Domains 1. Introduction 2. Isometric Mappings . 3. Complexes 4. Construction of j-algebras 5. Homogeneous Imbeddings of Bounded Domains in the Siegel Disk Kn 6. Algebraic j-algebras

199 199 200 204 208 211 216

Appendix Introduction 1. Siegel Domains of Genus 1 and 2 2. Decomposition of a j-algebra Associated with a Commutative Ideal 3. Algebraic j-algebras . 4. Decomposition of a j-algebra Associated with a Commutative Ideal (continuation) 5. Representation of a Homogeneous Domain in the Form of a Siegel Domain of Genus 2

219 219 225 231 238 242 253

References

254

Index.

257

Introduction The theory of automorphic functions of one complex variable was created at the end of the nineteenth and beginning of the twentieth centuries by Klein, Poincare, Koebe and others. The theory of automorphic functions of several complex variables began to develop at the same time. It was only after the work of C. L. Siegel, however, that the theory of automorphic functions of several complex variables became an independent discipline. _ Methods from the theory of Lie group representations, especially the theory of infinite dimensional representations, may playa fundamental role in the theory of automorphic functions. 1. M. Gel'fand's paper [1] contains a survey of applications' of representation theory to the theory of automorphic functions. We will not make direct use of methods from the theory of represent ations, but it is nonetheless possible to see the connection with the theory of representations at many points. This book does not pretend to complete coverage of all the fundamental trends in the contemporary theory of automorphic functions. Its aim is c~nsiderably more modest-to discuss the group of problems associated with: (1) the theorem on algebraic relations for fields of automorphic functions; (2) the geometry of homogeneous and, in particular, symmetric domains in n-dimensional complex spacet; (3) the theory of abelian modular functions. This book makes no pretense at

-r A domain in en is said to be homogeneous if there exists a one-to-one analytic onto mapping that maps any pair of points onto each other. The domain is said to be symmetric if for any of its points Zo there exists an analytic one-to-one onto mapping Po with the following properties: (1) rPo(z) = z only if z = zo; (2) rP~ is the identity mapping. As E. Cartan [1] showed, every symmetric domain is homogeneous. 1

2

THE GEOMETRY OF CLASSICAL DOMAINS

all to completeness even in its treatment of this group of problems. To a considerable extent, the choice of material was governed by the author's personal taste. Let f» be some n-dimensional complex manifold and let r be a discrete group of one-to-one analytic mappings (analytic automorphisms) of the manifold f». We will say that the functions that are meromorphic in f» and invariant with respect to the group rare automorphic functions. The set of such functions obviously forms a field. The object of the theory of automorphic functions of complex variables is to study the algebraic structure of this field. The first problem that we encounter is the following. What is the degree of transcendence of this field? In the most important cases, it is equal to the complex dimension n of the domain f». More exactly, this field contains n+l functions/o,/!, ... ,};, such that: (1) any function in the field is a rational function of these n + 1 functions; (2) they are related by one polynomial relationship. In other words, a field of automorphic functions is a finite extension of a field of rational functions of n unknowns. In what follows we will call this theorem the theorem on algebraic relations. If f» is a bounded domain in e", and this, for the most part, is the case with which we will deal in this book, it is easy to use Poincare series to show that the degree of transcendence of the field of automorphic functions is at least n. Thus, the fundamental difficulties are associated with proving that the degree of transcendence is at most n. C. L. Siegel [7] proved the theorem on algebraic relations for the case in which the factor space f» jr is compact. But in the most important and interesting cases such as the case in which r is Siegel's modular group, the factor space f» jr is not compact. The theorem on algebraic relations was proved for this group, although in a somewhat weaker form, in C. L. Siegel's paper [3], which actually began the development of the modern theory of automorphic functions of several complex variables and is even now becoming a classic. The methods available at present for proving the theory on algebraic relations can be divided into two groups. The first of these methods is based on the following well-known theorem. The degree of transcendence of a field of meromorphic functions of a compact analytic norrp.al space is not greater than the space's complex dimension (Remmert [1 D. As a result, in order to prove that the degree of transcendence of a

INTRODUCTION

3

field of automorphic functions is not greater than n, it is sufficient to imbed q; jr in a compact analytic normal space M in the form of an everywhere dense set and to show that every meromorphic function on q; jr extends to all of M. In order to prove this last, it is sufficient to show that the complex dimensions of M' = M - q; jr is no greater than n - 2, where 11 is the dimension of q;Jr. An analogous and independent result was obtained in the work of W. L. Baily, Tr. A.Borel [1].

In Chapter 3 we will, for any arithmetic group (in the sense of A. Borel) acting on a symmetric domain, explicitly extend the factor space q; jr to a space M with all the necessary properties. t For the case in which r is Siegel's modular group, a construction for such an extension was first given by Satake (1. Satake [1]). Satake's extension coincides with the one obtained by applying the general construction to the special case of Siegel's modular group. Constructions ofthis extension were given by W.L. Baily, Tr. [4] andI.I. PyatetskiiShapiro [11] for the modular groups associated with the set of all abelian varieties with a given ring of endomorphisms. All present examples of the construction of the analytic normal extension M lead to the same result as the general construction of Chapter 3. Another method of proving the theorem on algebraic relations is a consequence of C. L. Siegel's remark that a somewhat weaker formt of the theorem on algebraic relations follows from the fact that the dimension Am of a space of automorphic forms of weight m increases no faster than l1'Z" as m ---700, where n is the complex dimension of the domain q;. As a result, the problem reduces to estimating Am. This estimate requires that we use some form of the principle of the maximum for analytic functions. Such a method has been used to prove the theorem on algebraic relations for the case in which q; jr is compact. For the case in which q;jr is not compact, it was this method that was first used to prove the theorem on algebraic relations for Siegel's modular group. In this case, the apparatus afforded by Fourier series

-r The unit disk Izl < 1 is an exception; here the dimension of M' = M- ~/r cannot be less than 11 - 1, as a result of which the definition of automorphic functions must, in this case, be subjected to additional constraints that will make it possible to analytically extend them to all of M. t This weaker form of the theorem asserts that the field of all functions representable in the form of ratios of automorphic forms of one weight is a finite algebraic extension of a field of rational functions of 11 unknowns.

4

THE GEOMETRY OF CLASSICAL DOMAINS

was used, while more complex cases (other discrete groups) required use of Fourier series whose coefficients were theta functions (FourierJacobi series). In Chapter 4 we will use this method to prove the theorem on algebraic relations for arbitrary arithmetic groups. This method was considerably improved in the remarkable paper [1] by Andreotti and Grauert. In this new form, first of all, it requires no special expansions of the Fourier-series type, and, second, it makes it possible to prove the theorem on algebraic relations for the field of all automorphic functions, and not only for those functions representable in the form of a ratio of automorphic forms. The above-noted paper by Andreotti and Grauert describes a particular class of groups, which the authors call pseudoconcave groups, for which the theorem on algebraic relations is valid. In Chapter 3 we will show that except for certain trivial exceptions, all arithmetic groups acting on symmetric domains are pseudoconcave. Thus, in this book we will give three proofs for the theorem on algebraic relations for arithmetic groups in symmetric domains. All of these proofs make implicit use of the notable work by A. Borel and Harish-Chandra on the structure of the fundamental domains of arbitrary arithmetic groups. Their work is an extension of the classical investigations of G. Minkowski and C. L. Siegel. The description given by A. Borel and Harish-Chandra (especially the description given in Borel's paper [3]) of the fundamental domains of arithmetic groups are no less definitive and complete than the corresponding descriptions given by G. Minkowski and C. L. Siegel for the fundamental domains of groups of integral unimodular matrices. Let us recall the definition of an arithmetic group (A. Borel [3]). Let G be a semisimple algebraic group defined over the field of rational numbers Q. Let k be some ring. We agree to let Gk denote the set of all matrices in G with elements in the ring k and determinant equal to the identity of the ring k. In particular, GR is the set of all real matrices in the group G, Gz is the set of all integer matrices in G with determinant equal to ± 1, etc. A discrete subgroup r of the group GR that is commensurablet with the group Gz is called an arithmetic group. Let K be a maximal compact subgroup of the group GR, and let X = GRi K be the corresponding symmetric space. By an arithmetic subgroup of the group GR we will mean an arithmetic group of transformations of the

t Two subgroups r 1 and r 2 of the group GR are said to be commensurable if their intersection is of finite index in each.

INTRODUCTION

5

symmetric space. If the space X is a symmetric domain in Gil, we call such groups arithmetic groups of analytic automorphisms of the domain X. In the same paper, A. Borel gave a construction for a compactification of the fundamental domains in symmetric spaces for arbitrary arithmetic groups. This construction was proposed earlier by 1. Satake [3] for classical symmetric spaces. As a rule, Satake's and Borel's extensions of fundamental domains do not have a complex structure. This will be clear if we use the following interpretation of the maximal Borel and Satake extensions. As a preliminary, we need the following definition. We will say that a subgroup B of a semisimple algebraic group G defined over Q is solvable and splitt (over Q) if it is a Satake subgroup and it is a maximal solvable and split (over Q) normal subgroup divisor in its normalizer. As above, let K be a maximal compact subgroup of the group GR and let X = GR/K be the corresponding symmetric space. Let L be the set of all orbits {B R x}, where x E X and B is some Satake subgroup of the group G. The identity subgroup is always a Satake subgroup, so that X C L, and it is clear that the group GQ and, a fortiori, Gz operate on L. We sett S = L/Gz . Borel's basic result consists of the fact that the space S is compact in some natural topology . Unfortunately, even in the case in which S has a complex structure, it is impossible to introduce the structure into the space S "naturally". Here we are using the word "naturally" in the following sense. The space S contains X/G z as a closed everywhere-dense subset. If X has a complex structure, then X/G z also has a complex structure. We will say that the complex structure in X is natural if it induces the same complex structure in X/G z • It is easy to show that S itself has more than one natural structure. In Chapter 3 the Borel-Satake construction is changed so that it leads to a complex normal space M when the initial space X has a complex structure. This change consists of replacing a space of orbits by a space of bounded holomorphic hulls of orbits. Definition: Let :»

t An algebraic group B defined over a field k is said to be solvable and factorable (over k) if it contains a sequence of normal subgroup divisors Bk(k = 1, ... , m) such thatBl B, Brn E, and the factor group Bk/Bk+l is isomorphic as an algebraic group to the additive or multiplicative group of the field k. t For the sake of simplicity, we wi111imit the compactification to the case in which r = G z •

6

THE GEOMETRY OF CLASSICAL DOMAINS

be a bounded domain and let Y be some subset of~. The bounded holomorphic hull O( Y) is the set of all z E ~ such that

1¢(z)1 ;£ sup 1¢(z)1 ZEY

for any function ¢(z) that is bounded and regular in ~. Let be the set obtained when the elements of a space L are replaced by their holomorphic hulls. We set M = S)}(fG z • The fact that the mapping of S onto M is continuous implies that M is compact. The proof that the space M is an analytic normal space uses essentially one theorem that is valid for arbitrary bounded homogeneous domains. Let ~ be some bounded homogeneous domain in e", B be some group of analytic automorphisms of the domain ~, B(B) be the normalizer of the group B in the group of all analytic automorphisms of domain~, and let SJc(B) be a maximal subgroup of B(B) that is solvable and split over R. If m(B) = B, we will say that the group B is a Satake group. We have the following theorem. A fibering of a domain into bounded holomorphic hulls of orbits of any Satake groups is always a homogeneous analytic fibering. t Our proof of this theorem is based on a description of all homogeneous fiberings of bounded domains ~; this description is given in Chapter 2. We should also note the following problem, which is clearly related to possible arithmetic applications of the theory of automorphic functions. Let r be an arithmetic group of analytic automorphisms of a symmetric domain ~ c: C l • Let P be the field of functions automorphic with respect to the group r. We will call a number field k with the following property the field of definition of the field P: P contains 11 + 1 functions io, ib ... ,ill such that: (1) every function in P is a rational function of these n + 1 functions, and (2) the coefficients of the polynomial relationship between the functionsio,il' ... ,ill belong to k. Although it is quite likely, the author does not know of any proof of the proposition that if r is an arithmetic group, the field of definition

we

t We will say that a fibering of a domain !!fi is analytic if its base is a complex analytic manifold and the projection onto the base is an analytic mapping. As a lule, the fiberings discussed in this book are locally non-trivial, and, in this case, the analytic (both in the Complex and real senses) fiberings are usually direct produ::ts. We will say that a fibering is homogeneous if the fibering-preserving set of analytic automorphisms of domain!!fi is transitive on!!fi.

INTRODUCTION

7

of the automorphic functions is a finite extension of the field of rational numbers Q. The second theme to which this book is devoted is certain problems in the theory of complex domains, which problems appear in connection with the theory of automorphic functions and, in particular, in connection with the problem of describing the fundamental domains for the case in which fYfi/r is not compact but has a finite volume. In Chapter 1 we ·will describe a general construction for Siegel domains-natural multidimensional analogs of the upper halfplane 1m z > o. We should note that Siegel domains of genus 2, which are the most important special class of Siegel domains, appeared in connection with the problem of constructing the analog of Fourier-series expansions of functions automorphic in the ball IZl12 + IZ212 < 1. It turns out that for this purpose it is convenient to map the ball onto the following domain: (1)

This domain can be treated as an analog of the upper halfplane 1m z 1 > O. The following transformations play the role of parallel translations in this domain:

+a+2iz2 E+ Ibl2 --+z2+ b

Zl --+ Zl Z2

(2)

where a is some real number and b is an arbitrary complex number. We will also call transformations of the form (2) parallel translations. Let t1 be a discrete subgroup of transformations of the form (2) with compact fundamental domain in the group of all transformations of the form (2). As we can show with little difficulty, the functions that are invariant with respect to the group t1 can be expanded in series of the following form: (3)

where A is some real number uniquely defined by the group t1 and ¢k(Z2) is a Jacobi function. We will call series of this type FourierJacobi series. In Section 1 of Chapter 4 we will discuss the general theory of such series. In Chapter 4 we will use these series to prove the theorem on

8

THE GEOMETRY OF CLASSICAL DOMAINS

algebraic relations for functions that are automorphic with respect to arithmetic groups. This proof makes no appeal to the theory of analytic normal spaces or considerations related to analytic convexity. In the end, however, the method of Chapter 3 is preferable. It seems to us that the apparatus provided by Fourier-Jacobi series and Siegel domains may be of interest in problems lying between the theory of automorphic functions and the theory of numbers. For example, it can be used to find the field of definition of a field of automorphic functions and in the theory of generalized Hecke operators. Chapter 6 and, in part, Chapter 2 of this book are devoted to a classification of the bounded homogeneous domains in e". Here the principle of reducing geometric problems to the study of Lie algebras with the same or additional properties proves to be very effective. We will usei-algebras (see Section 2, Chapter 2) to study bounded homogeneous domains. Ai-algebra is a Lie algebra having the additional structure of a complex Hermitian space. The system of axioms for i-algebras was chosen so that we have the following theorems. (1) If a group ® is transitive in some bounded homogeneous domain ~, its Lie algebra is ai-algebra. (II) If G is a j-algebra, the corresponding Lie group ® is transitive in some bounded domaint ~. We should note that the first of these theorems is a corollary of two well-known results: (1) the conditions that occur when a homogeneous manifold has an invariant complex structure, and (2) the work of Koszul on evaluation of the Bergman metric in terms of Lie algebras (1. L. Koszul [1]). Conversely, the proof of the second theorem requires relatively complex algebraic apparatus. We will obtain the following proposition incidentally. Let M be a homogeneous n-dimensional complex manifold, and let dv = k(z)dz1A ... dzllAdz1A ... dzn

be an invariant volume in some local coordinate system. We agree to call the manifold M a homogeneous Bergman manifold if the form

a2 1n k I-a a- dzadzp Za zp is positive definite.

t The group @ defines the domain!!} uniquely.

(4)

INTRODUCTION

9

We have the following theorem. Every homogeneous Bergman manifold can be analytically mapped by a one-to-one mapping onto some bounded domain in C". The most important class of j-algebras is the class of j-algebras that are Lie algebras that are simultaneously solvable and split over R. Suchj-algebras are called normalj-algebras. In Section 3 of Chapter 2 we will study the systems of roots of normalj-algebras. In so doing we will prove that every normal j-algebra G can be represented in the form of a sum H + K, where K IS a nilpotent subalgebra and H is a commutative subalgebra; and the representation of H onto K is semisimple. As a result, the space K can be decomposed into the sum of root spaces Ka, where a is a linear form on H (Ka consists of all x E K such that [h,x] = a(h)x for any hEH). The system of roots of any j-algebra always has the following form: (5)

where a 1 , ... , a p are linearly independent and form a basis for the space of all linear forms on H. The dimension Na of the root spaces Ka is, generally speaking, different from unity and, in particular, may be equal to zero. We should note that the number Na does not uniquely determine a j-algebra. Additional invariants are subjected to thorough study in Chapter 6. In Section 5, Chapter 2 we will describe a construction for Siegel domains of genus 2 corresponding to a given normalj-algebra. The article by E. B. Vinberg, S. G. Gindikin and 1. 1. PyatetskiiShapiro in which it is shown that every bounded homogeneous domain :» has a transitive solvable Lie group is a supplement to this book. This theorem implies that the correspondence between normalj-algebras and bounded homogeneous domains is one-to-one. The system of axioms defining normal j-algebras is such that the existence of even one normal j-algebra is not immediately obvious. In addition, the problem of describing them completely is not clear. In Chapter 6 we will provide a universal and rather convenient method for constructing normal j-algebras. The notion of a complex plays a fundamental role in this chapter. We will now define a complex. First we must introduce the notion of an isometric scalar product in the tensor product of two Euclidean spaces. Let X and Y be Euclidean

10

THE GEOMETRY OF CLASSICAL DOMAINS

spaces. A nonnegative scalar product in their tensor product Xx Y is said to be isometric if (x x y, x x y) = (x, x)(y, y)

(6)

for any XEX, yE Y. Generally speaking, the isometric scalar product is degenerate. For example, assume that a law of multiplication Xl, X2 -7 Xl x 2 with the following property is defined in the space xt: (7)

We then define an isometric scalar product in Xx X in the following way: (Xl X X2, X3

x x4)

=

(Xl X2, X3 X4)

If one of the spaces X or Y has a complex structure, the space X x Y also has a complex structure and an isometric scalar product must, in this case, be invariant with respect to multiplication by i. We will now define a complex. A complex of rank p is a set consisting of: (1) Euclidean spaces XknI' 1 ~ k < m ~ p; (2) Hermitian spaces Zk' 1 ~ k ~ p; (3) isometric scalar products in the spaces X km X Xms and X km X Zm. In addition, it is required that the isometric scalar products be compatible. We define compatibility in the following manner: let n~bX2 be a linear transformation of the space X klll such that for any X, x' EXkm (8)

so that if (1) t < k, then Xl' X 2 E X tk , or (2) if t = 111, then Xl' X 2 EZII" or (3) if t > m, then Xl' X 2 E X l1It • The compatibility condition consists in the requirement that the transformations n~1,X2 and n~1'Y2 commute if t < k < m ~ s. In Chapter 6 we will describe construction of the complex corresponding to a given normal j-algebra. We will also show that the correspondence established in this manner is one-to-one. We will also find a connection between the notion of a complex and E. B. Vinberg's [5] theory of generalized matrix algebras (T-algebras). T-algebras are

t As we know, multiplication can be introduced into X in one of the following four cases: (1) X is the field of real numbers; (2) X is the field of complex numbers; (3) X is the field of quaternions; (4) X is the algebra of Cayley numbers.

11

INTRODUCTION

bigraded algebras with an involution. Generally speaking, T-algebras are not associative. The central result of the theory of T-algebras is the theorem that states that it is possible to represent every convex homogeneous cone in the form of the set of all positive definite Hermitian matrices of some T-algebra. We will now give several examples. (1) The n+ 1 dimensional ball

Iz d + ... + IZII+ 2

112

<1

corresponds to the complex of rank 1 consisting of an n-dimensional Hermitian space. (2) The complex of rank p in which dimXkk + 1 = 1, 1 ~ k ~p-1, dimXklll = 0, if 111-k ~ 2, dimZk = 0, 1 ~ k ~ p is the complex corresponding to the so-called Siegel circle K p , i.e., the set of all complex p x p symmetric matrices Z such that

ZZ<E where E is the identity matrix. One of the most interesting and important cases of arithmetic groups is the class of groups for which the theory of abelian functions is valid. First of all, we should recall the classical connection between elliptic functions and ordinary modular functions, which, in essence, led Gauss to discover these last. Modular functions and modular groups naturally appear in the study of the manifold of all nonisomorphic fields of elliptic functions. Modular functions are functions of this manifold. As we know, two fields of elliptic functions are isomorphic if and only if the period lattices corresponding to them are obtained from each other by a linear transformation (z ---7 CiZ) of the complex plane. As a result of this, we can construct the manifold F of all nonisomorphic fields of elliptic functions in the following manner. Consider the set Q of pairs of complex numbers (W1' ( 2 ) such that the ratIO W 2 /W 1 is not real. Two fields of elliptic functions with fundamental periods (Wb ( 2 ) and tRecalI that an abelian function is a merom orphic function in CP with 2p periods that are linearly independent over the field of real numbers.

12

THE GEOMETRY OF CLASSICAL DOMAINS

(w~, w;) are isomorphic if and only if there exists a complex number a and an integral matrix A with determinant ± 1 such that

(9)

The manifold F can obviously be obtained by identification of all pairs (Wl' ( 2) that correspond to isomorphic fields of elliptic functions. First we identify the pairs (Wl' ( 2) and (awl' a( 2), where a is an arbitrary

nonzero complex number. This leads to a set representing the complex plane with the real axis removed. Let K be a connected component of the set thus obtained. For example, assume that K is the upper halfplane 1m L > O. It is easy to verify that two points L 1 and L2 correspond to isomorphic fields of elliptic functions if and only if there exists an integral matrix A with determinant equal to unity such that L..,

~

=

aL 1

eLl

+ b, A = +d

(ac

db)

(10)

Thus, F can be treated as the factor space Kjr where K is the upper halfplane and r is a discrete group of analytic automorphisms of the domain K. The group r obtained in this manner is called a modular group. Unfortunately, the scheme given above does not generalize to the case of a large number of variables. The difficulty results from the fact that the set of nonisomorphic fields of abelian functions of a given number of variables beginning withp = 2 is not a manifold. The reason for this is that no 2p vectors independent over the reals of a p-dimensional space can be periods of a nondegeneratet abelian function. It is also not difficult to show that the system of periods for abelian functions forms an everywhere dense set in the manifold of all systems of 2p vectors independent over the reals in a p-dimensional complex space. The changed scheme is as follows. Certain analytic manifolds Q are chosen in the set of systems of periods, and a construction similar to the above construction for an ordinary modular group is applied to each manifold Q, but with certain necessary changes. The changes consist of

t An abelian function of p variables is said to be non degenerate if its set of shifts, i.e., functions of the formf(z + r), where rE CP contains p analytically independent shifts.

INTRODUCTION

13

identifying not only the systems of periods that correspond to isomorphic fields of abelian functions, but systems of periods in which the isomorphism of the abelian functions extends to some neighborhood in Q. The manifold F obtained from Q as a result of this identification can be treated as a factor space K/r, where K is some classical domain and r is an arithmetic group of analytic automorphisms of domain K. This group r is also called a modular group corresponding to the manifold Q. Thus, an infinite number of modular groups correspond to the abelian functions of a given number of variables. In contrast to ordinary modular groups, some of these have compact fundamental domains. Chapter 5 contains a classification of all modular groups associated with abelian varieties. In conclusion, the author would like to express his gratitude to 1. M. Gel'fand for his friendly support. A. O. Gel'fond provided the initiative for writing this book. While the first edition of this book was being prepared, and especially while the English edition was being prepared, many of its sections were discussed in the seminars of I. R. Shafarevich and E. B. Dynkin. The author took part in these seminars. The many discussions with 1. R. Shafarevich were particularly important. In conclusion, the author would like to thank E. B. Vinberg and S. G. Gindikin, who proofread the manuscript and gave much valuable advice.

CHAPTER 1

Siegel domains In this chapter we will study certain unbounded domains in an affine complex space; these domains are the analogs of the upper halfplane for the case of one variable. We call these domains Siegel domains. As we will show in the next chapter, any classical domain can be mapped onto some Siegel domain. In this chapter we will discuss the general properties of Siegel domains, give a constructive definition, and give conditions sufficient for a Siegel domain to be analytically homogeneous. In Section 1 we will discuss the simplest Siegel domains, which we call Siegel domains of genus 1; in Section 2 we will discuss Siegel domains of genus 2; and in Section 3 we will discuss Siegel domains of genus 3. Note that Siegel domains of genus 1 may be treated as special cases of Siegel domains of genus 2, and Siegel domains of genus 2 may be treated as special cases of Siegel domains of genus 3. In Section 4 we will describe a class of complex manifolds that are similar in their properties to bounded domains. Througho'ut the book we will use the following notation: A' is the transpose of a matrix A; A is the conjugate of A; A* =A';

det A is the determinant of A; E,. is the identity matrix of order r; when we need not indicate the order, we will simply write E; A > 0, where A is a Hermitian matrix, indicates that the characteristic values of matrix A are positive; e" is a complex n-dimensional affine space. 15

16

Section 1.

THE GEOMETRY OF CLASSICAL DOMAINS

Siegel Domains of Genus 1

In this section we will define Siegel domains of genus 1, we prove that all Siegel domains of genus 1 may be mapped onto some bounded domain, and we will find the form of the analytic automorphisms that preserve the "point at infinity" of S. Let V be an open convex cone in an n-dimensional real spacet whose intersection with any line is either a segment or a half line. Henceforth we will discuss only such cones. Definition. We agree to call a set S of points of the form z = x + iy (y E V, x arbitrary, Z E en) a Siegel domain of genus 1. We will prove that any Siegel domain S of genus 1 is analytically equivalent to a bounded domain. Using the fact that the cone V does not contain a whole line, we can prove the existence of a coordinate system in which V lies inside the octant Yl > 0, ... 'YII > 0. As a result, S is contained within the domain Imzl > 0, ... ,ImzlI > 0, which is analytically equivalent to the product of n disks. We agree to call the part of the boundary of S that consists of points of the form Z = x the skeleton. Consider the set E of all functions that are analytic in St and attain their maximum in S. It is easy to show that for any function ¢(z)EE there exists a point in the skeleton on which the modulus of the function reaches a maximum. On the other hand, for any point in the skeleton there exists a function ¢(z) EE whose modulus reaches a maximum at this point. For the point ZO = (x~, ... , x~), in fact, the function sought is ¢(z) =

1

(1)

o. 0 .. (Zl-Xl +t) ... (zlI-Xn+ 1)

It follows from the hypothesis that the skeleton of domain S is mapped into itself under the automorphisms of S that are analytic in S. Theorem 1. Any analytic automorphism of the domain S that is continuous in S has the form Z -?

(2)

Az+b,

t A set of points in an n-dimensional real space is called a any point it contains every half line connecting it to the origin.

COlle

if together with

t S denotes the closure of a domain S in the natural topology of the affine space.

SIEGEL DOMAINS

17

where A is an affine transformation of the cone V onto itself and b is a real vector. For the proof of this theorem we will need Chebotarev's Lemma (Levin [1], p. 229), which says that a function g(A) that is analytic in the upper halfplane 1m A > 0, continuous in 1m A ~ 0, and takes real values on the real axis is a linear function. Without loss of generality, we may assume that S is contained in the domain Imzl > 0, ... ,Imzll > 0. Let z ~ cfJ(z) = (cfJl(Z), ... , cfJn(z» be an analytic automorphism of S that is continuous in S. The auxiliary function

g(A) = cfJixo + Ayo),

1 ~ k ~ n,

where satisfies the conditions of Chebotarev's Lemma and, consequently, is a linear function. Thus, z ~ cfJ(z) is a linear transformation in the n-dimensional complex space and may therefore be represented in the form cfJ(z) = Az+b, where A is some complex matrix and b is a complex vector. The skeleton of the domain S is mapped into itself by the mapping z ~ cfJ(z), so that A and b are real. If we separate the real and imaginary parts, we find that '

cfJ(z) = Ax + b + iAy, i.e., if y E V, then Ay E V. The inverse transformation z ~ cfJ -l(Z) has the form

As a result, yE V implies that A-lYE V. We have thus proved that A is the matrix of an affine transformation of the cone V onto itself. This completes the proof of the theorem. It is well known that any bounded domain has a volume that is invariant under the analytic automorphisms of this domain (see Section 4). We will find the formula for an element of the invariant volume in the domain S. Let

du = A(X, y) dx dy,

(3)

where dx = dX 1 ••• dx", dy = dYl ... dYIl" Since the domain S admits transformations of the form z ~ z+b, where b is any real vector, the

18

THE GEOMETRY OF CLASSICAL DOMAINS

coefficient;t must be independent of x. As a result dv

;t(y) dx dy.

(4)

Furthermore, if y ~ Ay is an affine transformation of the cone V, then z ~ Az is a transformation of S, and ;t(Ay)(detA)2 = ;t(y).

(5)

If the cone V is linearly homogeneous (i.e., for any pair of points Yb Y2 E V there is an affine transformation of V onto itse]f that maps Y1 onto Y2), then the domain S is analytically homogeneous. Such domains S are extremely interesting. Note that ;t(y) is uniquely determined by (5) in this domain. If V 1 is a homogeneous cone in an n1-dimensional real space and V2 is a homogeneous cone in an nrdimensional real space, then the set of points (Y1,Y2), Y1 EVi , Y2 E V2, forms a homogeneous cone in an n 1+n 2dimensional real space. Cones that cannot be obtained by this method are called irreducible. Below we give examples of irreducible homogeneous cones and the Siegel domains corresponding to them. 1. Consider the p x p Hermitian matrices Y = (Yks). Each matrix Y may be associated with a point in a p2-dimensional real space with coordinates

The set of points corresponding to the positive definite Hermitian matrices clearly forms a cone. The affine transformations of this cone have the form Y~A*YA,

where A is any nondegenerate p x p complex matrix. The corresponding Siegel domain may be conveniently described as the set of complex p x p matrices Z = X + i Y such that X is any Hermitian matrix and Y is a positive defimte Hermitian matrix. This domain is symmetric (see the footnote on p. 1). An involution at the pointZ = iEhas the formZ ~ _Z-l. As we will show in the next chapter, this domain is analytically equivalent to a domain of the first type when p = q (according to E. Cartan's [3] classification of symmetric domains; also see Siegel [1]).

19

SIEGEL DOMAINS

2. Consider the 2p x 2p Hermitian matrices Y for which (

YJ=JI',

. o

o

o

0

)

j

j=

J=

o

0

...

( 0 1) -1

(6)

0

j

We set Y = (Yks), k, s = 1, ... ,p, where Yks is a 2 x 2 matrix. Relationship (6) implies that As a result Ykk

=(

U kk

o

0) Ukk

(k

< s),

'

where Ukk' a ks ' bks ' Cks , and dks are real numbers that may be taken for coordinates. These matrices Yform ap(2p-l)-dimensional real space. The set of points in this space that correspond to the positive definite Hermitian matrices clearly forms a cone. The affine transformations of this cone have the form y

~A*YA,

where A is any nondegenerate 2p x 2p complex matrix that satisfies the condition AJ = flf. It can be shown that the cone obtained consists of all positive definite quaternion matrices. The corresponding Siegel domain may be described as the set of all 2p x 2p complex matrices Z = X + i Y, where XJ = JX,

X* = X,

YJ = Jr,

y* = Y

and

Y > O.

In other words, ZJ = JZ' and the matrixi l(Z -Z*) is positive definite. The domain obtained is symmetric. An involution at the point Z = iE has the form Z ~ - Z-l. In Cal'tan's classification this domain is analytically equivalent to a domain of the second type with even p. 3. Consider all real symmetric matrices Y = (Yks) of order p. With each matrix Y we may associate a point in a tp(p + 1)-dimensional real space with coordinates

20

THE GEOMETRY OF CLASSICAL DOMAINS

The set of points corresponding to the positive definite symmetric matrices forms a cone. The affine transformations of this cone have the form Y~A'YA,

where A is any nondegenerate real matrix of order p. The corresponding Siegel domain may be described as the set of symmetric complex matrices Z = X + i Y of order p, where X is any real symmetric matrix and Y is a positive definite real symmetric matrix. This domain is also symmetric. As above, an involution at the point iE has the form Z ~ - Z-l. This domain is analytically equivalent to a domain of the third type in E. Cartan's classification; in the literature it is frequently called "Siegel's generalized upper halfplane". 4. Consider the n-dimensional real space whose points are denoted Y = (Y1' ""YI/)' A cone is given by the inequality Y1Y2-Y~- ... -Y; > 0,

Y1 > O.

The affine transformations of this cone have the form

y~Ay,

A'HA=AH,

H=

[01. to 20

°1 -E

where A is any positive number. The corresponding Siegel domain consists of all points of the form Z = x + i Y in n-dimensional complex space, where x is arbitrary and Y is contained in the cone; it is symmetric. An involution at the point Z = (i, i, 0, ... ,0) has the form -Z2 -Zl

Z

Z3

Zn )

= (Zl, Z2' ... , zn) ~ ( A(Z)' A(Z) , A(Z)' ... , A(Z) , A( z) =

Z1 Z2 -

d - ... - Z~.

As we will show in the next chapter, this domain is analytically equivalent to a domain of the fourth type in Cartan's classification. It is not difficult to show that the cones described above are selfadjoint, i.e. the enveloping space contains a positive definite quadratic form H(x,y) such that: (1) H(x,y) > 0 for all X,YE V, (2) if H(xo, y) > 0 for all y E V~ then Xo E V. It is not difficult to show that the Siegel domain corresponding to a

SIEGEL DOMAINS

21

selfadjoint cone is a symmetric domain. In addition to the cones listed above, there is one more irreducible cone in 27-dimensional space. This cone can be realized by means of third-order Hermitian matrices over the Cayley numbers (E. B. Vinberg [1]). Work by M. Koecher [3], E. B. Vinberg, and Hertneck [1] has established the connection between homogeneous selfadjoint cones and Jordan algebras. The simplest example of an affine homogeneous non-selfadjoint cone is the following. Consider the set V of all 3 x 3 symmetric positive definite matrices Y = (Yks) such that Y31 = Y13 = 0. This example was provided by E. B. Vinberg [1]. E. B. Vinberg [5], Rothaus [1], and Koszul [3] have worked on classifying convex affine homogeneous cones. E. B. Vinberg has also found an interesting class of non-associative algebras (left-symmetric algebras and T-algebras) that are in one-to-one correspondence with the convex affine homogeneous cones. B. Yu. Veysfeyler has also worked on the theory of left-symmetric algebras. A particularly interesting study of E. B. Vinberg is the following: construct any convex affine homogeneous cone as the set of all "positive definite Hermitian matrices" of some T-algebra (E. B. Vinberg [5]). Section 2. Siegel Domains of Genus 2

In this section we will define Siegel domains of genus 2 and we will consider problems about these domains; the treatment is similar to that of Siegel domains of genus 1 in Section 1. The simplest example of a Siegel domain of genus 2 is the following domain: Imz-\u\2 > 0,

(1)

where z and u are numerical complex variables. The following problem about this domain naturally arises: map the sphere

onto some domain S in a manner such that all transformations of the sphere that leave a given boundary point fixed are linear transformations of S.

22

THE GEOMETRY OF CLASSICAL DOMAINS

Direct computation, which we omit, easily shows that the domain (1) is actually a solution to this problem. Later on (Chapter 2) we will prove a general theorem from which our assertion follows as a special case. Before we give a general definition of Siegel domains of genus 2, we will introduce the concept of V-Hermitian vector functions, which are generalized Hermitian positive definite forms. In n-dimensional real space cn, let V be a convex cone not containing an entire line, F(u, v) a vector function with domain (u, v), U, VE CIII (generally speaking, m :::f. n), and range contained in C II • The vector function F(u, v) is called V-Hermitian if (1) F(u, v) = F(v, u); (2) F(AUl + J.1Uz, v) = A(Fub v) + J.1F(uz, v), where A and J.1 are arbitrary complex numbers; (3) F(u, u) E V (Vis the closure of V); (4) F(u, u) = 0 only ifu = O.

Definition. We agree to call the set S of all points (z, u) E CII+m for which

Imz-F(u,u)EV.

(2)

a Siegel domain of genus 2. Here is an example of a Siegel domain of genus 2 in

1m z

-I

U

liz - ...

-I

U m IZ

em + 1 :

> 0,

(3)

where z, U j , ••• , Um are numerical complex variables. We will show that this domain is analytically equivalent to the sphere

We set

z-i z =-1

z+ i'

It is easy to show that m-+1

1-

2

I IZkl z=-,z+-./z(Imz-lu1Iz- ... -lumlz.),

k=l

1

whence our assertion follows at once.

23

SIEGEL DOMAINS

In Section 1 we noted that every Siegel domain of genus 1 is contained in a domain that may be mapped onto the product of n disks. Here we will prove that any Siegel domain of genus 2 is contained in a domain that may be mapped onto the product of n spheres. Without loss of generality, we may assume that V is contained in the cone Yi > 0, ... 'YlI > O. Then every component Fk(U, u,) k = 1, ... , n will be a positive definite Hermitian form in m variables Ul, ••. , U m • We express each of the forms Fiu, u) in the form of a sum of squares of linear forms (4) We now construct a vector function F(n, n), such that (1) the domain S is contained in the domain S which is given by the inequalities

(2) the domain S is analytically equivalent to the product of n spheres. We set F 1(u,u) = Fl(U,U). We define Fz(u,u) in the following manner. We eliminate each of the linear forms L Z1 ' ••• ,L 2s2 that is a linear combination of L l l , ... ,L is !. We set

Fz(u, u) =

I' IL

2 2s 1 ,

s

where the prime indicates that the sum is taken only over the forms that have not been eliminated. Furthermore, we eliminate the forms among L 31 , ... ,L3s3 that are linear combinations of the forms Lks(l ~ k ~ 2). We set

Flu, u) =

I' IL 3s1

2

,

s

where, as before, the prime indicates that summation is performed only for the forms that have not been eliminated. We define F4(U, u) etc. similarly. Clearly S c We must still show that the domain S is analytically equivalent to the product of n spheres. Section (4) of the definition of F(u, v) implies that the equations Lks = 0 (1 ~ k ~ n, 1 ~ S ~ Sk) have the unique solution u = o. As a result, the number of linear forms that have not been eliminated is equal to m. By construction, they are linearly

s.

24

THE GEOMETRY OF CLASSICAL DOMAINS

independent. We construct new variables u~, ... , u~J from them. In the variables Zb ••. , Zm u~, ... , u~, the equations defining S have the form

(5)

where m1' m 2, ... , mil -1 are integers. To complete the proof, it is sufficient to note that each of the inequalities obtained defines a Siegel domain of genus 2 that is analytically equivalent to a sphere (see (3)). We have proved that any Siegel domain of genus 2 is analytically equivalent to a bounded domain in C"+ m• For the domain S, the part of the boundary that consists of the points (z, u) such that 1m Z = F(u, u) is called the skeleton. It is possible to show that every function that is analytic in S and whose modulus has a maximum in S has at least one maximum-modulus point on the skeleton. On the the other hand, it is easy to construct an example of function whose modulus has a maximum at a preassigned point on the skeleton. For the domains discussed in Section 1, therefore, the skeleton is invariant with respect to the analytic automorphisms of the domain S that are continuous in S. In addition, under any analytic automorphism of S, a point in the skeleton may be mapped either again onto a point in the skeleton or onto the point at infinity. We should note that in contrast to Siegel domains of genus 1, the real dimension of the skeleton is larger than the complex dimension of the entire domain. The following transformations constitute, for Siegel domains of genus 2, an analog of parallel translations: Z

~

z+a +2iF(u, b)+ iF(b, b), }

u~u+b,

(6)

where a is any n-dimensional real vector and b is any m-dimensional complex vector. Sometimes we will also call these transformations "parallel translations". The set of these transformations forms a nilpotent group of class 2.t It is easy to verify that Imz-F(n,n)

t A group G is said to be nilpotent of class 2 if its 'commutant', i.e., the group generated by elements of the form gl g2g1- 1g2- 1 (gl, g2 E G) is commutative.

25

SIEGEL DOMAINS

remains fixed under these transformations. An arbitrary point (z, u) in the domain S may be mapped onto the point (iy,O), where y = Imz-F(u, u), by transformation (6). Generally speaking, the transformations (6) do not exhaust all of the linear transformations that are described by the following theorem. Theorem 1. Any linear transformation of a Siegel domain of genus 2 has theform z ~ Az+a +2iF(Bu, b)+ iF(b, b), } u

~

(7)

Bu+b,

where a is any real n-dimensional vector, b is any m-dimensional complex vector, A is a linear transformation of the cone V onto itself, and B is a complex linear transformation such that AF(u, v) = F(Bu, Bv) for any complex u and v. Proof Let (8)

be an affine transformation of S onto itself. We will use the fact that the skeleton is invariant under transformation (8). The point (0,0) is clearly contained in the skeleton. Its image (I', q) will also be a point on the skeleton, and then 1m r = F(q, q). Multiplying transformation (8) by a suitable transformation of the form (6), we obtain a new transformation such that q = 0 and I' = 0: z ~ R~ z+R~ u,

u ~ Q~ z+Q~ u.

(9)

A skeletal point of the form (x,O) (x real) is mapped onto the point (R~ x, Q~ x). As a result, for any x (ImR1)x = F(Q~ x, Q~ x).

The left side of this equation is linear in x, while the right side is of the second degree in x; this is possible only if 1m R~ = 0, Q~ = O. The point (iy, u), y = F(u, u), is mapped onto the point (iRly+R~ u, Q~ u), whence R~ y+lmR~

u=

F(Q~

u, Q~ u).

(10)

Substituting eiljJ u for u in the relationship thus obtained, we find that 1m eiljJ R'z u is independent of cPo As a result, R~ u = 0 and, because u is arbitrary, R~ = 0 as well. Substituting the expression for y in terms of u

26

THE GEOMETRY OF CLASSICAL DOMAINS

into (10), we find that Ri F(u, u) = F(Q; u, Q; u). We have shown that (9) and, consequently, (8), are of the form (7). The theorem is proved. Let Q be the set of all linear transformations A of a cone V that extend to linear transformations of all of a domain S, i.e., are such that for some complex linear transformation B, AF(u, u)

= F(Bu, Bv).

(11)

It is easy to verify that if Q is transitive on V, then the corresponding domain S is analytically homogeneous. We will now give several examples of homogeneous Siegel domains. A domain S can be uniquely described by giving a cone V and a V-Hermitian vector function F(U, V). The domain that we will discuss in the first example is the simplest non symmetric domain, its complex dimension being equal to 4. In the second example we will construct a set of one (continuous) parameter domains in 7-dimensional complex space. As we will show in Section 5 of Chapter 6, these domains are not analytically equivalent. We should also note that when 11 < 7, the number of bounded homogeneous domains in ell is finite. The domains of example 3 are symmetric. (1) In 3-dimensional space R 3 , let V be the cone given by the inequalities

(12) Moreover, assume that m = 1, and F(u,v) = (uv, 0, 0). In other words, our domain S 1 is given by the inequalities 2 2 Y1 u > 0, (Y1 u )Y2 - y~ > (13)

-l l

-l I

°

where Yk = Imzb k = 1,2,3. Here Zb Z2' and Z3 are complex coordinates. It is not difficult to show that the group Q consists of transformations of the form Y1 ~ Ai Y1 + fl2Y2 + 2A1 flY3 Y2 ~ A~ Y2

(14)

Y3 ~A2flY2 +A 1 A2 Y3 where A1 , A2 , and fl are arbitrary real numbers. Indeed, a transformation B: u ~ A1 u, such that F(Bu 1, Bu 1)

= AF(u l' U1)

corresponds to each such transformation A.

(15)

27

SIEGEL DOMAINS

The dimension of the domain Sl we have constructed is equal to 4. In the next chapter we will show that it is the only non symmetric domain in 4-dimensional space. We will now give a simple direct proof of the non symmetry of this domain. The proof is based on the following lemma. Lemma 1. If it exists, an involution of the Siegel domain S of genus 2 with stationary point (zo, 0) always has the form

(z, u) ~ (c/>(z), A(z)u)

(16)

where z ~ c/>(z) is an involution of the Siegel domain of genus 1 with cone V and A(z) is some analytic matrix function. Proof. We write an involution with stationary point (zo, 0) in the form (z, u) ~ (c/>(z, u), A(z, u))

(17)

It is clear that it must commute with any automorphism of domain S that maps the point (zo,O) into itself and, in particular, with the transformation (18)

Thus,

c/>(z, ei6 u) = c/>(z, u),

A(z, ei6 u) = ei6 A(z, u).

(19)

U sing the fact that the transformation is analytic with respect to u, it is easy to use (19) to show that an involution must have the form (16). In order to complete the proof, note that z -+- c/>(z) is an analytic automorphism of the Siegel domain of genus 1 with cone V and the unique stationary point Zo; its square is the identity transformation, and, therefore, this transformation is an involution of this domain. The lemma is proved. We will now show that the domain Sl is not symmetric. Proof. By lemma 1, if an involution exists at the point (i, i, 0, 0), it must have the form

(Z,. Z2, Z,. ll) -->

(~(:)' ~(:)' t.~:)' a(z" Z2. ~(z) =

Zl Z2

Z ,),,)

(20)

-z~

As we showed above, under an analytic automorphism of a Siegel domain, a skeletal point is either transformed into a point at infinity or into another skeletal point. Thus, in order to show that (20) is not an

28

THE GEOMETRY OF CLASSICAL DOMAINS

analytic automorphism of our domain, it is sufficient to find a skeletal point that is not mapped into a skeletal point under mapping (20). We can, for example, take the point (i, 1, 1, 1). Automorphism (20) transforms this point into the point

i -i 1- -1., -1 -1.,

1 ( -1 -1.,

) ail1 (,,)

which is obviously not a skeletal point. (2) Consider the 4-dimensional space R4 of all 2 x 2 matrices of the form y=

Y3+ iY4)

Yl (

Y3- iY4

Y2

Let V be the set of all positive definite matrices Y in this space. In other words, V is the set of points in this space such that (21) Let 111 = 3 and let uo, Ul, U2 be coordinate in a space u in which a vector function is given. We set

Fl(U,U) F(u, u) = ( F 3 (u,u)-iF 4 (u,u) F 1(u, u) = lu ol2,

Fiu, u) =

F 3 (u, u) = Reu l tio,

F 3 (U,U)+iF 4(U,U)) Fiu,u)

lu

l

12 + lu 212,

(22)

Fiu, u) = (Reu 2 uo)cos¢

+ (1m u 1 iio)sin¢

We will now verify that the domain we have obtained is homogeneous. In order to do so, it is sufficient to show that the group Q contains transformations of the following forms

Yl ~ Ai Yl' Yl ~ Yl'

Y2 ~AfY2'

Y3 ~ Y3 + 111 Y1'

Y3 ~ A1 A2 Y3'

Y4 ~ Al A2 Y4

Y4 + 112 Yl' Y2~(l1r + 11~)Yl + 2111 Yl + Y2

(23)

(24)

(Ak and 11k are arbitrary real numbers). Recall that in order to prove that a transformation Y ~ Ay belongs to Q, it is sufficient to find a transformation u ~ Bu such that F(Bu,Bu) = AF(u,u)

(25)

SIEGEL DOMAINS

29

We can use the transformation (26) as B in a transformation of the form (23). We can use the transformation

U2

~

U2

+ f12(COS 4»u o

(27)

as B in a transformation of the form (24). Thus, the group Q contains transformations of the form (24) or (23) or, in other words, transformations of the form Y~A*YA

(28)

where A is any matrix whose diagonal consists of positive numbers and whose left lower corner consists of zeros. As a result, the group Q is transitive in our cone V. It can be shown (see Section 5, Chapter 2) that the domains under discussion are equivalent if and only if 4>1 == 4>2(mod 2n). (3) Lei> V be a cone of Hermitian positive definite matrices Y of order p. The space in which the vector functions F will be defined may be conveniently realized as the space of all complex p x s matrices U. The dimension of this space is clearly equal to ps. We define the function F( U, V) with the formula F(U, V)

= UV*,

(29)

This function is a p x p matrix, so F( U, U) E V. Group Q, in this case, consists of all affine transformations of the cone V. Indeed, consider the affine transformations of the form U ~ BU of our space, where B is a nondegenerate p xp matrix. We have F(BU,BV) = BU(BV)* = BUV*B* = BF(U, V)B*.

(30)

It remains for us to recall the form given in Section 1 for all linear transformations of the cone V. We will now prove that our domain is symmetric. Indeed, the transformation

(Z, U) ~ (-Z-1, - iZ- 1 U) is an involution with the unique fixed point (iE, 0),

(31)

30

THE GEOMETRY OF CLASSICAL DOMAINS

We will now find the form of an element dv of an invariant volume. We set

dv = A(X, y, Ul, u 2 ) dx dy dU l du 2 , where

dx = dx l ... dx",

dy = dYl'" dy,p

U

(32)

l = Reu,

U2 = 1m u, while du 1 and dU2 denote the corresponding products of

differentials of coordinates. The existence of automorphisms of the form (6) in S clearly implies that A = J.(y-F(U, U)).

(33)

Furthermore, if z~Az,

u ~Bu

is an analytic automorphism of S, then A(Ay)(detA)2Idet BI2 = A(y).

(34)

When the domain S is affinely homogeneous, A(Y) is determined uniquely up to a numerical factor by (34). Section 3. Siegel Domains of Genus 3

In this section we will define and study certain properties of Siegel domains of genus 3. These domains appear as a result of the following fact. The boundary of a domain in an n-dimensional complex space, as is well known, is not homogeneous, i.e., it contains analytic "subvarieties" of various dimensions. In the theory of automorphic functions of several complex variables it is important to consider the passage to the limit that results in a point inside a domain approaching a boundary point belonging to some analytic "subvariety". Siegel domains of genus 3 are very convenient for studying such a transition to the limit. Chapter 5 contains applications of Siegel domains of genus 3 to the theory of bounded homogeneous domains. We now define Siegel domains of genus 3. First we will conside,r certain very simple concepts in linear algebra. Consider the scalar form (i.e., taking numerical values) L(u,v) of pairs of vectors u, v E em. We call the form L(u, v) semihermitian if it may be

31

SIEGEL DOMAINS

written in the form of a sum L o(u,v)+L i (u,v), where Lo(u,v) is a Hermitian form and L i (u, v) is a symmetric bilinear form. It is easy to verify that the semihermitian form L(u, v) has the following properties: (1) the form L(u, v) is complex linear in the first variable and real

linear in the second variable; (2) the difference L(u, v) - L(v, u) is purely imaginary. The converse is also true: in particular, a form with properties (1) and (2) is semihermitian, i.e., is the sum of a Hermitian form and a symmetric bilinear form. Indeed, (1) implies that L(u, v) may be represented in the form 111

L(u, v) =

L

111

a kr Uk

L

vr +

k,r=i

k,r=i

b kr Uk

Vn

whence 111

L k,r=

111

L(u, v)-L(v, U) =

(akr-ark)UkVr+ 1

Setting all of the variable except the expression

Uk

and

111

I bkrUkVr - k,r= L 1 bkrVkUr • k,r= 1 Vr

equal to zero, we find that

is a purely imaginary number. It is easy to verify that this relationship implies that

The proposition is proved. It is easy to verify that the representation in the form of a sum of a Hermitian form and a symmetric form is unique. Later on we will need semihermitian vector forms, which we define in the following manner: a vector form is said to be semihermitian if each of its components is a semihermitian form. From now on we agree to call a form L(u, v) nondegenerate if L(u, vo) = 0 for all u implies that Vo = O. We denote by !0 a bounded region in space Ck, whose points we agree to denote by the letter t. A nondegenerate semihermitian form LtCu,v) with domain CIII and range in C" correspond to each tE!0. In n-dimensional real space, V is the cone discussed in Section 1.

32

THE GEOMETRY OF CLASSICAL DOMAINS

DefinitlOn. In the space eN(N = m +n +k)the set of points w

= (z, u, t)

such that Imz-ReLrCu, U)E V,

tED,

(1)

forms some domain S. We call this domain a Siegel domain 0/ genus 3 if it is analytically equivalent to some bounded domain. We will now give the simplest nontrivial example of such a domain. 11 = m = k = 1; ~ is the unit disk 1 tl < 1 in the complex plane; Vis the half line y > O. We set (2) This domain is analytically equivalent to the Siegel disk, i.e., the set of all symmetric 2 x 2 matrices Z such that

ZZ<E It is clear that any Siegel domain S admits transformations of the following form: z

~

z+a,

u ~ u,

t ~ t,

where a is an arbitrary real vector. Let e(t) be a vector function that has its range in em and is analytic in f!). We agree to say that the vector function e(t) is consistent with the/orm Llu, v) if Llu, e(t)) is an analytic function of t E ~ for all u. The set of all vector functions consistent with a given form Llu, v) is, as we can easily see, a linear space over the field of real numbers. Later we will show that the dimension of the space does not exceed 2m, and in the most important and interesting cases it is equal to 2m. For example, in the domain defined by (2) with Liu, v) we can set e(t) = e - te, where e is an arbitrary complex number. Indeed, in this case

LiLt, e(t)) =

ue.

(3)

It is not difficult to show that for a function of the form (2) any consistent vector function has the form e(t} = c- te. This example shows that if e(t) is consistent with a form Liu, v), then e 1 (t) = ie(t) is, generally speaking, not consistent. The importance of consistent vector functions will become clear from the following theorem.

SIEGEL DOMAINS

33

Theorem 1. Let c(t) be some vector function (not necessarily analytic) with domain ~ and range in em. The transformation z ~ z + a + 2iL t (u, e(t)) + iLrCe(t), e(t)), ) u~

Ll

+ e(t),

(4)

1 ~t,

where a is all arbitrary real n-dimensional vector, is a one-to-one mapping of domain S onto itself. It is analytic if and only if c(t) and LrCu, c(t)) are analytic functions oftE~for any u. Proof Direct computation shows that a transformation of the form (4) leaves the difference Imz- ReLrCu, u) fixed and, therefore, maps S into itself. The property of being single-valued follows from the fact that transformation (4) has an inverse that may be obtained by substituting - c(t) for c(t) in (4). We now tUrn to the proof of the second part of the theorem. It is easy to show directly that if (4) is an analytic mapping, then c(t) and LcCu, c(t)) are analytic as functions of t E ~ for any u. We now prove the converse, i.e., if c(t) and LrCu, c(t)) are analytic as functions of t for any u, then mapping (4) is analytic. It is sufficient to verify that LcCc(t), c(t)) is analytic. Note that L t2 (c 1 (t 1), cit 2)), where c1 and c2 are fixed vectors, is an analytic function of t 1 when t 2 is fixed, and, similarly, of t2 when t1 is fixed. By Rartogs' theorem, therefore) L t2 (C 1 (t 1 ), cit 2)) is an analytic function of tl and t2 (see Fuchs [lD. Setting t1 = t 2, we find that LrCc 1 (t), cit)) is an analytic function of t. The theorem is proved. From now on we will call transformations of the form (4) "parallel translations". The set of such transformations forms a group, which we will denote by the letter~. Just as for Siegel domains of genus 2, the group ~ is a nilpotent group of class 2. It may be considered as the set of pairs (c, a), where a is an n-dimensional real vector and c is a ,u-dimensional real vector (/1 is the dimension of the space of all vector functions consistent with a given form Lt(u, v)). Note that the form (5) is analytic with respect to t for all fixed c 1 (t) and cit) and takes real B

34

THE GEOMETRY OF CLASSICAL DOMAINS

values only. As a result, it is independent of t. The law of composition in the group ~, as we can easily show, is defined by the formula (Cl'

a l ) x (C2' a 2 ) =

(Cl

+ C2 , a 1 + a 2 + Q(c l , c2 )).

(6)

The group of linear transformations of Siegel domains of genuses 1 and 2 plays an important role in the study of these domains. Quasilinear transformations play an analogous part in the study of Siegel domains of genus 3. Definition. A one-to-one transformation of a Siegel domain Shaving the form

z ~ A(t)z+a(u, t), u ~ B(t)u + bet), t~

J1

(7)

get),

where A(t) and B(t) are matrix functions analytic in f», a(u, t) and bet) are vector functions analytic in f», and t ~ g(t) is an analytic automorphism of the domain f», is called a quasilinear transformation. During the study of quasilinear transformations of a domain S, we will need the following general properties of analytic automorphisms of bounded domains: . (A) An analytic automorphism of a bounded domain with a fixed point is uniquely determined by its Jacobian matrix at this point. (B) A sequence of analytic automorphisms of a bounded domain is compact if there is at least one point whose sequence of images is compact in this domain. B. A. Fuchs' book [1] contains the proofs of these properties. We denote the following analytic automorphism of the domain S by ¢;.: (8)

where A is an arbitrary real number. The existence of a family of automorphisms ¢). is characteristic of Siegel domains. Note the following properties of quasilinear transformations: (1) The matrix A(t) is independent of t for any quasilinear transformation of the form (7). The linear transformation y ~ Ay is a one-to-one transformation of a cone V onto itself. Indeed, the point (z, 0, t) E S is mapped onto the point (A(t)z+a(O, t),

35

SIEGEL DOMAINS

bet), get)) by transformation (7). A condition for membership (of a point obtained) in S can be written in the following manner: 1m (A(t)z + a(O, t)) - Re LtCb(t), bet)) E V

(9)

for any tEf». The initial point (z, 0, t) belongs to S if and only if y = 1m Z E V. As a result, the points ().z, 0, t), where A is an arbitrary positive real number, belong to S, along with the point (z, 0, t). Substituting AZ for Z in (9) and passing to the limit as A --? 00, we find that for any t E f» 1m (A(t)z)EV ,

then

ImzE V

(10)

(V is the closure of V). From this it clearly follows that ACt) is a real matrix for any t. As we know, an analytic function with purely real range is constant. As a result, A(t) is independent of t. To complete the proof, we need only consider the inverse transformation. (2) Each component of the vector A(u, t) is a polynomial of degree not greater than 2 in u, which polynomial has coefficients that may depend on t. Let tf; be a quasilinear transformation of the form (7). Consider the family of automorphisms tf;;. = ¢; 1 tf;¢;.. It is easy to verify that tf;;. has the following form: Z--?

AZ+A- 2 a(Au, t),

1

u --? B(t)u + A-1 bet),

~

t--?g(t).

J

(11)

Applying an automorphism tf;;. to the point (z, 0, t), we can easily see that a sequence of analytic automorphisms tf;;. is compact when A --? 00. It is clear from the explicit form (11) of the automorphisms tf;;. that a sequence of them may be compact if and only if a(u, t) is a polynomial of degree not greater than 2 in u. (3) Together with transformation (7), Z --?

Az+a(u, t)

u --? B(t)u + bet) t --? get),

36

THE GEOMETRY OF CLASSICAL DOMAINS

a transformation of the form

Ll ~

(12)

B(t)u,

t ~ get) (azCu, t) is homogeneous part of degree two of a(u, t) is an analytic automorphism of the domain S. Indeed, we have proved that a sequence of automorphisms (11) is compact when ). ~ 00. On the other hand, it is clear from the explicit form (11) of these automorphisms that they converge to mapping (12) when A ~ 00.

°

(4) Let toE::2. If Rea(O, to) = and bCto) = 0, then a(u, t) only contains terms that are quadratic with respect to u. First of all, we prove that 1m a(O, to) = 0. The point (z, 0, to) belongs to S if and only if 1m Z E V. As a result, if yE V, then y+lma(O, to)E V, whence Ima(O, to) EV. We can easily show that - 1m a(O, to) E V by discussing the inverse transformation. As a result, a(O, to) 0. In this case a sequence of automorphisms (11) will be compact when A ~ 0. In fact, the sequence of images of a point of the form (z, 0, to) is compact. A sequence of automorphisms of the form (11) can be compact when }c -7 if and only if a(ll, t) contains no terms of zero- or first-degree in u. Let e(t) be a vector function that is analytic in f» and consistent with the form LrCu,'O). Property (4) implies that if e(t o) 0, then e(t) = for all tEf». For the proof, consider a transformation of the form (4), where e(t) is given and a = 0. It follows from (4) that such a transformation is trivial and, therefore, e(t) == 0. What we have proved implies that the dimension p of the space of vector functions e(t) that are consistent with LrCu, v) and analytic in f» does not exceed 2m. From now on, we will always assume that p = 2m. Throughout the remainder of this book we will consider only such Siegel domains of genus 3. (5) A quasilinear transformation such that A = E, B(t) == E, g(t) == t, i-; a "parallel translation" in a domain S, i.e., belongs to the group ~.

°

°

37

SIEGEL DOMAINS

It is clearly sufficient to show that if a quasilinear transformation of the form z~z+a(u,t),

u~u+b(t),

t~t

(13)

has the following property: for some to E f» Re a(O, to) = 0, b(to) = 0, then it is the identity transformation. We should note that (4) implies that a(O, to) = 0. As a result, the point (z, 0, to) is fixed under our quasilinear transformation. Furthermore, it is clear from (4) that the Jacobian matrix of transformation (13) is the same at a point of the form (z,O, to) as the Jacobian matrix of the identity transformation. As a result, (13) is the identity transformation of the domain S, i.e., a(u, t) == 0, b(t) == 0. It foliows at once from what we have proved that the group d is a normal divisor of the group of all quasilinear transformations of the domain S. (6) A transformation of the form z~Az+azCu,t),

I

u ~ B(t)u,

~

t~ t,

I

(14)

J

where azCu, t) is a homogeneous form of the second degree in u, is an analytic automorphism of a domain S if and only if

LtCB(t)u, B(t)v) = ALtCu, v)

(15)

for all u, v E C/t, t E f» and when

azCLl, t) == 0.

(16)

If we transform a transformation of the form (4) by means of (14) and we use the fact that according to (5), the transformation obtained must again be of the form (4), we obtain relationship (15). It is immediately clear that if (15) is satisfied, then the transformation z~Az,

u~B(t)u,

t~t

(17)

is an analytic automorphism of the domain S. Furthermore, the resultant of transformation (14) composed with the inverse of (17) is an automorphism belonging to the group d, whence we immediately obtain azCu, t) == 0.

38

THE GEOMETRY OF CLASSICAL DOMAINS

In conclusion we will give certain conditions sufficient for analytic homogeneity of a Siegel domain S. Let Q be the set of all linear transformations y ---* Ay of a cone V onto itself such that for each of these transformations there is a matrix functIOn B(t) that is analytIc in f» and such that

LiB(t)u, B(t)v) = ALtCu, v)

(18)

for all u, v E C"z, t E f». In addition, we denote by G the set of all analytic automorphisms of the domain f» for each of which there exists a matrix function B(t) that is analytic in f» and is such that (1) L~(t)(B(t )u, B(t)v) = Li(u, v) and (2) the function KtCu, v) = L;(u, iJ) - L;(t) (B(t )u, B(t )v) depends holomorphically on t, where Li(u, v) is the Hermitian part of the form LtCu, v) and L;(u, v) is the symmetric part of the form Lt(u, iJ). We have the following: Theorem 2. If the group Q is transitive on a cone V and the group G is transitive on the domain f», then the domain S is analytically homogeneous. Proof It is easy to verify directly that transformations of the form

z ---* Az,

U ---*

z ---* z- iKtCu, u),

B(t)u,

t ---*

u ---* B(t)u,

t,

(19)

t ---* get)

(20)

are analytic automorphisms of the domain S. Now let w1 = (Zl' Ul' t 1 ) and W2 = (Z2' U2, t 2) be two arbitrary points in S. We will now prove that there is an analytic automorphism of our domain S that maps one of these points onto the other. First we note that by means of an automorphism of the form (20) we can map the point W2 = (Z2' u 2, t 2) onto the point W3 = (Z3' U3, t 1 ), and that by means of an automorphism of the form (19) we can map W3 onto a point lV4 = (Z4' u4 , t 1 ) such that 1m Z4 -ReLtJ (u4, u 4)

= Imzl -ReLt (u 2, Ll2)'

Finally, point lV4 may be mapped onto Wl by means of some "parallel translation", i.e., an automorphism of the form (4). The theorem is proved. We will now find the form of an element dv of an invariant volume. We set

dv where

Lil = Reu,

;1.(z, u,t) dx dy dU 1 dU2 dtl dt 2, Li2 = Imu,

dx = dXl ... dx ll ,

tl = Ret,

t2 = Imt,

dy = dYl ... dYIl'

(21)

39

SIEGEL DOMAINS

The existence of "parallel translations" implies that

A(Z, u, t) = A(Im Z - Re Lt(u, u), t). If

Z -+

Az,

U -+

B(t)u,

(22)

t -+ t

is an analytic automorphism of the domain S, then

A(Ay, t)(detA)2IdetB(t)12 = A(y, t), and if

z -+ z- iKlu, u), u -+ B(t)u,

(23)

t -+ get)

(24)

is an analytic automorphism of our domain S, then (25) where iit) indicates the Jacobian of the transformation t -+ g(t). In case the conditions of Theorem 1 are satisfied, equations (23) and (25) uniquely determine A(Y, t) up to a numerical factor. All of the Siegel domains of genus 3 that are discussed in this book can be obtained by means of the following construction. In this construction we first define the notion of the principal Siegel domain of genus 3 corresponding to a given Siegel domain of genus 2. Let H be some Siegel domain of genus 2 corresponding to the cone V and the vector function F(u, v). Consider the set K of all antilinear transformations u -+ pu such that

F(pu, v) = F(pv, u)

for all

F(u,u)-F(pu,pU)EV

u, VE CIII

for any

UEC

F(u,u)-F(pu,pu):j:.O if u:j:.O

(26) III

(27) (28)

(Here V denotes the closure of the cone V.) Let P be the set of all antilinear transformations of the space CIII that satisfy (26). It is clear that P has the structure of a complex space. It is also clear that K is a bounded domain in P. We should also note that (27) implies that the transformation E + p is nondegenerate. Consider the following domain S in the space CIJ+III xP:

Imz-ReLiu,u)EV,

pEK

(29)

where First of all, we will show that the form Liu, v) is semihermitian, i.e., (1) the form Liu, v) is complex linear with respect to the first argument

40

THE GEOMETRY OF CLASSICAL DOMAINS

and real Imear with respect to the second argument, and (2) the difference Lp(u, v) - LP( v, u) is purely imaginary. The first of these statements is clear, while the second can be proved in the following manner. We set U1 = (E+p)-1 U, V 1 = (E+p) 1V. Then Lp(u, v) - Lp(v, u) = F((E + p)u 1, V1) - F((E + P)V1' u 1)

= F(u 1, v1)-F(V1' u 1) =

(30)

2iImF(u1' v 1).

We now show that the domain S is analytically equivalent to a bounded domain. It is not difficult to see that in order to prove this statement, it is sufficient to show that 2ReL/u, u)-F(u, U)EV.

(31)

Indeed, (31) implies that the domain S is contained in the domain Imz-tF(u,u)EV,

pEK

which is the product of two bounded domains. We must therefore prove (31). We have 2ReLp(u,u)-F(u,u) = 2ReF(u1 +pu1,ud-F(u 1+pU 1,U 1+pu 1)

= 2F(u1' ud-2F(pu 1, PU1)E V where u = U1 +pUl. We have thus proved that the domain S is a Siegel domain of genus 3. In what follows, we will call this domain a principal Siegel domain of genus 3. We will now write out the form of the consistent vector functions. We set C(p) =

where

CE

c+ pc

(32)

Cn!. Then Liu, c(p» = F(u, c)

so that C(P) is a consistent vector function. The dimension of the space of vector functions of the form (32) is clearly equal to 2m, and; consequently, all of the consistent vector functions have the form (32).

41

SIEGEL DOMAINS

The Siegel domain of genus 3 with base f», where f» is some bounded domain in C\ and fiber H can be obtained in the following manner. Let ¢ be an analytic mapping of the domain f» into a domain K. Consider the following domain Sin C"+ m + k :

Imz

ReLt/J(t)(u,U)EV,

tEf»

(33)

What we have proved above about principal Siegel domains implies that (32) defines a Siegel domain of genus 3, and that the dimension of the space of consistent vector functions for this domain is 2m. In the following chapters we will use the following definition. A directing subgroup of a Siegel domain S is a subgroup of transformations ¢). of the form z

---*

A2 Z,

U ---*

AU,

t ---* t

where A> O.

Section 4. Bounded H%l1lorphic Hulls Let f» be a bounded domain in C" and let X be some set in f». The bounded holomorphic hull O( X) is the set of all Z E f» such that /¢(z)/ ~ sup/¢(z)/

(1)

zeX

where ¢(z) is any function that is regular and bounded in f». In this section we will deal primarily with the bounded holomorphic hulls of Lie-group orbits. We will first consider this situation for the case of the unit disk Izl < 1. Here there are three types of one-parameter subgroups: hyperbolic, parabolic, and compact subgroups. The first, two may be realized in the upper halfplane 1m z > 0 in the following manner:

z

---*

AZ,

z---*z+a

A> 0

(2) (3)

It is clear that the bounded holomorphic hull of any orbit of a hyperbolic subgroup is trivial, i.e., coincides with the orbit itself. The bounded holomorphic hull of orbits of a parabolic subgroup is a set of the form Imz > b, where b is a positive constant. A compact subgroup is conjugate to the following group of transformations of the disk Izl < 1: (4)

42

THE GEOMETRY OF CLASSICAL DOMAINS

The orbits of this group have the form Izl = r, where 0 ~ r < 1. The principle of the maximum implies that the bounded holomorphic hull of the orbits has the form Izl < r. For the case of the unit disk, therefore, the bounded holomorphic hulls are nontrivial for compact and nilpotent subgroups. Neither of these propositions is true in the case of several complex variables, as the examples given below will show. It is very important, however, that the bounded holomorphic hull of the orbits of a group of parallel translations in a Siegel domain is always nontrivial. More exactly, we have the following proposition. Lemma 1. Let S be the Siegel domain of genus 2 corresponding to the cone V and the vector function F(u, v). Then the bounded holomorphic hull of any orbit of the group of parallel translations consists of the points z, u such that Imz-F(u, u) - rEV

(5)

where r is a fixed vector in V. Proof As we showed in Section 2, the orbits of a group of parallel translations consist of the points z, u such that Imz-F(u, u) = r

(6)

where r is a fixed vector in V. Let X/, be the set of points of the form (6). The bounded holomorphic hull of the orbits of a group of parallel translations is clearly invariant with respect to this group. It is therefore sufficient to prove that: (1) y - rEV implies that

!¢(iy, 0)1 ~ sup I¢(z, u)1

(7)

(z,lI)eX r

for any function ¢(z, u) that is regular and bounded in S, and (2) if y E V, Y - r ¢ V, there exists a function ¢(z, u) that is regular and bounded in S and is such that

I¢(iy, 0)\ > sup I¢(z, u)1

(8)

(z,ll)eX r

In order to prove the first statement, consider the auxiliary function = ¢(A(Y - r) + 1',0). This function is clearly regular and bounded in the upper halfplane 1m A > O. As a result,

f(A)

, 1¢

(i)

1

~ sup 1 ¢ (A) Im).=O

whence follows (7).

1

(9)

43

SIEGEL DOMAINS

In order to prove the second statement, We note the following fact. It is clear that there exists a linear function a of ell such that a(x) > 0 for any x E V and a(y) < a(r). As we can easily see, the function

.( " 1 cp z , u) -- a( z) + i

(10)

is bounded in Sand 1

1¢(iY, 0)1 1 + a(y) > 1 +

1

sup I¢(z, u)1

(11)

(z,u)eXr

This finishes the proof. We will now give an example of a semisimple Lie group in which the bounded holomorphic hull of the orbits is trivial. First we will show that the bounded holomorphic hull of the set Y consisting of all points of the form (iy,O), y E V, in the domain S is trivial. It is sufficient to show that for any point Zo = Xo + (vo, Yo E V, Xo =f. 0, there exists a function J(z) that is bounded and regular in the domain ImzE V and is such that 11(zo)1 > sup IJoy) 1

(12)

yeV

Note that there exists a linear function a(y) defined on RII so that a(y) > 0 when y E V and a(xo) =f. O. Moreover, there clearly exists a function ¢(A) that is holomorphic and bounded in the upper halfplane 1m A > 0 and is such that 1¢(a(xo))1 > sup I¢(A)I

(13)

Re,1,=O

The functionJ(z) = ¢(a(z)) is the function sought. The principles of constructing a semisimple Lie group in which the holomorphic hulls of its orbits are trivial is clear from what we have proved. Let V be a cone in which a semisimple Lie group G is transitive. Consider a Siegel domain S of genus 1 corresponding to this cone. Assume that a point of the form (Yo, 0) is contained in an orbit Y of the group G. What we have proved above implies that the bounded holomorphic hull of the orbit Y is trivial. It is also not difficult to show that the bounded holomorphic hull of any orbit of the group G is trivial. It is also possible to show that the bounded holomorphic hulls of the orbits of all subgroups of the group G are also trivial.

44

THE GEOMETRY OF CLASSICAL DOMAINS

In conclusion, we will consider the two following problems. (1) What domains !& contain groups G such that any bounded holomorphic hull O( Y) of any orbit Y coincides with!&. Homogeneous domains provide a trivial example of such domains. Inhomogeneous Siegel domains of genus 1 or 2 provide examples of inhomogeneous domains with this property. Indeed, let S be a Siegel domain of genus 2 and let G be its group of transformations containing all parallel translations and the following transformations

z ---* A?Z,

u ---* AU

(14)

Then the lemma that we proved at the beginning of this section implies that the bounded holomorphic hull of any orbit coincides with S. The author knows of no examples of domains other than Siegel domains of genus 2 with this property. It would be very important and interesting for the theory of functions of several complex variables to find whether there exist domains other than Siegel domains in which there is a group G such that the bounded holomorphic hull of any orbit coincides with the domain. (2) Let Yo = Gz o be an orbit of a group G, and let H be a subgroup of G. It is clear that

I

O(Hz) c: 0(1"0)

(15)

zeYo

By a "guilty" group we will mean a minimal subgroup H such that

I

O(Hz) = 0(1"0)

(16)

zeYo

When G is the group containing all parallel translations of a Siegel domain, the commutative subgroup consisting of transformations of the form z---*z+a,

u---*u

where a is an arbitrary real vector, is "guilty". It would be interesting to find out how to construct "guilty" subgroups in the general case. For example, it is worthwhile to find out whether every solvable group G has a commutative "guilty" subgroup.

CHAPTER 2

The geometry of homogeneous domains Section 1. Statement of Fundamental Results

The central result of the present chapter is the theorem about description of realizations of bounded homogeneous domains in the form of Siegel domains of genus 3. For symmetrical domains all their realizations in the form of a domain of the third family are described in the work of A. Kozanyi, T. Wolf Generalised Cayley Transformations of Bounded Symmetric Domains (to be published). Every homogeneous Siegel domain of genus 3 contains a natural homogeneous fibering, so a homogeneous fibering of a bounded homogeneous domain f» is associated with every model of the domain f» in the form of some Siegel domain of genus 3. The basic result of the present chapter consists of the statement that the converse of this statement is also true. In Section 7 of this chapter we will show that a model in the form of some Siegel domain of genus 3 corresponds to every homogeneous fibering of a domain .@. In the first edition of this book, models in the form of Siegel domains were obtained computationally for classical domains. These computations are given in Sections 10 and 11 for classical domains of the first, second, and third types. We should note that here it can be observed that a base of the corresponding Siegel domain of genus 3 is an analytic boundary component. In the first edition of this book, this fact was used in applications to the theory of automorphic functions. Although the notion of boundary components is not used in those chapters of this book that are devoted to the theory of automorphic functions, we will retain the fundamental definitions associated with this notion. They are given in Section 9. Here we will present a brief survey of the contents of the present chapter. Section 2 contains a summary of the 45

46

THE GEOMETRY OF CLASSICAL DOMAINS

fundamental definitions associated with the notion of j-algebras, i.e., Lie algebra containing an additional structure associated with the fact that the corresponding group acts analytically on a complex manifold. Section 3 is devoted to the theory of normal j-algebras, i.e., j-algebras that are simultaneously solvable and split over the field of real numbers as Lie algebras. As was shown (see the appendix), there is a transitive group whose Lie algebra is solvable and split over R defined in each bounded homogeneous domain. Moreover, it is known that all maximal solvable and split (over R) subalgebras in any Lie algebra are conjugate (see Borel [4] and E. B. Vinberg [2]). Thus, there exists a one-to-one correspondence between normal j-algebras and bounded homogeneous domains in CII • In Section 3 we will describe the root systems of all normalj-algebras. Chapter 6 contains a more detailed study ofnormalj-algebras and more general methods for constructing them. Section 4 is devoted to study of the ideals of j-algebras that are simultaneously j-subalgebras. We will show that every normalj-algebra has only a finite number of such ideals, and that they can all be easily listed. In Section 5 we will present an explicit construction of Siegel domains of genus 2 with the aid of normal j-algebras (1. 1. Pyatetskii-Shapiro 1[2]). Sections 6 and 7 are devoted to proving that some realization in the form of a bounded Siegel domain of genus 3 corresponds to each homogeneous fibering of a bounded domain. We will also note the following result without proof. For any analytic automorphism of a bounded homogeneous domain !?2, there exists a realization of !?2 in the form of a Siegel domain in which this automorphism is a linear transformation. Section 2. j-algebras

In this section we will define certain fundamental entities, j-algebras, simplectic representatioris, etc. Let G be a Lie algebra with a distinguished subalgebra Go. We will also assume that G has an endomorphismt j such that j(G o) = 0 and j2(X)+XEG o for all xEG. Definition. The triple {G, Go,}} is said to be a)-algebra if

tj

is an endomorphism of G as a vector space, but not as a Lie algebra.

47

THE GEOMETRY OF HOMOGENEOUS DOMAINS

(0:)

[x, yJ + j([j(x), y]) + j([x,j(y)]) - [j(x),j(y)] E Go

(1)

for any x, yE G. ([3) There exists a linear form w(x) defined on G in such a manner that for any x, y E G w([j(x),j(y)])

= w([x, y]),

w([jx, x]) > 0

for

x ¢= Go.

(2)

(y) If Kis a compact sub algebra of G andj(K) c K + Go, then K eGo.

Thus, a j-algebra is a Lie algebra in which some subalgebra is distinguished and an endomorphism is defined. Two j-algebras {G, Go,j} and {G', Gb,j'} are said to be isomorphic if there exists an isomorphism ¢ of the Lie algebra G onto the Lie algebra G' such that ¢j(x) == j' ¢(x)

(mod Gb)

In other words, j-algebras distinguished only by the forms ware not treated as different. Let G 1 be a subalgebra of the algebra G. If j(G 1) C Gl + Go, then in the space G1 it is possible to define an endomorphism jl such that j1 (x) == j(x)(mod Go) for any x E G. Set G Ol = Go n G l • We define the linear form Wl on G1 as the restriction of the form w. There are no difficulties in proving that {G 1 , GOl ,jl' Wl} is a j-algebra. Such a sub algebra is called aj-subalgebra of the algebra G. Aj-subalgebra that is simultaneously an ideal of an algebra is called a j-ideal. Let {G, Go,j} and {G', Gb,j'} be two j-algebras. A homomorphism ¢ of the algebra G into the algebra G' is called a j-homomorphism if ¢}(x) ==j'¢(x) (modG~) for any xEG. The value of j-algebras is explained by the following theorem. Theorem 1. Let!:» be some bounded homogeneous domain in en, ® be some transitive group of analytic automorphisms of!:», and let ®o be a stationary subgroup at some point Zo E !:». We denote the algebras of the Lie groups ® and ®o by G and Go, respectively, and by j an endomorphism induced in G by an endomorphism of the complex structure. Then the triple (G, Go,}) is aj-algebra.

In order to prove this theorem it is sufficient to verify conditions 0:, [3, and'}' of the definition of j-algebras. Condition 0: is the well-known condition for a manifold to have an invariant complex structure, while [3 follows from results of Koszul [16] in the following way. Let ds 2 be the

48

THE GEOMETRY OF CLASSICAL DOMAINS

Bergman metric in f0. The space tangent to !!2 at the point Zo can be identified in a natural way with the factor space GIGo. Let h(x,y) be the Hermitian form induced by the Bergman metric on GIGo. This form can be extended to all of G by setting hex, y) = 0 for x E Go and Y E G. As Koszul showed, we have the following relationship: 1m h(x,y)

= w([x,y])

(3)

where w is some linear form on G. This form w clearly satisfies condition f3 of the definition of j-algebras. We can prove condition}, in the following wanner. If K is a j-subalgebra of the algebra G, then the orbit of the group exp K containing the point Zo is a complex submanifold of the domain!!2. If the group exp K is compact, any of its orbits are compact. As a result, if there exists a compact subalgebra K that is also a j-subalgebra, its orbit containing the point Zo is a compact complex manifold. It is well known that a bounded domain cannot contain a nontrivial complex compact submanifold. As a result, the orbit of the group exp K that contains the point Zo coincides with this point, and, therefore, K eGo. lt is important to note that the converse is also true. Converse. Let {G, Go,j} be aj-algebra. There always exists a bounded homogeneous domain f0 in 'which there is a transitive group of analytic automorphisms @J whose Lie algebra is isomorphic to G, and the group @J o corresponding to the Lie algebra Go is a stationary subgroup of some point Zo E f0 (see the appendix). In contrast to the direct part of the theorem, the proof of the converse requires algebraic apparatus that is, in essence, quite complex. We should note the following corollary of the theorems we have formulated. Let M be a complex homogeneous manifold with an invariant volume given by the form. dv

= k(z)dz 1 A ... dz Adz 1 A ... dz Il

lI

(4)

If the Hermitian form 2

'" 8 ln k d Z(I. dz- p

L., - - -

8z(l.8z p

(5)

is positive definite, then the manifold M is called a Bergman manifold. We will now show that any homogeneous Bergman manifold M is analytically homomorphic to a bounded homogeneous domain. Indeed, the proof of the first theorem is precisely the same for homo-

THE GEOMETRY OF HOMOGENEOUS DOMAINS

49

geneous Bergman manifolds. As a result, some j-algebra corresponds to each homogeneous Bergman manifold. It remains to note that some bounded homogeneous domain corresponds to eachj-algebra. A special role in the theory of bounded homogeneous domains is played by the so-called Siegel disk, i.e., the set ~l of all complex symmetric 11 x 11 matrices Z such that (6)

ZZ<E.

Here Z denotes the complex conjugate of Z. Inequality (6) implies that all of the characteristic roots of the Hermitian matrix ZZ are less than 1. As we know, the analytic automorphisms of the Siegel disk are described in the following manner. Consider the set @:>Il of complex 2n x 211 matrices P such that

P'lIP

H=

= H,

(-Eo 0)

E '

P'IP = I

1=

(0 E) -E

0

(7) .

With each matrix P E @:>Il we associate the following analytic automorphism of the Siegel disk: Z ~ (AZ + B)(CZ + Dr 1,

P- (Ac B ). D

(8)

The proof that transformations of the form (8) map the Siegel disk into itself is immediate. Toward this end, we must use the relationships between the matrices A, B, C, and D that follow from (7). We will now show that the group @:>Jl is isomorphic to a real symplectk group. Toward this end, we set

}vI = L- 1 PL,

L = (E

-iE)

E

iE

F or the matrices M, (7) imply that

M'IM

= I, lvI'IM = I,

(9)

whence it follows that the matrices M are real and that the set of such matrices forms a so-called symplectic group.

50

THE GEOMETRY OF CLASSICAL DOMAINS

The j-algebra corresponding to the Siegel circle can be described in the following manner. Let U be an n-dimensional Hermitian space. We denote the Hermitian scalar product in U by h(u1' u2 ), and the operator for multiplication by i by I. A transformation p that is a linear transformation of the space U over the reals is said to be simplectic if

p(PU1,U2)+P(U1,PU2)=0

(10)

where The set 8 11 of all symplectic transformations of the space U is clearly a linear Lie algebra in which commutation is defined, as usual, by the equation [p, q] = pq qp. We define the endomorphism j in the algebra 8 11 by means of the formula j(p) =

t[1, pJ

(11)

where I is the operator for multiplication by i. We denote the set of all p E 8 11 such that j(P) = 0 by All" It is not difficult to show that All is a subalgebra of the algebra 811" We define the form w in the following manner. As we can easily show, each transformation p E All is linear over the complex numbers and skew Hermitian. As a result, its trace is a purely imaginary number. We set w(P) = (l/i)rpp for pEA n , and then extend w(p) to all of 8 11 in such a manner that w(p) 0 if Ip = --pl. We leave the proof that {811 , A,,,j, w} is a j-algebra to the reader. So-called symplectic representations will play an important role in what follows. We will now define them. Let G be some j-algebra. A symplectic representation of the algebra G is a j-homomorphism g ~ Pg of the algebra G in the j-algebra 8 w A complete reducibility theorem holds for symplectic representations. More accurately, if a complex subspace U o of a space V is invariant with respect to the operators Pg , g E G, then its orthogonal complement U1 is also invariant with respect to the operators Pg. Indeed, U 1 consists of all li1 E U such that P(U1' uo) = 0 for each U o E Vo. The operators Pg are symplectic, so for any U 1 E V 1 and lto E Uo we have

p(piiI 1), u o) = - p(u l ' piuo)) = 0, whence follows the proposition.

51

THE GEOMETRY OF HOMOGENEOUS DOMAINS

In concluding this section we will consider the following problem. Let !!2 be some bounded homogeneous domain in C", and let ® be its full group of analytic automorphisms. As a rule, there are many nonisomorphic subgroups in !!2 that are not transitive in!!2. Thus, there exist many nonisomorphic j-algebras corresponding to this domain. In the appendix to the book we will show that for a group ®, the maximal subgroup ®1 that is solvable and split over the real numbers is transitive in!!2. As we know, such a subgroup is defined uniquely up to conjugacy. The j-algebras corresponding to such subgroups are called normalj-algebras. Their value consists of the fact that the natural correspondence between normal j-algebras and bounded homogeneous domains is one-to-one. A large part of the present chapter and nearly all of Chapter 6 are devoted to a detailed study of normalj-algebras. Section 3. Normalj-algebras

This section is devoted to study of the root systems of normal j-algebras. For convenience in reference we will now give a system of axioms for normalj-algebras. Let G be a solvable Lie algebra with the following properties: (I) The operator adg 0: g ~ [g 0, g] has only real characteristic roots for all go E Gt. (II) G has an endomorphismj such thatj2 = -1 and [x, yJ + j([j(x), yJ) + j([x,j(y)J) - [j(x),j(y)] =

°

for all x, yE G. (III) There exists a linear form (f) on G snch that (f)([jx, xJ) > 0,

if x =I- 0,

(f)([jx,jyJ) = (f)([ x, yJ).

A Lie algebra G with all of these properties is called a normalj-algebra. Note that, generally speaking, a given normal j-algebra G has no one form (f) for which III is true. We will not treat these algebras as different. In other words, two normal j-algebras G and G' are said to t This property means that in some basis all of the operators adg can be de3cribed by triangular matrices.

52

THE GEOMETRY OF CLASSICAL DOMAINS

be isomorphic if there exists an isomorphism ¢:G""""""* G' that commutes withj, We can define a Hermitian positive definite scalar product h(gbg2) = (W([jgl,g2])+iw([g1,g2]) in a natural manner and a real positive definite scalar product k(gl,g2) = W([jgl,g2J) = Reh(g1,g2)

A subalgebra of a normal j-algebra G that is invariant with respect to the endomorphism j is clearly also a normalj-algebra, For the rest of this section we will assume that allj-algebras discussed are normal. For the sake of brevity, therefore, we will omit the word "normal" in this section. First we will give some examples of j-algebras. We will first describe the structure of all j-algebras G of dimension 2, Let Xo E G, It follows from III that Xl = [jxo,xo] i= 0, We will show that [jxbxd = AX 1 , where ), > 0, Indeed, Xo and jxo are linearly independent, so Xl = ajxo + [3x 0, where rx and [3 are real numbers, and, therefore, [jx 1,xd = (a2+[32)([jxo,xoD = (a 2 +[32)x1' We now set r = A-1Xt; then, aswe can easily show, [jr,r] = r. Wehaveprovedthateveryj-algebra G of dimension 2 contains an element I' such that [jr, 1'] = r, This immediately implies that all j-algebras of dimension 2 are isomorphic, As we will show below, any j-algebra contains aj-subalgebra of dimension 2, Suchj-algebras play an important role in the theory of j-algebras; their role is similar to the role of three-dimensional simple algebras in the theory of semisimple Lie algebras. We will now give other examples of j-algebras, Let G be a complex Hermitian space with scalar product h(gl' g2)' We denote the operator for multiplication by i by j, Let 1'0 be some vector in G of length 1. We denote the set of vectors Z E G such that l1(z, l'o) = by Z. Set

°

[jro,

1'oJ =

[jro, z] = !z [z,z'J = (1m h(z, z'))r o

}

1'0

where

z, z'

E

z,

(1)

It is not difficult to show that formula (1) introduces the structure of a j-algebra into G. In what follows we will call suchj-algebras elementary j-algebras. It can be shown that the domain corresponding to such a j-algebra is the ball IZ112+ ... lzmI2 < 1, where m is the complex dimension of the space G. We denote the one-dimensional space

THE GEOMETRY OF HOMOGENEOUS DOMAINS

53

generated by the vector 1'0 by R. It is clear that R is an ideal of our j-algebra. We will show that the one-dimensional ideal R is unique. Indeed, let T be a one-dimensional ideal different from R, and let tE T. Then [t, ro] E T and [t, 1'0] E R. As a result, [t, ro] = 0. Thus, tAro + z, where A is a real number and z E Z. Furthermore, [jz, t] [jz, z] = h(z, z)l'o. Since T is an ideal, [jz, t] E T. We have obtained a contradiction. We will now show that an elementary j-algebra contains no proper j-ideals. Indeed, let H be a j-ideal; then R cHand, therefore.iR c H, whence it follows that H = G. It follows from Lemma 1 below that the only j-algebras that do not have j-ideals are elementary. Note that the commutator of an elementary j-algebra G is equal to R + Z, while the orthogonal complernent to it is jR [in the sense of the scalar product k(x, y) = Re hex, y)]. Thus, in an elementary j-algebra the dimension of the orthogonal complement to the commutator is equal to 1. In what follows we agree to call the dimension of the orthogonal complement to the commutator [in the sense of the real scalar product k(x,y)] the rank ofthej-algebra in question. The value of elementary j-algebras lies in the fact that any j-algebra is the semidirect sum p of elementary j-algebras, where p is the rank of the algebra G. This follows from Lemma 1. In particular, any j-algebra of rank 1 is elementary. Thus, elementary j-algebras can be defined either as j-algebras having no proper j-ideals, or as i-algebras of rank 1. Lemma 1. Let G be ai-algebra, and let R be some one-dimensional ideal of G. We have the following decomposition: G = R+jR+Z+G'

where

(2)

(1) G' is aj-subalgebra of the algebra G,

(2) R +iR + Z is an elementarY.i-algebra,

= 0,

(3) [R +.iR, G'] (4) [G',Z] c Z.

Proof. Set (3)

where U denotes the orthogonal complement of R. It is clear that = U. By 1'0 we denote an element of R such that [jro, "0] = 1'0 It is clear (by axiom III) that

jU

m(ro) = m(Liro, roJ) > 0.

54

THE GEOMETRY OF CLASSICAL DOMAINS

First of all we will show that [R, UJ = o. Indeed, w([ro, uD = 0 by orthogonality. On the other hand, [ro, u] E R because R is an ideal of the algebra G. As a result, [ro, u] O. R + U is the centralizer of the element 1"0 and, therefore, a sub algebra of the algebra G. We will now show that U is invariant with respect to the operator adjro. We have (4)

and, therefore [jr(}, u] E R + u. Furthermore, by axiom II, [Jro,ju] =j[jro,u]Ej(R+U) =jR+U. As a result, [jro,u]E(jR+U)n(R+U) = U. At the same time, we have proved that the operator adjro is linear over the complex numbers on U, i.e., it commutes withj. We will now show that the characteristic roots of the operator adjro on U are equal to zero or t. Indeed, let Ao be a characteristic root of the operator adjro on U. Then there exists a U o E U such that

[jro, LloJ = Ao Uo

and

[jro,juoJ

(5)

Aoju o.

As a result

[jr o, [juo, LioJJ = 2Ao(Liuo, uoJ)·

(6)

R + U is a subalgebra, so

[juo,uo]=r+u,

fER,

UEU.

(7)

It follows from (6) and (7) that

2Ao(r+u) = [jro,r+u] = [jro,rJ+[jl'o,uJ whence (2Ao -1)r 0 and, therefore, either Ao = show that in the latter case Ao = O. We have

r+[jro,ll],

t

or r =

o.

We will

2Ao w([juo, LtoJ) = w([jr o, [ju 0, Li oJJ) =

w([ro, - j[juo, uoJJ)

w(O)

= O.

(8)

Here we have used axiom III and the fact that [R, U] = O. It follows from (8) that if [juo, lto] E U, then Ao = 0, since, by axiom Ill, w([ju o, lt o]) > O. We have proved that the characteristic roots of the operator adjro on U are equal to zero or t. Now we will denote the set of all g E U such that (adjroylg =0 for o me integer m by G'.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

55

Also, we will denote the set of ZE U such that (adjro--t)mz = 0 for some positive integer 111 by Z. It is easy to use the Jacobi identity to prove that (a) G' is aj-subalgebra, (b) [G',Z] c Z, and (c) [Z,Z] c R. To complete the proof, it remains to show that R+ jR+Z is an elementaryj-algebra and that [jR, G'] = O. [n order to do this, it is sufficient to show that the operator adjro is semisimple on U. We wBl first show that the operator adjro is semisimple on G', i.e., that [jro, g] = 0 when g E G'. It follows from the Jacobi identity and the fact that G' is a subalgebra that

w([[jr o,g1J,g2])+W([g1, [jr o,g2J]) = w([jr o, [g1,g2J]) = O. This equation and the fact that adzj 1'0 commutes with the operator j shows that the operator ad G , jro is skew Hermitian. It remains to note that skew Hermitian operators are always semisimple. The semisimplicity of the operator adzjro follows from the fact that, as we can show with no difficulty, the operator ¢(z) = [jro, z] --tz, is skew Hermitian. This completes the proof of Lemma] . Let G be an arbitrary j-algebra. Since the algebra G is solvable, it contains commutative ideals. Let R1 be a minimal commutative ideal. By axiom 1 it immediately follows that the ideal R1 is one-dimensional. By Lemma 1, G = G1 +G',

(9)

where G1 = R1 +jR1 +Z1 is an elementary j-algebra, while G' is some j-subalgebra of the algebra G, and [jRl + R 1, G'] = O. The algebra G' contains a one-dimensional ideal R2 and, therefore, by Lemma 1, G' = G2 + G", where G2 = R2 +jR2 = Z2 is an elementary j-algebra, G" is aj-subalgebra, and [jR2 +R2' G"] = O. Continuing this process, we ultimately obtain the decomposition (10) where

(1) Gk = Rk+jRk+Zk is an elementaryj-algebra, (2) [jRk+R k, Gs] = 0 for k < s,

and

(3) [Gs,Zk]

c

Zk for k < s.

It is not difficult to show that decomposition (10) with the indicated properties is unique up to the order of the factors. This decomposition is naturally called the semidirect decomposition, i.e., [G k, Gs] c Gk for

56

THE GEOMETRY OF CLASSICAL DOMAINS

k < s. The result we have obtained demonstrates the importance of elementary j-algebras in the theory of arbitrary j-algebras. As we noted above, the commutator of an elementary j-algebra R+jR+Z is equal to R+Z and, therefore, the commutator of G contains

Moreover, for k < s, and, therefore,

[G, GJ =

p

L (Rk+Z k)· k=l

As a result, the orthogonal complement of the commutator is equal to

and, therefore, its dimension p is equal to the number of terms in decomposition (10). In particular, if the dimension of the orthogonal complement of the commutator of aj-algebra is equal to 1, thej-algebra is elementary. By rk , we will proceed in our study of j-algebras in the following manner. We denote an element Rk such that [jrb rd = rb 1 ~ k ~ p. Let k < s. Then We setp(z)

[jr s' ZkJ c Zb [1's, ZkJ c Zk. = [jrs'z], q(z) = [rs,z] for ZEZk • It is clear that pq-qp = q,

(11)

i.e., the mapping irs --* p, rs --* q is a representation of the j-algebra of dimension 2 that is generated by the elements rs andjrs • We will prove that this representation is a symplectic representation (see Section 2). We set p(z, z') = m([z, z'J) = Imh(z, z'). The operators p and q are symplectic operators. Indeed, p(p(z), z') + p(z, p(z')) = m (Ur s, [z,

z'JJ) = o.

(12)

Similarly, it can be shown that the operator q is symplectic. Moreover, it follows from axiom II that

q+jqj+jp-pj = 0

or

tU,[j,qJJ = [j,p].

(13)

,

THE GEOMETRY OF HOMOGENEOUS DOMAINS

57

Relationships (11, 12 and 13) imply that the operators p and q generate a j-subalgebra of the algebra Sn (see Section 2), i.e., that the mapping rs -+ q,jrs -+ p is a symplectic representation. This symplecticrepresentation has the following property: The characteristic roots of the operators p and q are real. Symplectic representations with this property are said to be normal. They can be described in the following manner. Let G be a j-algebra of dimension 2, and let r be an element of the algebra G such that [Jr, r] = rand r -+ q, jr -+ p is a normal symplectic representation of the algebra G in the space Z; then the space Z can be represented in the form Z

X+jX+Z',

and (1) the spaces X +j X llnd Z' are orthogonal, (2) p(x, x') = 0 for any X,X'EX, (3) p(z) = AZ, where A = -t when ZEX, A = t when ZEjX, and A = 0 when ZEZ', and (4) q(x) =jx for XEX, while q(z) = 0 when zejX +Z'. Proof Choose some orthonormal basis in Z. Any real linear transformation ¢(z) of the space Z can be written in the form

¢(z) = Az + Bz, where A and B are complex matrices and z is the complex conjugate of z. It is not difficult to show that the transformation ¢(z) is a symplectic transformation if and only if the matrix A is skew Hermitian and the matrix B is symmetric. Under a coordinate transformation induced by a unitary matrix U in Z, the pair of matrices (A, B) is carried into (UAU*, UBU'). We now set q(Z)

Az+Bz,

p(z) = Cz+Dz.

(14)

The fact that the mapping r -+ q, jr -+ p is a symplectic representation implies that

D= iB,

(15)

CA-AC+2iBB = A,

(16)

CB+BC'-i(AB+BA') = B.

(17)

It is well known that any symmetric matrix can be reduced to diagonal form with non-negative elements on the diagonal by means of a transformation B -+ UBU', where U is a unitary matrix. As a result,

58

THE GEOMETRY OF CLASSICAL DOMAINS

we can assume without loss of generality that the matrix B is diagonal. We denote its diagonal elements by Ai, ... , Am, and we assume that As > 0 for 1 ~ s ~ v, where As = 0 for v + 1 ~ s ~ m. We set A = (a ks ), C = (cks ). It follows from (17) that Cks As

+ Csk Ak =

s =f. k,

i(aksAs+ ask A k ),

2Ai Ckk -

ia kk )

(18) (19)

= Ak·

The matrices C and A are skew Hermitian, so it follows from (19) that (20)

It follows from (16) that spA

= 2ispB 2 = 2iLAi.

(21)

Multiplying (16) by the matrix A, we obtain the relationship A2 = 2iB2A+CA2_ACA

from which we obtain v

sp A 2 = 2·1 '" ~

v 12 a kk Ak

= -

k= 1

'" ~ Ak2 k= 1

(22)

[here we have used relationship (20)]. It follows from (21) and (22) that spAA* =

spA

2

1

= 2ispA.

(23)

It follows from relationship (16) that (24)

It follows from (24) and (20) that

= -2i1 L a kk -::::;;-4'v

-1 sp A 2i whence we see that

* 4' v

spAA ~

i.e., that aks =f. 0 if k = s = v. We have therefore shown that A=

~(:' ~).

(25)

THE GEOMETRY OF HOMOGENEOUS DOMAINS

59

It follows from (18) and (25) that the matrix C is of the form

(Co

C

1

0 ),

(26)

C2

where C 1 and C 2 are v x v and (n1.-v) x (m-v) matrices, respectively. Substituting the expressions we have obtained for A and C into (16), we find that Ak = t when 1 ~ k ~ v, i.e., that

B=t ( Ev o

0) .

(27)

0

We conclude from (18) and (27) that the matrix C 1 is skew symmetric. Thus, we have shown that in some basis

q = tiNz+!Nz,

p(z) = CzHNz,

N

= (:":).

(28)

where the matrix C is of the form (26) and the matrix C 1 is skew symmetric. We denote the set of vectors of the form jq(Z) , ZEZ, by X, and we denote the orthogonal complement of X +j X by Z'. It follows from (28) that

p=

and

C!E

on

X

0

on

Z'

tE

on

jX

j

on

X

0

on

jX+Z'.

q={

(29)

(30)

It is clear that for x, x' E X,

p(x, x') = 1m hex, x') =

o.

(31)

This finishes the proof. In what follows we agree to call a linear subspace X of a space U for which (31) is true a Euclidean subspace. It is clear that jx 1= X if x EX. In what follows we will need to describe all normal symplectic representations of j-algebras which are the direct sum of j-algebras of dimension 2.

60

THE GEOMETRY OF CLASSICAL DOMAINS

Let G be aj-algebra that is the sum ofj-algebras Gb k = 1, ... ,p, that are each of dimension 2. By fk we denote an element of Gk such that [jrb rd = rk. Since the algebra G is the direct sum of the algebras Gk, we have [jrb rs] = 0. As a result, at each point in such aj-algebra there exists a system of elements rk such that (32) where bks is the Kronecker delta. It is not difficult to show that property (32) uniquely defines the system of elements rk • It is also clear that aj-algebra G of dimension 2p that contains a system of elements rk with property (32) is always the direct sum of j-algebras of dimension 2. All normal symplectic representations of suchj-algebras are described in the following lemma. Lemma 2. Let g -,>-g(z) be a normal symplectic representation in the space Z of aj-algebra G that is the direct sum ofj-algebras of dimension 2. Then the space Z can be represented in the form of the direct sum of orthogonal Euclidean subspaces X k, jXk, k = 1, ... ,p, and the complex subspace Zo in such a manner that

= -tbksz, riz) = bksjz, ZEXs , zEjXs, r,lz) = 0, jrk(z) = tb ks z, jrk(z) = 0, riz) = 0, ZEZ o·

jrk(z)

(33).

Proof As we showed above, the space Z contains Euclidean subspaces X for which (33) is true. We need only show that these spaces are orthogonal. It follows from (32) that the transformationsjrk(z) and rsCz) commute when k =1= s. Thus, (34) As we showed above, the spaces X k andjXk are Euclidean, i.e., the form p(z, z') is equal to zero on each of them. It follows from the properties of decomposition (12) that if p(rsCu) , u) = 0, then rsCu) = 0. As a result, it follows from (34) that (35) It follows from (35) and the properties of decomposition (12) that the

spaces X k are orthogonal. This completes the proof of Lemma 2. The basic result of the present section is given in the following.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

61

Theorem 2. Let G be some i-algebra, K its commutator, and let H be the orthogonal complement of K in the sense of the form k(x, y). Then (1) H is a commutative subalgebra of the algebra G and its representation onto K is completely reducible, i.e., K can be represented in the form of the sum of root spaces Ka that each consist of all x E K such that [h,x]

cx(h)x

forall

hEH.

We agree to call the linearforms cx(h), for which dimK =f. 0, roots. (2) Let Kal , ... , Ka p be all root spaces such that jK~t c

Then p = dim H and, when the roots the roots are of the form t(cxk+cx m ), t(cx,,-cx m ), 1;;:;; k < (3) Let

(36)

H; CX k

are appropriately labelled, all of

111 ~

p, tCXl" cx k , 1 ~ k ~ p,

(37)

p

xEKt(ak-alll)' then

zEK tall1 +

I Kt((JIIl-a s=m+ 1

s );

[x,jz] =j([x,z]) h([x,z], [x,z]) = Am h(x,x)h(z,z),

°

(38)

Am> 0.

(39)

(4) Let x E Ktcak -am)' r E Kalil; if X =f. and r =f. 0, then [x, r] =f. 0. (Note that the dimension of some of the spaces Ka where cx is of the form (37), may be equal to zero: in this case it follows from (38) and (39) that if dim K t (ak- all1) =f. 0,

dim Kt(alll-ad =f. 0,

then dim Kt(ak-at) =f. 0.) Proof We showed above that every i-algebra G can be represented in the form (40) where Gk = Rk +iRk + Z" is an elementary j-algebra, 1 ~ k ~ p, and [Gs,jRk+R k] = 0, (41)

[G s' ZkJ c Zk' for k < s. As we showed above, the representation of the i-algebra

(42)

p

IURs+Rs) s=k+l

in the space Zk is a normal symplectic representation. As a result, by

62

THE GEOMETRY OF CLASSICAL DOMAINS

Lemma 2, the space Zk can be represented in the form of the sum of orthogonal Euclidean spaces X ks ' j X ks , 1 ;?3 k < s ;?3 p, and the complex space ZOk, so that [jrs, x] = t(6 sk -6 st )x, [jrs,Y]

t(6 sk + 6st )y,

if XEX kt if YEjX kt if

(43)

ZEZ Ok •

We also noted above that the commutator K of the j-algebra G is equal to

while its orthogonal complement H is equal to

It follows from (41) that H is a commutative subalgebra, while it follows from (43) that its representation 6nto K is completely reducible. The root spaces KrJ. such that jKrJ. c H are spaces R k , 1 ;?3 k ;?3 p. It follows from (43) that every root is of the form (37). This proves 1 and 2 of Theorem 2. Statement 4 is an obvious corollary of Lemma 2. It remains to prove statement 3 of Theorem 2. Let x E X km and Z E Z,w It follows from axiom II of the definition of j-algebras that

[x, z] + j([jx, z]) + j([ x,jz]) [jx,jz J. We will prove that

(44)

[jX,z] = 0

if XE X kll" ZEZ,w It follows from (43) that

[jr,U' [jx,z]] = [[jrm,jx],z]+[jx, [jrlll'z]] = [jx,z]. As a result, [jx, z] E R'Il" On the other hand, it follows from (42) that [jx, z] EZk and, therefore, [jx, z] = o. It therefore follows from (44) that

j([x,z])

=

[x,jz]

63

THE GEOMETRY OF HOMOGENEOUS DOMAINS

Relationship (38) of statement 4 is therefore proved. It remains to prove (39). As a preliminary, note that if Zu Z2 EZIIP then [Zl' Z2] E Rm. Set [Zl' Z2]

Then

cx(zu z2)rm •

w([ z u Z2]) = cx(z l' z2)w(rm ) = 1m h(z 1, Z2)'

As a result (45) We will now show that

1

(46)

h([x, z], [x, z]) = 2 - - hex, x)h(z, z) w(rl/J

for XEXkm , ZEZm • Setting Zl=jZ, Z2=Z in (45) and taking the commutator of both sides of the equation and x E X kllP we obtain the relationship h(z,z)

- (-) [I'm' x] w I'm

.

.

= [[]Z,x],z]+[]Z, [z,x]J.

(47)

Now, using the fact that [I'm, x] = jx and again taking the commutator of both sides of relationship (47) and x, we obtain h(z, z) [ . ] JXX w(rm) ,

--

[[[jZ,x],z],x]

+ [[jZ,[z,x]],x].

It is easy to verify that [[z, x], x] = 0 for any x

E

X km ,

Z Ezm.

(48)

Thus, each

term of the right side of (48) is equal to

[[jz, x], [z, x]], whence follows (46). This completes the proof of Theorem 2. In concluding this section, we will consider classification of j-algebras of dimension 2n for small values of n. If n = 1, all suchj-algebras are isomorphic, as we showed above. The domain corresponding to them is the unit disk. If n = 2, the rank p does not exceed 2. When p = 2 and n = 2, comparison of dimensions shows that the algebra G is the direct sum of j-algebras of dimension 2. Thus, when n = 2, there exist two nonisomorphic j-algebras, one of rank 2, the other of rank 1. Now

64

THE GEOMETRY OF CLASSICAL DOMAINS

consider the case 12 3. For simplicity, we will consider description of only the irreducible j-algebras, i.e. j-algebras that do not decompose into a direct sum of j-subalgebras. For an irreducible j-algebra, n = 3 implies that p ~ 2. If p = 2 and the algebra G is irreducible, then dim X 12 > 0, and lZ= 3 implies that dim X 12 = 1 and dimZ01 = dimZ02 = 0. It is not difficult to show that all j-algebras of rank 2 for which dim X 12 = 1 and dimZ01 = dimZ02 = are isomorphic, and that they correspond to the Siegel circle [(2' Moreover, for n 3 there exists still one irreducible j-algebra of rank 1. We will now consider the case 12 = 4. If n = 4 and the algebra G is irreducible, then rank p ~ 2. Moreover, it follows from irreducibility that dim X 12 > O. Two cases are possible: (1) dim X 12 = 2 and (2) dim X 12 = 1. In the first case we obtain a symmetric domain of the first type, i.e., the set of all 2 x 2 complex matrices Z such that

°

ZZ<E.

(49)

In the second case dimZ2 = 0. Indeed, it follows from statement 3 of Theorem 2 that dimZ 1 ~ dimZ2+dimX12' As a result, ifdimZ2 > 0, then dimZ1 ~ 2, whence we find that dim G ~ 2(2+dimZ1 +dimZ2) ~ 10. Thus, in our case dimZ2 = 0, dimZ o l = 1. It is not difficult to see that all such j-algebras are isomorphic. The domain corresponding to them is a nonsymmetric domain in C 4 • Section 4. j-ideals

In this section we will prove that every normal j-algebra has only a finite number of j-ideals, and that the orthogonal complement of a j-ideal is always aj-subalgebra. Let Go be aj-ideal of a normalj-algebra G, and denote the orthogonal complement of Go by G'. Since Go is j-invariant, G' is also j-invariant. It remains to show that G' is a subalgebra. Let g 1, g 2 E G'. In order to prove that G' is a subalgebra, it is sufficient to prove that w([g 0, [g 1, g 2]]) = 0 for any go E Go. We have

Here we have used ,the fact that Go is an ideal and, therefore, w([[go,g],g']) = 0, if goEG and g'EG'. We have proved that the orthogonal complement G' of aj-ideal Go is always aj-subalgebra.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

65

We will now consider the natural mapping ¢ of the j-algebra G onto thej-algebra G'. This mapping is clearly aj-homomorphism. Let p

G=

I G k=l

k

be the decomposition of the j-algebra G into a semidirect sum of elementary j-algebras. As we showed in Section 3, a j-homomorphism of an elementary j-algebra is either an isomorphism or a homomorphism onto zero. As a result, ¢(Gk ) is either equal to Gk or 0. Thus,

I

Go =

Gki

•


We have therefore proved that any j-ideal of an algebra G is the semidirect sum of some number of elementary j-algebras contained in the decomposition ofthej-algebra G. It is clear from this that every normal j-algebra contains only a finite number of j-ideals. It is not difficult to prove by induction that if Go is a givenj-ideal of a normalj-algebra G, then the order of the elementary j-algebras contained in the semidirect decomposition of the algebra G can always be changed so that the algebras contained in Go will always be first in order. The theorem about the finiteness of the number of j-ideals no longer remains true for the case of arbitrary (not necessarily normal)j-algebras. We will give an example of a j-algebra containing a continuum of nonisomorphicj-ideals. Let G1 =R+jR+Z be an elementary jalgebra, and let K be a commutative Lie algebra consisting of skew Hermitian transformations of the space Z. We now define aj-algebra G in which G1 jis an ideal and the factor algebra GIG 1 is isomorphic to K. Consider the direct sum G of the spaces Gland K. We define commutation in G so that on G1 and Kit coincides with the one already available. Moreover, we set [K, R +jR] = 0, [k, z] =k(z). We define the endomorphism j and the form w in the following manner:

. {j

J=

°

o

on

G1

on

K

w(g) = for gEjR+Z+K, w(ro) = 1 where ro is an element in R such that [jr o, "0] = 1'0' It is easy to verify that {G, K,j} is a j-algebra. The j-ideals of the algebra G can be described in the following manner. Let Tbe an arbitrary subalgebra of the algebrajR+K, and assume that T is not contained in K. Then GT = R + Z + T is clearly a j-ideal. It is not difficult to see that G contains a continuum of nonisomorphic c

66

THE GEOMETRY OF CLASSICAL DOMAINS

algebras G T and, therefore, a continuum of nonisomorphic i-ideals. The situation does not improve if we restrict the discussion to algebraic j-ideals, because there are also infinitely many nonisomorphic i-ideals here. However, if we restrict the discussion to the so-called complete j-ideals, the theorem about finiteness becomes true. Let {G, K,i} be a j-algebra. By i-differentiation we mean a differentiation of the algebra G that commutes withi. We will say that ai-ideal Go of the algebra Gis complete if any element go of the algebra G such that the mapping x ~ [go, x] is a differentiation of the i-algebra G belongs to Go. It is easy to see that any i-ideal can be embedded in a complete i-ideal. It can be shown that any i-algebra contains only a finite number of complete i-ideals (see Chapter 6, Section 6). We should also note the notion of a complete i-algebra, which notion is important in the theory of arbitrary i-algebras: a i-algebra G is said to be complete if all of its differentiations are interior, i.e., have the form x ~ [go, x], where go E G. It can be shown that if ai-algebra {G, K,i} is complete, then the Lie algebra G is algebraic (see the appendix). Section 5. Homogeneous Siegel Domains of Genus 2

In this section we will describe how a given normal i-algebra B can be used to construct a Siegel domain of genus 2 in which the group exp B is transitive. Let B be a normali-algebra, K its commutator, and H the orthogonal complement of K. As we showed in Section 3, the algebra B can be represented in the form (1)

where the KrI. are root spaces. We denote the roots such thatiKrl.t c H by CX l , ... , cx p • Recall that, as we showed in Section 3, every root is of the form

t(CXk±CXm), 1;;;; k < m ;;;; p,

tcx k, CX k, 1;;;; k ;;;; p.

(2)

We define (3)

Y km = Kt(rl.k-rl. m )'

We set Then

L

= IR k +I k

Y km ,

Zk

= Ktrl.k·

V

G = L+jL+V.

= IZk' k (4)

THE GEOMETRY OF HOMOGENEOUS DOMAINS

67

Henceforth we will call this decomposition the canonical decomposition of a normalj-algebra. It is not difficult to see that L is a commutative ideal of the algebra B, jL is a subalgebra, [u, u] c L, and [j, U] c U. Consider the natural "representation of the algebra jL in L that associates the operator (5) with each element P EjL. The linear representation of the Lie algebra jL uniquely determines the linear representation of the corresponding Lie group @o expjL. We set p

r = Irk' k=l

where J'k denotes an element of Rk such that [jr k , rd = rk • It is not difficult to see that

[jr, I]

=

I,

for any

l E L,

(6)

whence, by means of axiom II of the definition of normal j-algebras, we can easily show that eli, r] = I.

(7)

It follows from (7) that the set V of points of the form gr, where g E@o, is a domain in L. It follows from (6) that V contains, along with any point I, the point AI, where A > 0, i.e., it follows that V is a cone. Denote the complex span of the space L by Co. The group @o is naturally defined in Co. Consider the set So £ Co of points of the form z = x+ iy, where yE Vand x is any real vector. We will show that in So a group of affine transformations (i.e., transformations of the form z ~ Az + c) whose Lie algebra is isomorphic to L +jL is transitive. We first define affine representations of the algebra L+jL in Co in the following manner:

where

cPp(z) = [j12' z] + Ii' P = Ii +j12' [jl,z] = [jl, X] +i[jl,y].

(8)

Recall that an affine representation of a Lie algebra G in a linear space is a mapping that associates an infinitesimal affinet mapping cP g

t The transformation corresponding to each point in a vector is called an infinitesimal affine transformation.

68

THE GEOMETRY OF CLASSICAL DOMAINS

with each g E G, where

cp[gj, g2] = $91 CP92 - $92 CP9l

(9)

and $g denotes the linear part of the transformation cPg. In our case, verification of (9) is immediate. We will now consider the corresponding representation of the Lie group ® 1 whose Lie algebra is isomorphic to L+ jL. It is easy to see that the transformations in ®1 have the form z -+ Az+b,

(10)

where z -+ Az is a linear transformation in ®o and b is any real vector. Transformations of the form (10) are clearly transitive in the domain So. The space introduced in (4) is naturally a complex space. Let C 1 be the complex space consisting of the pairs (z, u), z E Co, U E U. We define a Co-valued vector function on U: F(u 1, U2) = ![ju 1, U2] +!i[Ul' u 2]

(11)

Consider the domain S 1 £ C 1 consisting of the pairs (z, u) such that Imz-F(u,

U)E

V.

(12)

We will now show that in Sl there is a simply transitive group ® whose Lie algebra is isomorphic to G = L+ jL+ U. We associate a linear transformation cPg z-+

[j12,z]+t[b,u]-ti[b,ju]+11'

U

-+

[j12,u]+b,

(13)

where z = x+iy, with each g = Ii +Jl2 +b, where IkEL, bE U. It is not difficult to see that the transformation cPg is complex linear in C 1 • The pro0,fthat the mapping g -+ cPg is an affine representation of the Lie algebra G presents no difficulties. We will now write out the group ® = exp G. The transformations expjl clearly have the form z -+

Az,

U -+

(14)

BLI

B = exp(ad u ;1). The transformations exp (l + b), where I EL, bE U are of the form

where

A = exp(adLjl),

z-+z+2iF(u,b)+iF(b,bj+l,

u-+u+b.

(15)

The transformations '(15) preserve Imz-F(u,u). It is easy to verify that transformations (14) and (15) form a group that is simply transitive in Sl' As we can show with little difficulty, the endomorphism} of the

69

THE GEOMETRY OF HOMOGENEOUS DOMAINS

algebra G coincides with the endomorphism induced by the complex structure of the domain S l' It remains to show that the domain Sl is a Siegel domain of genus 2. We will now give two entirely different proofs of this proposition. The first of the proofs is based on certain general propositions in the theory of functions of several complex variables. It has a conditional nature. That is, we assume that it is known that a domain in which the group exp G is transitive, where G is a normal j-algebra, is bounded. The second proof is purely algebraic and understandably unconditional. We should also note that by complicating the method used in the first argument, we can give it an unconditional character. First proof We must prove the following three propositions: (1) The cone V contains no lines, (2) the cone V is convex,

(3) F(u, u) E V for any u =1=

°(V

denotes the closure of the cone V).

Consider the j-algebra G1 = L +jL. As we have already proved, this j-algebra corresponds to a domain So in the space Co: ImzEV.

(16)

If the cone V contained a line, the domain So would contain the homogeneous complex space C 1 • It follows from the Liouville theorem that an analytic function that is bounded in So is constant on C 1 . As a result, not all of the points of So can be separated by bounded analytic functions, i.e., it is not possible to find a function h(z) that is bounded in So and is such that h(zo) =1= h(Zl) for all pairs of points zo, Zl E So. Thus, So cannot be analytically mapped onto a bounded domain. This contradicts our assumption that a domain in which the group exp Gis transitive, where G is a normalj-algebra, is always bounded. We will now prove that the cone V is convex. As we know, any bounded homogeneous domain is a domain of holomorphy. Thus, So is a domain ofholomorphy. A cylindrical domain (i.e., a domain of the form ImzE W, where Wis some set) is a domain of holomorphy only when the set W is convex. As a result, the cone V is convex. It now remains to prove that F(u, u) E V. It is sufficient to prove that the domain T

y-F(u, U)E V

(17)

is convex. Indeed, let Yo E V; then the points (Yo, 0) and (Yo+F(u,u),u)

70

THE GEOMETRY OF CLASSICAL DOMAINS

belong to T. It follows from the convexity of the domain T that the half-sum (Yo+tF(u,u),tu) of these points also belongs to T, and, therefore, Yo +tF(u, u)-tF(u, u) E V.

Taking the limit as Yo ~ 0 in this relationship, we find that F(u, u) E V. Convexity of the domain T is proved in the same way as convexity of the cone V, namely, we prove that some normalj-algebra Gis transitive in the cylindrical domain 1m WET. The j-algebra G is constructed in the following manner. We denote the complex span of the space U, i.e. We set the set of vectors Ul + iu 2 , where u l , U 2 E U, by

a.

G = L+jL+ a. We define an endomorphism] in

(18)

G in the following manner:

_ {j J=

on

i on

L+jL U.

(19)

We introduce commutation in the following manner:

[gl,g2]H = [gl,g2]

for

gl,g2 EL +jL,

(20) + iU2JH = [g, u l ] + i[g, u 2], g E L+ jL, [U l + iu 2, U 3 + iU4]H = [Ul,jU4] + [jU2, U3J, UkE U, k = 1, .. .4.

[g,

Ul

Second proof This proof is based on one lemma, which is of independent interest and will also be used in Section 7. Lemma 1. The cone V coincides with the set of all vectors YEL such that w(Ay) > 0 for all A E@o' This lemma is proved by induction on the rankt p of the cone V. For p = 1 the lemma is obvious, because the cone V is, as we can easily see, a set of vectors of the form Aru where A > O. We assume that the lemma is true for p -1, and we will prove it for p. Consider the j-subalgebra G' p-l

G' =

I k=l

p-l

I

(jRk+R k )+ (X km + k<m

Y km ) +

I

(21)

Zk

k=l

ofthej-algebra G. Moreover, set p-l

L' =

I R + l;E;k<m

p-l Yklll'

U' =

I

(Zk+XkP+

k=l

t The rank of a cone V is the rank of the j-algebra L + jL.

Y kP )'

71

THE GEOMETRY OF HOMOGENEOUS DOMAINS

Decomposition (4) for thej-algebra G' is of the form G' = L' +jL' + U'. It is clear that p-l

L

L' +Rp+ Y p , where

Yp =

L Y kp .

(22)

k=i

We need the following auxiliary formula: (exp(ad Lxp))(Arp + [' + Yp) = Arp+ [' + Yp+ [xp,y p] -Ajxp+tA[jxp, x p] (23) where A is an arbitrary real number and p-l

XpEXp =

L X kp ,

['EL',

YpE Yp.

k=l

Transformations of the form (23) preserve the expression (24)

Now denote the corresponding cone in the space L' by V'. The cone V coincides with the set Vof alll = }crp + l' + Y P' such that (25) This follows from the facts that: (1) the domain

V contains the point

(2) the group of transformations of the form exp (adLx), where x EjL, maps V into itself and is simply transitive in V. The first of these statements is obvious. The second statement can be verified in the same way as the analogous proposition for the domain S. We will now prove the lemma by the induction hypothesis, we may assume that the lemma is true for V'. Let l = Arp+l' +Y p. Then w(1) = Aw(rp) + w(l'). If lEV, then, by (25), we have A > 0, and, therefore, w(l) > w(l'). It follows from (25) that 1 w(1') > 2A w([jyp,y p]);?; 0.

We have proved that l E V implies that w(1) >

o.

72

THE GEOMETRY OF CLASSICAL DOMAINS

The cone V is homogeneous with respect to the entire group @o, so > 0 for all A E @o' We will now show that if w(AI) > 0 for all AE@o, then IE V. We apply a transformation of the form (23) with xp = (ljA)jyp to such an I. Then I = Arp + l' + yp is carried into w(AI)

1

[" = Ar p+ [' - 2A [jyp, yp]. Moreover, consider the vector I;' = (exp (ad L ijrp))l". It is not difficult to see that 1 [~ = [tArp+ [' - 2A [jyp, yp]. By hypothesis,w(l~) > 0 for all t, and, therefore, w(l') ~ (lj2A)w([jyp,yp])' The domain V is open, so w(!') > (lj2A)w([jyp,yp])' We have proved that 1= Arp + l' +Yp is such that w(AI) > 0 for alIA E@o, then w(k(l)) > 0, where k(1) = AI' - tUyP' yp]. It is not difficult to show that k(exp (adLx)l) = (exp (adLx))k(l)

for all x EjL'. As a result, if weAl) > ofor alIA E@o,then w((exp (adLx)) k(/)) > 0 for all x EjL'. By the induction hypothesis, this implies that k(l) E V' and, therefore, lEV. This completes the proof of the lemma. We will now prove that the domain Sl is a Siegel domain of genus 2. This requires us to verify statements 1, 2 and 3 of the first proof. Statement 1 follows from the fact that the set of linear forms of the form weAl), where A E@o, contains a complete system of linearly independent forms. Indeed, if the forms wCAI) were linearly dependent, this would also be true for the forms w([a, I]), where a EjL. As a result, in this case there exists an ao =f:. 0 such that w([a o, l]) = 0 for all l. This is clearly impossible, because w([ao,jaoD < 0 for ao =f:. O. Statement 2 is obvious. It remains to prove 3. We have F(Bu, Bu)

= AF(u, u), A = exp(adLjl)

B = exp(aduj[),

[EL.

As a result, w(AF(u, u)) = w(F(Bu, Bu))

= w([j(Bu), Bu]) > 0

for all A E@o' By our lemma, this implies that F(u, u) E V. This completes the proof that Sl is a Siegel domain of genus 2.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

73

In concluding this section, we will present a method permitting explicit representation of polynomial inequalities defining the cone V. We will prove that if V is a cone of rank p, there exist p polynomials Pl(l), ... ,Pi!) such that the cone V coincides with the set of points Pk(l) > 0,

k

= 1, ... ,p.

°

If A > 0, p = 1, the cone V consists of vectors of the form Ar l , A > We can take A for Pl. We will now present an inductive construction of the polynomials Pl(l), ... ,Pil). As in the proof of the preceding lemma, we will use relationships (25). Assume that Pl(l'), ... ,Pp_l(l') are polynomials that define a cone V' by means of the inequalities Pl(l') > 0, ... ,Pp-l(l') > 0.

We substitute AI'--![jyp,yp] for 1', and denote the resulting polynomials in lby P 2 (l), ... ,Pil). We also setPl(l) = A. It follows from considerations stated in the proof of the lemma that the inequalities P 1(l) > 0, ... ,Pi!) >

°

describe the cone V. It is not difficult to show that the degree of Pk(l) is no greater than 2k - l • The considerations we have given make it possible to explicitly write the polynomials corresponding to the given j-algebra. When the cone V is the set of all positive definite symmetric matrices, the polynomials P ll) are the successive principal minors of these matrices. The polynomials we have constructed have the following important property: This property is important in the theory of special functions (S. G. Gindikin [2] and [3]). Section 6. Universal j-algebras

Let B be a normalj-algebra. In this section we will describe use of the algebra B to construct a j-algebra with the following properties: (1) A admits the decomposition A=B+G

(1)

74

THE GEOMETRY OF CLASSICAL DOMAINS

where B is a j-ideal and G is a semisimple j-algebra. (2) Every normal j-algebra N with the ideal B and the factor algebra G' can be represented in the form

W=B+G',

(2)

and there exists aj-homomorphism ¢: G' ~ G such that [g:

bJ =

[¢(g'),

bJ.

(3)

It is natural to call such an algebra A a universalj-algebra. We will first describe the construction of the algebra G. Let B be a normalj-algebra, and let

B=L+jL+U

(4)

be its canonical decomposition. Consider the set G of all real linear transformations u ~ pu, such that

U

E U, (5)

(uadjL)p = p(adujl)

for all

lEL.

(6)

Here and in what follows P(Ul' uz) = CO([Ul' uz ]). It is clear that if Pb pz E G, then [pl,Pzl = PI pz - PZPl belongs to G, i.e., G is a Lie algebra. The transformations adujl commute withj, as a result of which it follows from axiom III thatjEG. We now define the endomorphismj in G by means of the following formula: j(p) = t[j, pJ.

(7)

Moreover, we denote the set of all pEG such thatj(p) = 0 by Go. It is easy to verify thatjz(p) +P E Go for allp E G. The form co is defined in the following manner. Any real linear transformation P of the space U can be written in the formpu = au + 13ft, where a and 13 are complex linear transformations of U and ft is the complex conjugate of u. It is not difficult to show that if pu satisfies (5), a can be described by a skew Hermitian matrix in any orthonormal basis, while 13 can be described by a symmetric matrix. We set co(P) = (lji) spa. It follows immediately that {G, Go,j, co} is aj-algebra. We now show that the algebra G is completely reducible. Indeed, let U0 be a subspace of the space U invariant with respect to G; we will

75

THE GEOMETRY OF HOMOGENEOUS DOMAINS

show that its orthogonal complement U 1 is also invariant with respect to G. Indeed, if Uo E U, U 1 E U 1 and pEG, then p(uo, PU 1) = - p(puo, Ul) = 0.

As a result,pul E U1. We now define the algebra A in the following manner: A =B+G.

We introduce a commutation operation so that: (1) on Band G it coincides with the commutation already defined, (2)

(8)

[G,L+jL] = 0,

(3)

[p,

u]

(9)

= pU.

It follows immediately, as the reader can easily show, that A is a j~algebra.

It remains for us to prove universality ofthej-algebra A, i.e., that any normal j~algebra N in which B is an ideal can be represented in the form (2), and that there exists aj~homomorphism ¢: G' -?- G for which (3) is true. Let N be a normalj-algebra in which the algebra B is an ideal. As we showed in Section 4, the orthogonal complement G' of B is aj-subalgebra of the algebra N. Using the theorem of Section 3 on the form of the roots of any normalj~algebra, we can show with no difficulty that

[G', L+ jLJ We set

Pg=adug,

0,

[G', u]

where

c

U.

gEG'.

(10) (11)

We now show that (11) defines a j-homomorphism of the algebra G' into the algebra G. First of all, we must verify that the operators Pg belong to G. We have P(piUl), u 2) + P(Ub piu2)) = w([g, [u 10 U2]]) = 0.

(12)

It follows from this and (10) that Pg E G. We now show that the mapping g -?- Pg is a j-homomorphism. It follows from the fact that [G', u] c Uthat P[91,92J

= P91 P92 -

P 92 P 91 •

(13)

It remains to verify that (14)

76

THE GEOMETRY OF CLASSICAL DOMAINS

It follows from axiom II of the definition of normalj-algebras that pg+jPjg+jpgj-Pjgj or

0,

[j, Pjg] = j{j, [j, pg]],

(15) (16)

which is equivalent to (14).

Section 7. Canonical Models of Bounded Homogeneous Domains 1. This section is devoted to describing realizations of bounded homogeneous domains in the form of Siegel domains. Recall that a Siegel domain is said to be homogeneous if its group of quasilinear transformations is transitive in it. Let ~ be some complex manifold. A fibering of the manifold ~ is said to be analytic if the base ~ 1 of this fibering is a complex manifold and the projection ¢ of the manifold ~ onto ~ 1 is a complex analytic mapping. All of the fiberings encountered below are analytic. Note that, as a rule, the fiberings discussed below are not locally trivial as complex analytic fiberings, and, simultaneously, as real analytic fiberings they are direct products. Henceforth, we agree to say that a fibering of a manifold ~ is homogeneous if the set of analytic automorphisms of ~ that preserve the fibering is transitive in ~. There is a natural homogeneous fibering for every homogeneous Siegel domain of genus 3. Some homogeneous fibering is therefore associated with each realization of a bounded domain in the form of a homogeneous Siegel domain. The fundamental result of the present section consists in the fact that the converse is also true, namely, that the following theorem is true. Theorem 3. Let ~ be a bounded domain in C". With each homogeneous analytic fibering of the domain ~ lve can associate a realization of the domain ~ in the form of a homogeneous Siegel domain of genus 3 whose base is the base of the given fibering of the domain ~. The plan of the proof for this theorem is as follows. First we describe the construction of the Siegel domain of genus 3 that corresponds to a given universalj-algebra. Then we prove that with each homogeneous fibering of the bounded domain ~ we can associate a j-ideal of the normal j-algebra associated with the domain ~. Theorem 3 then follows from the theorem of Section 6 on universality.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

77

2. Let B = L+ jL+ Ube a normalj-algebra. We denote the universal j-algebra corresponding to the algebra B by A. In this paragraph we will describe the Siegel domain S of genus 3 in which the transformations in the group whose Lie algebra is A are quasilinear transformations. Let V be a cone, and let F(u, v) be the vector function corresponding to the algebra B. A group whose Lie algebra is isomorphic to the algebra B is transitive in the domain H 5; C"+ m defined by the relationship

Imz-F(u, U)E V.

(1)

We denote by ® the group of affine transformations of the domain H of the form Z -1-

where

exz,

ex = exp(adLjl),

U -1-

f3u

(2)

f3 = exp(adujl).

As we showed in Section 5, transformations of the form z -1- exz are transitive in the cone Vand they satisfy the following relationship:

F(f3u, f3u) = exF(u, u),

U E

U.

(3)

Consider the set K of all antilinear transformations u -~ pu of the space U that possess the following properties:

F(pu, v) = F(pv, u)

(4)

(V is the closure of the cone V)

F(u, u)-F(pU,pU)E V F(u, u) =J. F(pu, pU),

f3p

pf3 for

f3

(5)

if u =J. 0

of the form

adujl.

(6)

(7)

We denote the set of all antilinear transformations u -1- pu for ¥
1m z- ReF(u, (E+ p)-l u) E V,

pEK.

(8)

We now show that a group whose Lie algebra is isomorphic to the algebra A = B + G is analytic in S. We first define, in S, the transformations in a group whose Lie algebra is isomorphic to B. Transformations of the form exp (lol- u) are "parallel

78

THE GEOMETRY OF CLASSICAL DOMAINS

translations" of the domain S (see Chapter 1, Section 3). The transformations expjl are defined by formula (2). We should note that (7) implies that they map the domain S into itself. It remains for us to define the action of the group exp G. First we show that this group is a group of analytic automorphisms of the domain K. Recall that the algebra G consists of all real linear transformations u -1- pu, U E U, such that

P(PU1,U2)+P(Ul,PU2) = 0, (adujl)p = padujl

for any

I EL.

(10)

As a result, the group exp G consists of all transformations which

P(¢Ub ¢U2) = P(Ul' u 2), (adujl)¢ = ¢(adujl)

(9)

Ul,U2EU

¢u for (11)

ttl' Ll2 E U

for any

U -1-

(12)

I EL.

We set ¢u = yu+bu, where the transformation yu is linear and bu is antilinear. It is not difficult to show that if p is antilinear and satisfies (9), and if ¢ satisfies (9), then ¢(p) = (yp+b)(bp+y)-l also satisfies (9); moreover, if h(pu,pu) < h(u, u), then h(¢(P)u, ¢(P)u) < h(u, u). (h(u, v) is a Hermitian matrix in U. It is also clear that (aduj/)¢(p) = ¢(p)(aduj/), if p E K and ¢ satisfies (12). As a result, for any a of the form exp (adLjl)

w(aF(¢(p)u, v)) = w(aF(¢(p)v, v)),

U,VEU

w(aF(¢(p)u, ¢(p)u)) < w(aF(u, u))

(13)

if p E K and ¢ satisfies (11) and (12). It follows from Lemma 1 of Section 5 that then (3) and (4) are satisfied by ¢(P) and, therefore, ¢(P)EK. We now define the action of the group exp G everywhere in the domain S. With each ¢ E exp G we associate the following transformation: Z -1-

z+ iK(u, Lt, p)

u -1- B(p)u

(14)

p -1- ¢(p) = (yp+b)(bp+y)-l where

, B(p) = (bp+y)-l K(u,v,p) = F(B(p)u, (E+¢(p))-lB(p)v)-F(

(E+p)-l V). (15)

THE GEOMETRY OF HOMOGENEOUS DOMAINS

79

It is clear that transformations of the form (14) map the domain S into itself. Difficulties arise only in verifying that these transformations are analytic, i.e., in proving that K(u, v,p) is an analytic function of u, v, and p. We will not present the details-only a plan of the proof. First of all, it is necessary to verify the formula

IX(K(u, v, p)) = K(f3u, f3v, p)

(16)

where IX and f3 are the same as in (2). These formulas imply that it is sufficient to prove that the function w(K(u,v, p)) = h(B(p)u, (E + ¢(p)) -lB(p)v) - h(u, (E + p)-l V)

is analytic. This can be done by direct substitution. 3. Let ~ be a bounded homogeneous domain in e" and let B be the corresponding normalj-algebra. In this paragraph we will show that the following decomposition of the algebra B corresponds to each homogeneous fibering of the domain: B = Bo+B'

(17)

where Bo is a j-ideal that is transitive on the fiber and B' is the complementary subalgebra corresponding to the base of the fibering. The theorem of Section 6 on universal j-algebras, (17), and the construction of the preceding paragraph immediately imply that the main assertion of the present section-Theorem 3-is valid. Decomposition (17) is almost obvious. There are no difficulties connected with proving it, and the proof may be conducted by means of standard techniques for operating with algebraic Lie algebras. We consider some homogeneous fibering of the domain ~ with the base ~'. We denote the p'"ojection of ~ onto ~' by ¢. Let ® be the set of all analytic automorphisms of the domain ~ that preserve the given fibering. Moreover, denote the subgroup of the group ® that consists of the automorphisms that map each fiber into itself by ®o. It is not difficult to show that the group ®o is transitive on the fibers of the given fibering. Let G and Go be the Lie algebras of these subgroups. It is clear that Go is an ideal of G. The algebra G is uniquely determined, because it is the maximal sub algebra of the Lie algebra of all analytic automorphisms of the domain ~ in which Go is an ideal. This implies that G is an algebraic Lie algebra. It is not difficult to show that Go is also an algebraic Lie algebra. Let Bo denote the maximal subalgebra of the algebra Go that is solvable and split over the field of real numbers,

80

THE GEOMETRY OF CLASSICAL DOMAINS

and denote a subalgebra of the algebra G that contains B o by B. The algebra Go, which is an algebraic algebra, is transitive on the fibers of our fibering. Therefore, B o is also transitive on the fibers of our fibering, and, therefore, is a j-ideal of the algebra B. As we showed in Section 4, the orthogonal complement of a j-ideal is always a normal j-subalgebra of the algebra B. This proves decomposition (17). 4. In Chapter 3 we will use the following terminology. Consider some homogeneous fibering of a bounded domain ~ and let S be the homogeneous Siegel domain corresponding to this fibering: 1m z-ReLlu, U)E V,

tE~'

(18)

where ~' is the base of the fibering. By a directing subgroup ¢)., A > 0, of this domain we will mean a directing subgroup of the corresponding fibering of the domain S. The group of parallel translations of the fibering is defined analogously. It follows from the proof of Theorem 3 that the mapping of the domain ~ onto the Siegel domains corresponding to the given fibering is uniquely defined, with the trivial exception of the case in which the base of the fibering is a point. t In what follows, we will, without special mention, assume that a fibering is given together with a directing subgroup and a subgroup of parallel translations. As a result of this convention, a uniquely defined mapping onto a Siegel domain corresponds to each homogeneous fibering. We should also note that the group B of parallel translations of a fibering consists of all of this region's automorphisms g such that lim cp;l g¢). = 1, ). .... +00

where ¢). is a directing subgroup. This implies, in particular, that the group of parallel translations is uniquely defined by a directing subgroup. The converse is also true, namely that a group of parallel translations uniquely defines the corresponding directing subgroup. Section 8. Canonical Models of Symmetric Domains

This section is devoted to certain refinements of Theorem 3 that are valid for symmetric domains. We will use these refinements in Section 6 of Chapter 3.

t This exception appears only if P2 is the direct product of domains such that the base of the fibering in one consists of one point.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

81

Theorem 4. (1) The base of a homogeneous jibering of a sYl1unetric domain ~ is always a symmetric domain. (2) The jiber of a homogeneous jibering is a homogeneous Siegel domain S of genus 2 for which the group n of all affine transformations of the cone V that extend to affine transformations of the domain S is reductive. (3) A maximal solvable and split (over R) normal subgroup of the group of analytic automorphisms of the domain ~ that preserve a given jiberingt is generated by a directing subgroup and the subgroup of parallel translations of the given jibering. Proof Let G be the Lie algebra of the group of all analytic automorphisms of the domain ~. Let H denote the maximal commutative

sub algebra of G that consists of semisimple elements with real characteristic roots. The representation of H in G is completely reducible and, consequently, G can be represented in the form of the sum of spaces Ga on each of which the representation H is scalar, i.e., Ga consists of all g E G such that [/z,g]=O'.(h)g

forall

hEH.

(1)

Here O'.(h) denotes a linear form in hE H. It is clear that Go (the centralizer of H) is of the form Go=H+C,

(2)

where C is a compact subalgebra of the algebra G. The nonzero forms O'.(h) such that dim Ga =f. 0 are called the roots of the algebra G. It is well known that along with 0'., the form - 0'. is also a root. We now introduce a lexicographical ordering into the space oflinear forms defined on H. Then either the root 0'. or the root - 0'. will be positive. Let

(3)

The algebra B is a maximal solvable and split (over R) subalgebra of the algebra G. Consequently, the algebra B is transitive in the domain ~ and, therefore, the structure of a normalj-algebra can be introduced into B.

t By automorphisms preserving a given fibering when the base of the fibering is a point we will mean the automorphisms that are linear on the corresponding Siegel domain of genus 2.

82

THE GEOMETRY OF CLASSICAL DOMAINS

The commutator of the algebra B is clearly equal to

while its orthogonal complement is equal to H. As we showed in Section 3 (Theorem 2), the set of root spaces of the algebra B contains p spaces Gak , k = 1, ... ,p such that

(4) where p = dim H. Every root of the algebra B has the form (5) for some enumeration of the roots We define

O'.k,

1

~

k

~ p.

(6)

P

Uk = Zk+

L

m=k+l

(X km + Y km )·

P

Then

B=

L (jR k+ Uk+R k) k= 1

(7)

and thej-algebrajRk + Uk+R k = Bk is an elementary j-algebra. Let n be some homogeneous fibering. We denote the Siegel domain of genus 2 corresponding to the fiber of this fibering by S. As we showed in Section 7, some j-ideal Bo of the algebra B corresponds to each fibering of the domain ~, and the group expBo is a maximal subgroup of the group exp B mapping each fiber of the given fibering into itself. In Section 4 we proved that any j-ideal Bo of the algebra B (after admissible relabeling of the algebras B k ) is representable in the form Po

Bo =

L jR k+ Uk+R k· k=l

(8)

Let rk denote an element of Rk such that [jrk' I'd = rk. The Lie algebra of the directing subgroup of the given fibering is generated by the vector

THE GEOMETRY OF HOMOGENEOUS DOMAINS

83

The Lie algebra L of the group of parallel translations of the fibering is of the form

L=2+U, where

2

=

L

Po

Ykrll'

U

1 ;;:;k<m<po

=

L1

Zk+

k=

L

(X km + Y km )·

(9)

1 ;;:;k;;:;po po;;:; m;;:; p

The Lie algebra G of the group of automorphisms preserving the given fibering is the normalizer of the algebra L. It is therefore clear that

(10)

The Lie algebra Go preserving the fiber is of the form

Go = Bo+Co+

L

Gtcam-ak)'

(11)

po'iJ;m>k'iJ; 1

where Co is some subalgebra of the algebra C. It is not difficult to show that the Lie algebra of the group generated by transformations of the cone V (corresponding to the domain S) is of the form (12)

and, consequently, is a reductive Lie algebra. The algebra G' = GIGo, which, as we can easily see, is semisimple, corresponds to the base of the fibering, whence it follows that the base is a symmetric domain. The maximal solvable and split (over R) ideal of the algebra G is clearly equal to

2+ U +{jr} where {jr} denotes the one-dimensional subspace generated by the vector Jr. This completes the proof of the theorem.

Section 9. The Geometry of Classical Domains Below we will assume that the classical domains under discussion are realized, for example, on page 120 in Siegel's book [1]. We will call this realization a "canonical realization of the 'unit circle' type", or simply a canonical realization. The concept of components that we introduce

84

THE GEOMETRY OF CLASSICAL DOMAINS

below is formally related to imbeddings in ell. As a matter of fact, moreover, this concept can be given an invariant form, i.e., one independent of the selection of the imbedding. Below we will briefly show how to do this. Definition 1. Let A be some set in en. It is said to be analytic if in a neighborhood of each of its points it may be given as a set of common zeros for a finite number of functions analytic in a neighborhood of this point·t Definition 2. An analytic set is said to be regular ifin a neighborhood of each of its points in the appropriate coordinate system it may be given as a set of common zeros of a finite number of linear functions. We will now formulate the notion of a boundary component of a domain, which notion is fundamental in this chapter. Formally, we will define it for arbitrary domains, but we will use it only for canonical realizations of classical domains. Definition 3. Let ~ be some domain, and let F be its boundary. An analytic set !F G: F is called a boundary component if any analytic curvet ¢(t), I t I < e that is entirely contained within F and intersects !F is completely contained in !F. We agree to say that a component that is a regular analytic set is regular. Note that from the definition it is not at all clear that every boundary point is contained in some boundary component. It is only clear that, at least for the case of regular components, every boundary point is contained in no more than one component. Example. Let ~ be a polycylinder, i.e., a domain defined by the inequalities

IZ11 <

1, ... , IZIII < 1.

The boundary F clearly consists of the points in

IZ11

e" such that

~ 1, ... , IZ/Il ~ 1,

where the sign of equality occurs in at least one of the inequalities. The set !F v of points of the form (1, ... , 1, zv+ l' ... , Z/I),

Izv+

11 <

1, ... , \z/ll < 1

(1)

t This definition is somewhat different from that generally accepted, which we will give in Section 10. t An analytic vector function cp(t) = (CPl(t), •.•. , cpn(t)) that is given in some disk I t I < e in the ordinary complex plane is called an analytic curve.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

85

is, as we will now show, a boundary component. It is clear that :¥v is a regular analytic set and :¥v c F. We will prove that any analytic curve in F that intersects :¥v is entirely contained in :¥v' Indeed, let ¢(t) = (¢l(t), ... , ¢,It)), Itl < e be an analytic curve lying in F, and let ¢(O) E :¥v' Membership of the curve ¢(t) in the boundary:¥ of domain ~ means that l¢l(t)1 ;;;; 1, ... , 1¢,lt)1 ;;;; 1 for all t, Itl < e. In addition, ¢(O) E:¥ v implies that ¢1(0) = 1, "" ¢v(O) = 1,

\¢v+ 1(0)\ < 1, ... , \¢nCO) \ < 1

and, therefore, by the maximum modulus principle for analytic functions, ¢l(t) == 1, .. " ¢v(t) == 1, I¢v+l(t)\ < 1, .. " \¢nCt)\ < 1 when It I < e.

The curve ¢(t) is therefore entirely contained in :¥v' It is not difficult to verify that any component of boundary F is analytically equivalent to a component:¥v for some v, 1 ;;;; v ;;;; n. Theorem 5. Let

~

be a classical domain, Then:

(1) any boundary point is contained in some regular component; (2) every component is analytically equivalent to some classical domain in a space of lower dimension; (3) if the domain ~ is the product of irreducible domains ~1' ... , ~h then any component:¥ is equal to the product ofthe components:¥ 1, '''':¥k of the corresponding domain'). t We will assume that the statements of the theorem are true for irreducible classical domains and show that they are true for all classical domains. In order to do this, it is clearly sufficient to prove the following: Let ~ = ~1 x .. , X ~k' where ~1' ... , ~k are irreducible domains. We will prove that any component :¥ is a product :¥ 1 X ... x :¥k, where :¥1' "·,:¥k are the components, possibly "ideal", of the domains

~l'''''~k' Indeed, let :¥ be some boundary component of the domain ~ and consider the projection of ~ onto domain ~ io(1 ;;;; io ;;;; k). In this case, the component :¥ clearly corresponds to some analytic subset :¥ io of the boundary of the domain ~ io or to the domain ~ io itself. It is easy to verify that :¥ io is either a boundary component of the domain ~ io or

t We will treat the set of all points inside a domain boundary component of this domain.

.@i(1 ;;;;

i ;;;; k) as an "ideal"

86

THE GEOMETRY OF CLASSICAL DOMAINS

all of domain ~ io. The product ff ff 1 X .•• X ffk is an analytic subset of the boundary of the domain~. It is easy to show that ff c ff. If ff and g; were not the same, then it would not be difficult to construct an analytic curve that is contained in F, intersects ff, and is not entirely contained in ff. From this it follows that j/: = ff. Statements 1 and 2 of the theorem follow from statement 3. The proof of the assertions of the theorem for irreducible classical domains is given in Sections 10 and 11. Let ~ be some classical domain. As we show later on, its boundary F consists of a finite number v of sections that are homogeneous with respect to the full group of analytic automorphisms of the domain~. It is clear that the points of each such part are contained in analytically equivalent components. As a result, there are exactly v typical (analytically nonequivalent) boundary components for the domain ~. The following theorem describes the structure of a domain close to the boundary. Theorem, 6. For every classical domain ~ there exist v Siegel domains (from now on we will call them canonical) for which the typical components of domain ~ serve as bases. For any point zo, there is an analytic mapping of the domain ~ onto one of these v domains, where the analytic automorphisms of the domain ~ that map a component ff(zo E ff) into itself become quasilinear transformation of this domain, while the analytic autol1wrphisms of the domain ~ that leave the point Zo fixed become linear transformations of this domain. The Proof of this theorem is given in Section 10. In Sections 10, 11, and 12 we will separately construct the above-mentioned canonical Siegel domains for each type of irreducible classical domain. For domains that are products of irreducible domains, these canonical realizations may be obtained as products corresponding realizations of irreducible domains. We will now give an invariant definition of a component, i.e., one that is independent of the method of imbedding ~ in a complex affine space. As we know, for every classical domain ~ there is a dual complex compact symmetric manifold Dt in which ~ is imbedded in the form of some domain such that the analytic automorphisms of ~ are analytic automorphisms of D. The set F of points D\~ that are limit points for ~ is called, as usual, the bqundary of the domain~.

t See Section 10, p. 91.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

87

As we will show in Sections 10 and 11, the boundary introduced by this method is the same as the boundary (in the usual sense of this word) of the canonical realization of ~ in an affine complex space. Thus, there is a unique imbedding of the domain ~ in C", A component defined by . means of this imbedding is invariant. We shall give a different method for defining a component by means of the "distance" between boundary points (1. 1. Pyatetskii-Shapiro [l1])·t The domain ~ is a symmetric Riemann space, so there is a Riemann distance in it that is invariant under the analytic automorphisms. We denote the distance between points Z E ~ and w E ~ by p(z, w). We will show that it is possible to introduce a "distance" between the points in the boundary of domain ~. Let a and b be two points in the boundary F, and let A and B be any of their neighborhoods in D. We set PA,B

=

inf p(z, w),

zeA()D weB()D

pea, b) = sup PA,B' A,B

f (2) j

Here, the upper bound of the numbers PA,B is taken over all possible pairs of neighborhoods A and B. We will call the quantity pea, b)(a, b EF)the distance between the points a and b in the boundary F of the domain ~, We are most interested in studying the boundary subvarieties on which the distance pea, b) is finite. It is important to note that they coincide with the components in the sense of the earlier definition. Definition 4. A set of points !iF c F that is located at a finite distance from a fixed point in F is called a component. As an example, consider an elementary domain-a polycylinder, i.e., the set ~ of all points, in an n-dimensional complex space, such that IZll < 1, ... ,Iznl < 1. As we have already seen, the point sets !iF v of the form (1, ... , 1, Zv+ b

... ,

zn),

Izv+ 11 < 1, ... , IZnl < 1

are typical boundary components for a polycylinder. We will prove

t The idea of introducing a "distance" between boundary points arose during a discussion of these problems with F. 1. Karpelevich (see F. 1. Karpelevich [1]).

88

THE GEOMETRY OF CLASSICAL DOMAINS

that these sets are components in the sense of the new definition, The distance p(z, w) between two points z, wED, as we know, is 1k, /( L" In -I+P) -Pk 2

p(z, w) = \

Pk = I 1z-w k _~ I. -ZkWk

where

k=i

(3)

Let a = (ab .. " a,,) and b = (b i , .. " bTl) be two points in the boundary of the domain~, We will prove that the distance between them is finite if and only if lakal = 1 implies that bka = aka' and that if this condition is satisfied for all k, the distance pea, b) may be computed with the formula

Jet,

pea, b)

where

'_ Pk -

Indeed, if pea, b) <

In'

~ ~ ~:).

. If

Iak I < 1,

,if

Ia I = I,

{

ak - bk I' l-ak I

l.

0

00,

(4)

k

then there exist two sequences of points in

~,

(zylI>, .. " Z~II»),

Z(11I)

=

W(III)

= (Wl11l ) , .. " W~11I»),

that respectively converge to a and b in the affine topology of a complex space and lim

pC Z(I1I), W(11I») <

00,

m .... oo

It follows from (3) that

lim sup p(Z~II~ Wk(III») <

00

111 .... 00

for all k and, consequently, either lakl < 1, Ibkl < 1, or ak = bk,t Conversely, if either lakl < 1, Ibkl < 1 or ak = bk, then, as we can easily see, there are two sequences m ) and wim ) of points on the disk

zi

t

It is easy to verify that if

.' I

Zk(m) -

Wk(m)

I

11m 1 - Zk(m) ih(m) < 1, then either I ak I < 1, I bk I < 1, or ak = bJ,;.

89

THE GEOMETRY OF HOMOGENEOUS DOMAINS

IZkl < 1 that converge to ak and bk in the topology of an affine complex space, and p(z£m), w~m») = p(ab bk ). Then, by (3), p2(a, b)

~ limp2(z(m>, W(III») =

f p2(zfll), Wf"») f p\a =

k= 1

k= 1

b

bk ). (5)

On the other hand, let z(m) = (Z~III\ ... , z~m») and w(m) = (w~/I), ... , w~m») be two sequences of points that respectively converge in the topology of an affine complex space to a = (al' ... , all) and b = (b l , ... , bl/)' then II

lim inf p2(z(m), w(m») /11->00

~

II

L lim inf p2(Z~III), wim») k=lm->oo

~

Lp2 (a k , bk ).

(6)

k=l

Expression (4) clearly follows from (5) and (6). Our statement is proved. Again let ~ be an arbitrary classical domain. We denote some geodesic by z(s). We will show in Sections 10, 11, and 12 that the topology of an affine complex space contains a limit belonging to F for z(s) as s -1- + 00. Let ZE~ and aEF. We agree to say that points z and a may be connected by a geodesic if there is a geodesic z(s) such that z(o) = z and z( + (0) = lim Z(8) = a. s->

+ 00

Generally speaking, two arbitrary points z E ~ and a E F cannot be connected by a geodesic, as follows from Theorem 7. We should note that any two interior points of ~ may always be connected by a unique geodesic. Theorem 7. Let ZE~. Any component!F contains exactly one point a that may be connected to z by a geodesic. The set ~a of points in the domain ~ that may be connected by geodesics to a given point a E!F, is analytically equivalent to some Siegel domain of genus 2. A general proof applicable to arbitrary symmetric domains could be constructed for Theorems 4, 5, and 6 at the present time. The author, however, found it desirable to preserve the original computation, for it may prove useful upon a first reading and for construction of examples. Sections 10 and 11 give proofs of Theorems 4, 5, and 6 for classical domains of the first, second, and third types.

90

THE GEOMETRY OF CLASSICAL DOMAINS

In the past (I. I. Pyatetskii-Shapiro [11 D, this theorem replaced the more powerful Theorem 6 in applications to the theory of automorphic functions. Let ff be some boundary component. We denote the set of all points in domain ~ that are connected by geodesics to a point a E!F by ~ It follows from Theorem 6 that domain ~ "fibers" into sections ~ a' a E ff. This fibering is not an analytic fibering in the usual sense of the word, because it does not have the local structure of a direct product. As we will show in Sections 10 and 11, this "fibering" coincides with the natural fibering in the corresponding canonical realization of ~ in the form of a Siegel domain. From now on we will use the following terminology: A component consisting of one point is said to be zero-dimensional. A component ff of an irreducible classical domain is said to be a component of genus one if the corresponding fibers ~ a are Siegel domains of genus 1. A component ff of an irreducible classical domain is said to be a component of genus 2 if the corresponding fibers ~ a are Siegel domains of genus 2. In Sections 10 and 11 we wi11list all components and the fibers ~ a corresponding to them for classical domains. It will become clear from this enumeration that a component of genus 1 is always zerodimensional for irreducible domains. The converse is not true. For example, all components of the balllz l 12 + IZ212 < 1 are zero-dimensional, but they are components of genus 2. In Sections 10 and 11 we will show that ~ is analytically equivalent to some bounded homogeneous (generally speaking, non symmetric) domain. The existence of nonsymmetric bounded homogeneous domains was first discovered in this way. We will use the following terminology for reducible domains. Q'

Q

~

Let where the

~1' .. " ~P

=

~ 1 X ...

x f'2 P'

(7)

are irreducible domains. A component ff=ff 1 x, .. xff p

(8)

is said to be a component of genus 1 if each factor ffil ;;;; k ;;;; p) is either a component of genus 1 or an "ideal" component, i,e., coincides with ~k'

THE GEOMETRY OF HOMOGENEOUS DOMAINS

91

Components (8) of genus 2 for domains of type (7) are similarly defined. The remaining components of the form (8) are called components of genus 3. The following important subgroups of the group G of all analytic automorphisms of the domain ~ may be associated with every component ff: G 1 (ff)-the set of all transformations of G that map ff into itself; GzCff)-the set of all transformations of G that leave every point of !F fixed; G3 (ff)-the set G of all transformations g of G that leave every point of ff fixed in the sense determined by the interior Riemann geometry of the domain~. This means that for any geodesic z(s) such that lim z(s) = a EF, s .... +00

the limit relationship lim p(z(s), gz(s)) = s .... +00

o.

holds. We agree to denote the maximal commutative normal subgroup of the group G3 (ff) by Giff). We agree to denote the centralizer of the group Giff) in the group G 1 (ff) by Gs(ff). It is clear that GV+ 1 (!F)(l ~ v ~ 3) is' a normal subgroup of the group GvCff). In the following sections we will show that the group G3 (ff) coincides with the group A of "parallel translations" that correspond to the canonical realization of the domain ~ in the form of a Siegel domain. Section 10. Classical Domains of the First Type

Classical domains of the first type are described in the following manner (Siegel [1]). Let p "?:q > be an integer. We will consider p x q matrices C as points in a pq-dimensional complex space. The domain that interests us, ~, consists of the matrices Z, such that

°

Eq-Z*Z > 0, where Eq denotes the identity matrix of order q. The group G of affine transformations of an m = p+q-dimensional

92

THE GEOMETRY OF CLASSICAL DOMAINS

complex space that preserve a Hermitian form with p minuses and q pluses is a group of analytic automorphisms of the domain in question. More accurately, there is a correspondence between each square matrix M of order 111 = p+q such that M*HM=H,

H=(

-Ep

0 ),

o

(1)

Eq

and an analytic automorphism

M=(~ ~)

Z ..... (AZ+B)(CZ+D)-"

of the domain. t The boundary F of the domain

~

Eq-Z*Z ~ 0,

(2)

consists of all Z such that

det(Eq-Z*Z)

= O.

(3)

We will begin by enumerating all the boundary components of the domain ~. We prove the following lemma as a preliminary. Lemma 1. Let cP1(t), ... , cPlI(t) be a/unction analytic on the disk It I < 8: M = sup (lcP1(t)12 + ... + IcP,lt)12). It I <e

IcP1(0)12+ ... +lcP,lO)12 ~ M,

Then

(4)

where the equality occurs if and only if all a/the/unctions cPk(t)(1 ~ k ~ n) are constant. Proof The following equations are a consequence of the Cauchy integral formula: cPk(O) =

~J.2l! cPk(pe

2n

1

11

thus

iO

)

de,

k = 1, ... , 11; P < 8;

0

k~1 IcPk(0)12 ~ 2n

I2l! 0

11

k~1 1cPk(pe

2

iO )I

de < M.

If we have the sign of equality here, then

l¢k(OW

= 2~

f:'

l¢k(pe"W d8, k

= 1, """' n,

(5)

and, consequently, cPk(t) = const, k = I, ... , n. The lemma is proved.

-r It can be shown that all analytic automorphisms are of the form (2) if p =f- q. If, however, p = q, we also have the automorphism Z --+ ZI (Klingen [1]).

THE GEOMETRY OF HOMOGENEOUS DOMAINS

93

The following theorem contains a description of all components of the domain ~. Theorem 1. The set of points in F of the form

o )r Z r

,z*z < E q - n

1~ r

~ q,

(6)

p-r

q-r

forms a component $71' that is analytically equivalent to a classical domain of the first type with parameters p - rand q - r. Any boundary component may be mapped by some analytic automorphism of the domain ~ into the component $7", 1 ~ r ~ q. Every point F belongs to some component. Proof It is easy to verify directly that !FI' is a regular analytic subset of points of the boundary F of the domain ~. As a result, in order to prove that !F r is a component, it is sufficient to prove that any analytic curve contained in F and intersecting !FI' is entirely contained in !Fr. Let Z =Z(t), 1tl < e be some analytic curve, Z(O) E!F" Z(t) E F for all t, It I < e. We represent Z in the form

Z12 Z22 r

The inclusion Z

c

)r p-r.

q-r

Fimplies that Eq-Z*Z

~

0, whence, in particular,

Er-zt1 Z11 -Z~\ Z21

~ O.

(7)

It follows from (7) that if Z(t) E F, then p

r

L L \Zij(t)\2 ~ r, It I < e.

i= 1 j= 1

Furthermore, Z(O) E!FI' implies that P

I'

L L1 IZij(0)12 = i= 1

1'.

j=

As a result, by the lemma we proved above, the Z ij(t)(l ~ j ~ r, 1 ~ i ~ p) are constant, and, therefore, Z11(t) == En Z21(t) == O. In order to prove that Z 12(t) == 0, it is sufficient to note the equivalence of inequalities Eq-Z*Z ~ 0 and Ep-ZZ* ~ 0 (Siegel [1], p. 135). Using the same arguments as above, it is easy to show that Z12(t) == 0

94

THE GEOMETRY OF CLASSICAL DOMAINS

by means of the inequality Ep - ZZ* ~ O. It remains to show that ZIit)Z22(t) < Eq_r for all t, It I < 8. The inequality ZI2(t)Z22(t) ~ Eq_r follows from the fact that Z(t) EF. We must show that det(Eq_r -ZI2(t)Z22(t) =1= 0 for all t, Itf < 8. If this is not so, then there exists a to, Itol < 8, such that Eq-r-Z1'2(tO)Z22(tO) ~O,

Then there is a column vector

~o,

det(Eq_r-Zi2(to)Z2ito)) = O.

such that

~~(Eq-r - ZI2(tO)Z2ito»~0 = O.

We set ~(t) = Z(t)~o. It is clear that for all t, It/ <

8,

~*(t)~(t) ~ ~6' ~o

(8)

~*(O)~(O)

(9)

and that

<

~; ~o.

It follows from the lemma proved above that if equality occurs for at least one point in (8), then the vector function ~(t) is constant. This contradicts (9), which shows that

ZI2(t)Z22(t) < Eq - r. Thus, ff'r is a boundary component. ff'r is mapped onto a classical domain in the obvious manner, Er (

0)

o Z

-+Z.

(10)

We will now prove that any component may be mapped into a component ff'n 1 ~ r ~ q by an analytic automorphism of the domain. Let Zo E F. As we know, there are unitary matrices Uland U 2, such that 0 0 Xl

U 1 Z0 U2

= (E,0

;) x=

0

X2

0

0

0

0

0

0

0

0 J

0

..

Xq -

r

(11)

THE GEOMETRY OF HOMOGENEOUS DOMAINS

95

The point (11) clearly belongs to the component fFr and, therefore, the initial point belongs to the component cp(fFr), where cp is an analytic mapping of ~ onto itself: Z

-+

ViI ZVl1.

We have thus proved that any component may be mapped into a component fFr by means of an analytic transformation of domain ~ that leaves a given interior point fixed. At the same time, we have proved that every boundary point is contained in some component. Thus, the Theorem 4 (Section 9) is completely proved for domains of the first type. We will now give a criterion for membership of two points Zl and Z2 in one component. We set (12)

It is easy to verify that any analytic automorphism Z -+ Z of the domain ~ transforms the matrix W 12 according to the following rule:

(13)

where Q(Z) is a non degenerate q x q matrix that depends on Z and on the analytic automorphism Z -+ Z. We denote the rank of the matrix W 12 by r(Z1,Z2)' It is clear from (13) that r(Zl,Z2) is an invariant of the pair of points Zl and Z2 under the analytic automorphisms of the domain ~. The following lemma presents a criterion in terms of r(Zl,Z2) for membership of the points Zl and Z2 in one component. Lemma 2. Two points Zl' Z2 EF are contained in one component if and only

if (14)

Proof Necessity. It is easy to verify that every point of F may be mapped onto a point of the form

(~ r

o)r l~r~q o p-r'

(15)

q-l'

by an analytic automorphism of the

domain~.

Let Zl and Z2 be

96

THE GEOMETRY OF CLASSICAL DOMAINS

contained in one component. Without loss of generality, we may assume that Z1 is of the form (15). Then Z2 must have the form

(

EOr

zO),

Z*Z < E q _,.•

Direct computation shows that

W= W ll

12

=

(0 0);

° Eq-r

W 22

=( 0

°

° )

Eq_,.-Z*Z

in this case. It immediately follows from this that (14) is satisfied. Sufficiency. Let (14) be satisfied for Z1' Z2 EF. Without loss of generality, we may again assume that Z1 is a point of the form (15) We write Z2 in. the form

q-r

r

It follows from the relationship r(Z1,Z1) = r(Z1,Z2) that the ranks of the matrices fV ll

=

(0 0), °

(Er-Z11 . W 12 =

°

Eq-r

are equal and, therefore, Zl1 = E,.. Furthermore, Z2 E F implies that

E -Z*Z = ( q

-Z* Z 21 21 * -Z22 * Z21 -Z12

whence Z21 = 0, Z12 = 0, Eq_,.-Z'i2Z22 ~ 0. It follows from the relationship r(Z1,Z1) = 1'(Z1,Z2) that the ranks of matrices

W

ll

GE~J'

W

22

(~ Eq-,-~f2ZJ

are equal and, therefore, Eq-r > Z'i2Z22' The lemma is completely proved. We will now turn to 'the proof of Theorem 5. We must construct canonical realizations in the form of Siegel domains for domains~.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

97

According to the theorem stated in Section 5, there must be as many realizations as there are typical components (in our case, q). We will first describe a compact complex symmetric manifold D that has the same relationship to ~ as the Riemann sphere has to the unit disk. The manifold D is called the dual manifold of the manifold ~. We denote the set of all complex m x q (m = p +q) matrices U of maximal rank by Q. If UEQ, then UREQ, where R is a nondegenerate square matrix of order q. We agree to say that the matrices U and UR are equivalent. The set of all equivalent matrices forms a class. The set of classes admits a natural complex structure. It is not difficult to verify that it is the compact symmetric manifold D. We will now prove that ~ may be realized in the form of a domain in D. Let H be the Hermitian matrix defined by relationship (1). We will consider the set Q H of matrices U such that U*HU > O.

(16)

It is easy to verify that if U E QH, then UR E QH' where R is any nondegenerate q x q matrix. Thus, every class of equivalent matrices is either entirely contained in QH' or its intersection with Q H is empty. The set of classes contained in Q H is clearly a domain in D. We set

u=(~:):.

(17)

q

Condition (16) may be written in the form U*HU= -U[U 1 +U1U 2 >0.

(18)

As a result, if U E QH, then the matrix U 2 is nondegenerate. Thus, it is possible to select a unique representative of the form

GJ for every class contained in QH' Condition (16) implies that Eq-Z*Z> O.

We have proved that ~ may be realized in the form of a domain in D. Note that the classes belonging to the boundary F of the domain D

98

THE GEOMETRY OF CLASSICAL DOMAINS

!!2 consist of the U for which the determinant of the matrix U* HU is equal to zero and U*HU ~ O. In other words, F consists of those Z satisfying the following conditions: (1) the determinant of the matrix Eq-Z*Z is equal to zero, and (2) Eq-Z*Z ~ O. We will now turn to describing the canonical realizations in the form of Siegel domains. A general method for finding the canonical realizations consists in the following. Let H be an arbitrary Hermitian matrix of order 111 with p positive and q negative characteristic roots. We denote the class of those 111 x q rectangular matrices U such that U* HU > 0 by QH' We agree to say that two matrices U and UR, where R is a nondegenerate q x q matrix, belong to the same class. We will now prove that the set DH of classes belonging to Q H is a domain in D (the dual manifold) that is analytically equivalent to our domain ~. Indeed, there is a non degenerate matrix M of order 111 such that

M*HM = Ho.

Here Ho indicates the Hermitian matrix defined in (1). The mapping maps Q H into Q Ho ' It induces an analytic automorphism of D that maps DH into DHo' In order to realize DII in an affine complex space, it is sufficient to find a method for setting up a correspondence between each class of matrices U E Q H and a point in the affine complex space. The analytic automorphisms of DH may be described in the following manner. Let G be the set of all 111 x m matrices A such that A *HA = H. Every matrix A corresponds to an analytic automorphism U -+ A U of the manifold Q. It is easy to see that an analytic automorphism of D that maps DH into itself corresponds to it. The boundary of DH in D clearly consists of those classes of matrices U such that (1) det(U* HU) = 0,

(2) U* HU ~ O.

The invariant r(Z1' Z2) introduced above for a pair of points (see Lemma 2), is the same as the rank of the matrix ut HU2 • First of all, in fact, the rank of the matrix ut HU2 is independent of the choice of

THE GEOMETRY OF HOMOGENEOUS DOMAINS

99

matrices U 1 and U 2 and depends only on the classes to which they belong; furthermore, it is clear that we have W 12 = V; HU 2

for appropriately chosen U l and U2 • The last conclusion may be stated in the form of the following lemma, which extends Lemma 2. Lemma 3. We denote the rank of the matrix ut HU2 by r(Ul , U2 ). The matrices U 1 and U2 are mapped onto a point in one boundary component of the domain DH by the mapping n -+ D if and only if (19)

We will now write the canonical realization Sq corresponding to a zero-dimensional component of the boundary of the domain~. Consider a matrix H of the form

o ,

PI

= p-q.

o As we can easily verify, p characteristic roots of this matrix are equal to - 1, and the remaining q are equal to 1. We partition U in the following manner:

V=

VI

q

V2

Pl.

U3

q

Condition (16) may be written in the form

W

= V*HV = i(V; v 3 -vj Vl)-V; V 2 > o.

(20)

We will show that if U E nH then the matrix U 3 is nondegenerate. Indeed, otherwise there would be a nonzero vector b such that U 3 b = O. Then b* uj = 0 and, therefore, b *W b = i( b *V t V 3 b - b *V 3 V 1 b) - b *V; V 2 b = - b *V; V 2 b ~ O. We have been led to a contradiction that proves that the matrix U 3

100

THE GEOMETRY OF CLASSICAL DOMAINS

is nondegenerate for U E QH' It follows from this that every class of equivalent matrices V E Q H contains a unique matrix of the form

Substituting a matrix V of this form into (20), we obtain the inequality 1:- ( U 1 -U *1 ) - U'~i U 2> 0 .

(21)

l

This inequality defines some unbounded domain in pq-dimensional complex space (whose coordinates are the entries in the matrices Vb k = 1,2). The domain constructed is the Siegel domain of genus 2 that was described in Chapter 1, Section 2. "V-,Te must now prove that the transformations preserving an "infinitely distant" zero-dimensional component are linear. It follows from Lemma 3 that membership of a matrix V in. some zero-dimensional component is a consequence of the equations U*HU

= i(Ui

u 3 -UiV 1 )

V;V2

= O.

U sing this fact, we can easily verify that the class of matrices V that contains a matrix of the form

(22)

is carried into a zero-dimensional component under the mapping Q -+ D. We naturally assume it to be "infinitely distant". We will now find the automorphisms of the domain that leave point (22) fixed. The matrices A corresponding to such automorphisms satisfy the condition

E

A 0

(23)

o where R is some nondegenerate q x q matrix that depends, generally speaking, on A.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

We partition A into blocks

A=

[AU

A12

A2l

A22

A13 A 23

A31

A32

A33

PI

q

q

101

r Pl'

q

and write (23) in the explicit form

Au

A12

A13

E

All

A21

A22

A 23

0

A21

A3l

A32

A33

0

A3l

nR.

(24)

whence A21 = 0, and A31 = O. Furthermore, we can easily show that A 32 = 0 by using the fact that A* HA = H. A transformation with such a matrix A is linear. Indeed,

A12

A13

Au U I +A12 U 2 +A13

A22

A 23

A22 U 2 +A 23

o

A 23

E

A33

(25)

E It is easy to verify (see Chapter 1, Section 2) that transformations of the form (25) form the full group of affine transformations of Siegel domain (21). We now turn to describing the remaining canonical realizations. Consider a matrix H of the form

o

0

o

o o

, Pl=p-r,

Ql=q-r.

(26)

102

THE GEOMETRY OF CLASSICAL DOMAINS

It is easy to see that p characteristic roots of the matrix H are equal to - 1, and that the remaining q are equal to + 1. We partition U into blocks in the following manner:

u 11 U12 u=

r

U 21

U 22

P1

U 31

U 32

q1

U41

U 42

r

r q1 We now write condition (16) in the form

"I I I where (27)

; (U12 U 12 - U;2 U 42) + U: 2 U 32 - Ut2 U 22' I

j

We will prove that if U E OH then the matrix U 31 U32) (28) ( U 41 U 42 is nondegenerate. Assume that this is not true for some U satisfying (27). We multiply U by a nondegenerate square matrix Q such that for = UQ the entries of the last column in the matrix

a

are all zero. Let e denote a ql-dimensional vector whose coordinates are all equal to zero, except for the last, which is nonzero. It is easy to see that a32 e = 0, a42 e = O. As a result, e* W 22 e = - e* Ut2 U 22 e ;£ O.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

103

We have obtained a contradiction. Thus, matrix (28) is always nondegenerate. As a result, any U satisfying (27) may be normalized so thatt (29) We write formulas (27) in the following manner:

W

(30)

* W 12 = W 21 W 22

*

1 12 - U 21 =-;-U I

U 22 ,

= E q1 -Ut2 U 22 ·

The relations (30) define an unbounded domain Sr, r = Q-Ql' in a pQdimensional complex space whose coordinates are the entries of the matrices Uu, 1 ~ i,j ~ 2. We will now prove that Sr is the Siegel domain of genus 3 for which the component fFr serves as a base. We first prove that (31)

if and only if (32) Indeed, it follows from (31) that Q*WQ > 0

t Normalization is the selection of a unique representative U for each class of equivalent matrices in QH.

104

THE GEOMETRY OF CLASSICAL DOMAINS

for any nondegenerate matrix Q. We assume, in particular, that

Q= (

E

- w2l

fV 21

r

o ) > o.

Then

(33)

W22

It is clear that inequalities (31) and (33) are equivalent and, therefore,

so are (32) and (33). We may therefore write relation (30) thus:

1

-:-(U 11 - U 11 ) - U 2l U 21 1 * * I

-CiU 12 +uf, U22 lWzlC:U 12 + Uf, U22l* >0, W 22 = E q1 - U 22 U 22 >

j

(34)

o.

After parentheses are removed, the first of these inequalities takes the form

-iU 12 Will U't2 U 21 +iU't1 U 22

w2l

UI2 > O.

(3,5)

In order to make it clear that the domain obtained is a Siegel domain of genus 3, we will introduce some new notation that is characteristic of Siegel domains. We set t = U 22 ,

Z

=

2U ll ,

U

=

(U 12 , U 2l ),

v = (V12' V2l ).

It follows from (30), as we should expect, that t may vary over a classical domain of the first type with parameters PI' q1. We define an operator

for separating the real part in the space of z: (36)

We take the cone of all Hermitian positive definite matrices of order r for the cone V.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

105

Finally, we set

w;l Vt2 + viI U 22 w;l ui2 U 21 +1-i(U 12 w;l V 21 + V 12 W;/ U 2*2 U 21 ).

LtCu, v) = viI U 21 + U 12

(37)

We can directly verify that our domain is the Siegel domain corresponding to the cone Vand the function Llu, v). It is easy to show that the following identity is valid: 1 (Epi - U 22 ui2)-1 = Eqi + U 22 U;2 Ui2.

This identity may be used to represent LtCu, v) in the form

L/u, v)

= V!l(E pi - U 22 U 2*2) -lU 21 + U 12 W;;l Vt2

+1-i(U 12

w;l ui2 V 21 + V 12 w;l ui2 U 21 )

The boundary F of this domain consists of those matrices U such that det U*HU = 0,

U*HU ~ 0.

(38)

Consider the matrices U E F of the following form:

z*z < E.

U=

(39)

It is easy to verify that a class of equivalent matrices may contain no more than one matrix of the form (39). Furthermore, if U and are defined by (39), then

a

° )

W=U*H(J= ( 0 *_ Eqi -Z Z

°

'

whence it is clear (see Lemma 3) that U is a component analytically equivalent to the component !Fr. This component is naturally treated as "infinitely distant". We will prove that the analytic automorphisms of our domain that map component (39) onto itself are quasilinear transformations

106

THE GEOMETRY OF CLASSICAL DOMAINS

(defined in Chapter 1, Section 3), while the analytic automorphisms leaving the point

l

Er

0

o

0 I

(40)

fixed are linear transformations. In fact, let A be a matrix corresponding to an analytic automorphism leaving component (39) fixed. Then for any U of the form (39) there is a of the same form and a nondegenerate q x q matrix B such that

a

AU=

aBo

(41)

We partition A and B into blocks in the following manner:

A=

All

A12

A13

Al4

r

A21

A22

A 23

A24

PI

A31

A32

A33

A34

ql

A41

A42

A43

A44

r

r

PI

ql

r

E12 )' B22 ql ql

B= (Ell B21

r

Formula (41) may be written in the form

[All A21 A31

l A41

A12Z+A13

I= A 32 Z+A" J A22Z+A23

A42Z+A43

whence

Since

Z is

arbitrary, we find that

B12

1

ZB 21

ZB22

I

E2l 0

B22

1 r Ell

l

0

r

THE GEOMETRY OF HOMOGENEOUS DOMAINS

Thus,

A-

I

All

A12

A13

A141

o

A22

A 23

A24

0

A32

A33

A34

.

I'

A*HA =H.

107

(42)

l

0 0 0 A44 J. It is easy to verify that transformation of the following type corresponds to each A of the form (42)

E 0

E 0

o o o

Z

0

Z

E

o

E

0

0

0

thus Z = (A22Z+A23)(A32Z+A33)-1. (43) It immediately follows from (43) that a matrix A of the form (42) corresponds to an analytic automorphism of the domain that leaves the point (40) fixed if and only if (44) A 23 = 0, A32 = O. We will now prove the converse, namely, that a matrix A of the form (42) and subject to supplementary conditions (44) corresponds to every automorphism of our domain that leaves the point (40) fixed. In order to do this, it is sufficient to prove that an analytic automorphism of our domain that leaves (40) fixed leaves every "infinitely distant" component fixed. This last immediately follows from the fact that (1) a component must be mapped into a component under an analytic automorphism, and (2) every boundary point, according to Theorem 4, is contained in a unique component. We will prove that the analytic automorphisms of our domain that leave point (40) fixed are linear transformations. The set of all matrices A of the form (42) and subject to supplementary conditions (44) is generated by two of its subgroups. The first of these consists of the matrices A of the form

o E o 0 o 0

(45)

o

E

lOS

THE GEOMETRY OF CLASSICAL DOMAINS

where (46)

i (At4 - A 14 ) = A!4A 24 - Ai4A 34' The second of these subgroups consists of the matrices A of the form

where

All

0

0

0

0

A22

0

0

0

0

A33

0

0

0

0

A44

AllA!4 = E,

A!2 A 22

E,

(47)

(4S)

Ai3A33 = E.

As we can easily show, a transformation of the form

~11 -+ U 11 +A12 U 21 +A 14 -

U 12 A 34 -A 12 U 22 A 34 -A13 A 34'

1

021-+ U2l+A24-U22A34, U 12

-+

U 12 +A 12 U 22 +A 13 ,

U 22

-+

U 22 ·

(49)

J

corresponds to each matrix A of the form (15). Expression (46) may be used to verify that the transformations obtained are "parallel translations" in the sense of Section 3 of Chapter 1. A transformation U 11

-+

All U 11 At1'

U 12 -+ All U 12 Ai3' U 21

-+

A22 U 21 A1~'

U 22

-+

A22 U 22 Ai3'

(50)

corresponds to each matrix of the form (47). It follows from (49) and (50) that a linear transformation of our domain corresponds to each matrix A of the form (42) that satisfies supplementary conditions (44). A quasilinear transformation of our domain corresponds to each matrix of the form (42). It is not difficult to see that in order to prove

THE GEOMETRY OF HOMOGENEOUS DOMAINS

109

this, it is sufficient to show that a quasilinear transformation of our domain corresponds to a matrix A of the form

E

0

0

0

0

A22

A 23

0

0

A32

A33

0

0

0

0

E

A=

(51)

Direct computation shows that the following transformation of our domain corresponds to a matrix A of the form (51) : V ll

---+

U ll -U 12 (A 32 U22+A33)-lA32 U 2U

U 12 ---+ U 12(A 32 U 22 + A 33 )-1, U 21

---+

(A 23 U 22 + A 22 )-1 U 21,

U22

---+

(A22 U 22 + A 23 )(A 32 U 22 + A 33 )-1.

(52)

Thus, every transformation of the domain S that maps an "infinitely distant" boundary component into itself is a quasilinear transfor~ mation. Proof of Theorem 5 for classical domains of the first type. Let D be a classical domain of the first type with parameters p and q, p ;;; q. As we have already seen, there are exactly q typical boundary components. We have constructed exactly the same number of Siegel domains having the typical boundary components as bases. We will prove that for any boundary point of domain ~ there is an analytic mapping of ~ that satisfies the requirements of the theorem and maps ~ onto one of the Siegel domains that we have constructed. Let a given point belong to a component analytically equivalent to the component ff r • It is clear that there is a mapping of ~ onto Sr under which the given point is mapped onto a point of the form (40). As we have already shown, the transformations that leave this point fixed are linear transformations of domain Sr and the transformations leaving the component of this point fixed are quasilinear transfor.. mations of the domain Sr. Theorem 5 is therefore completely proved for domains of the first type. In Chapters 3 and 4, which are devoted to the theory of automorphic

110

THE GEOMETRY OF CLASSICAL DOMAINS

functions, we will need a criterion for "convergence" of a sequence of points to a point in an infinitely distant component. We will now give a general definition of "convergence". Let S be some Siegel domain with base ff. As usual, we denote the points of S by w = (z, u, t), and we denote the points of ff by t. Let V and LtCu, v) have the same meaning as in Section 3 of Chapter 1. Definition 1. Let Q be some domain in ff, and let r be a vector in V. We will say that the set of all w (z, u, t), such that Imz-ReLtCu, U)-ZE V,

to E Q

(53)

is the cylindrical domain SeQ, r) in S. Definition 2. Let W, W2' ... be a sequence of points in S. We agree to say that lim Wy = to,

tEff,

(54)

v .... 0Cl

if for any cylindrical domain SeQ, r) (where to E Q) there is a that for v > Vo

WyES(Q,r).

Vo

such (55)

Let U 1 EQH' U 2 EQH; we set B(U 1,U2) = U~HU1(UiHU1)-1U:HU2' }

W(U 1, U 2) = U~ HU 2.

(56)

It is not difficult to see that (1) the matrix B(U1 , U2 ) depends not on U 1 but only on the class to which U 1 belongs; (2) when U2 is replaced by any other matrix in the same equivalence class, B(U1 , U2 ) and W(U1 , U2 ) are transformed in the same way: B

-+

R*BR,

W

-+

R*WR,

(57)

where R is a nondegenerate square matrix depending on U2 and on the given analytic automorphism; (3) the pair of matrices B(U1 , U2 ) and W( Uu U2 ) is the joint invariant (with respect to analytic automorphisms of the domain D H ) that corresponds to a pair of points in DH , i.e., they are transformed according to formula (57) under the analytic automorphism. The following lemma provides a criterion for convergence to some point in an "infinitely distant" component.

111

THE GEOMETRY OF HOMOGENEOUS DOMAINS

Lemma 4. Let Uo be some matrix mapped onto a point in an "infinitely distant" component under the mapping n -+ D. A sequence of points in our domain converges to the given point in the component if and only if

(58)

lim B(U n , U o) = W(U o, Uo),

n .... oo

where the Un En are arbitrary preimages of the given sequence of points. Proof It is clear that we may set

Vo

=

E

0

0

0

0

E

0

0

u(n) 11

0

0

U(II)

0

E

E

0

=

Vn

22

without loss of generality. Direct computation shows that

_(i(u~ni-U'!t»)-1 B(Um U o)-

o

W(U o• U o) =

0 (E-

)

utt) u~ni)-1

G~).

It is clear from the definition of "convergence" that the sequence UII converges if and only if (u~ni - U'!t)-1 -+ 0, u~ni -+ 0, which is equivalent to (58). Lemma 4 then follows. Without proof we will now give formulas for the Riemann distance and for geodesics. We will assume that domain ~ is realized as described at the beginning of this section. It can be shown that the Riemann metric, which is invariant with respect to analytic automorphisms, is given by the formula (Klingen [1]) ds 2 = a((Eq-Z*Z)-1 dZ*(E p -ZZ*)-1 dZ);

a(A) is the trace of the matrix A. The distance between two points Z1 and Z2 is given by the following expression: 2

p (Z1,Z2)

1 ~

= 4- k=f...;1 In

2

1 + rt

- 1t' -r k

rk

Al - 1

= --" Ak

where A1 , ... , Aq are the characteristic roots of the matrix R(Z1,Z2) = (Eq-Z,! Z1)-1(Eq-Z,! Z2)(Eq-Zt Z2)-1(Eq-Zt Z1)'

112

THE GEOMETRY OF CLASSICAL DOMAINS

Any geodesic in !!2 is analytically equivalent to a geodesic of the form (Hua Loo Keng [1]) th Ct 1 8

0

0

0

th Ct 2 8

0

0

0

th Ct q 8

0

0

0

(59)

Z(8) =

where Cti + '" + Ct: = 1, s is the arc length. Using (3), we can easily verify that a limit (that is a point in F) exists for any geodesic Z(s) when s-+

+00.

Let Zl E!!2 and Z2 E F. We agree to say that points Zl and Z2 may be connected by a geodesic if there is a geodesic Z(s) such that

Z(O) = Zl

lim Z(8) = Z.

and

s .....

+ CIJ

Generally speaking, two arbitrary points Zl and Z2 cannot be connected by a geodesic. Every point Zl E!!2 may, however, be connected by a geodesic with any component; this assertion is a consequence of the following lemma. Lenmw 5. Two points Z 1 E!!2 and Z 2 E F may be connected by a geodesic if and only if all of the characteristic roots of the matrix R12 =

w1l w 11 w2l

fV 22

(60)

[Wij is defined by (12)] are equal to zero or one. In particular, let Zl E!!2 and let!F be any component. There is a unique point Z2 E!F that lnay be connected by a geodesic to Zl' Proof. Assume that the points Zl E!!2 and Z2 E F may be connected by a geodesic Z(s), Z(O) = Zl' Without loss of generality,t we may assume that Zl = 0 and Z(s) has the form

(P~S)

~}

P(8)

=

th Ct 1 8

0

0

0

th Ct 2 8

0

0

th Ct r S

0 Ct 1 ~ Ct2 ~ ... ~ Ct r

(61)

> O.

t The characteristic roots of the matrix R, as (13) implies, are invariant under the analytic automorphisms of the domain PJ.

113

THE GEOMETRY OF HOMOGENEOUS DOMAINS

It is easy to verify that R(Z1) Z(s))

=

(

E- 0p2(S) 0E)'

and, therefore,

We have shown that if two points Z1 and Z2 may be connected by geodesics, the characteristic roots of the matrix R2 are equal to zero or one. We will now prove the converse, i.e., that a sufficient condition for points to be connected by geodesics is that the characteristic roots of the matrix R12 be equal to zero or one. Without loss in generality, we may assume that 0

0

°

X2

0

0

0

Xr

0

0

0

X1

Zl=(::

::}

Z2 =

Xk

~

o.

Direct computation shows that the characteristic roots of the matrix R are equal to 1-xi, ... , l-x;, i.e., Xk is equal to zero or one. It is not difficult to use (59) to show that the required geodesic exists. It remains to prove the second assertion of Lemma 5. Without loss of generality, we may assume that Z1 = 0 and ff has the form shown in (10). It is not difficult to show, if we use the criterion we have proved, that Z E ff may be connected by a geodesic to Z 1 = 0 if and only if

We denote the set of points in the domain ~ that may be connected with a given point Z E~ by ~z(Z Eff). It follows from Lemma 5 that ~ "fibers" into fibers ~z. It is not difficult to verify that this fibering is the same as the natural fibering of Siegel domains (34). In other words, a fiber consists of points in domain (34) with fixed U22 . It is clear from (52) that all fibers are analytically equivalent. As a

114

THE GEOMETRY OF CLASSICAL DOMAINS

result, it is sufficient to describe the fiber ~o corresponding to anyone value of U22 • For simplicity, we assume that U22 = 0. It follows from (34) that ~o is given by the inequalities

1 (U 11-:l

U·~i1 ) -U 21 * U 21 - U~· 12 U i'2>0,

whence it is clear that ~o is a Siegel domain of genus 2. The group G3 (.%) (see Section 5) is the same as the group A of "parallel translations" of the domain S. Let Z(s) ( - 00 < s < + (0) be a geodesic entering the component .%. As we know, Z(s) = g(s)Zo, where g(s) ( - 00 < S < + (0) is some oneparameter subgroup of analytic automorphism of the domain ~. Let a = Z( + (0) E.%. It is clear that g(s)a = a and, therefore, g(s) E G1(.%). We will prove that g(s) E G2 (.%). Consider the homomorphism of G1 (.%) onto G'(.%) = G1 (.%)JGzC.%). It maps g(s) into the compact group g'(s), since g(s) (-00 <s < +(0) leaves the point a fixed. The characteristic roots of the matrix of g(s) in representation (42) are real. As a result, g'(s) == F. Furthermore, let g E G3 (.%); then

limp(Z(s),g(Z(s))) = 0. for any geodesic Z(s), (Z( + (0) E .%). Assuming that Z(s) = g(s)(Zo), we find that lim p(g(s)Zo, g(g(s)Zo)) = 0, s-+

whence

+ ex:>

lim g-l(S)g(g(s)) = E. s-+

+ ex:>

It is not difficult to show by means of direct computations that the set of matrices defined by the last requirement coincides with the set of matrices A of the form (45). The assertion in question follows from (45) and (49).

Section 11. Classical Domains of the Second and Third Types In this section the results of Section 10 for domains of the first type are extended to classical domains of the second and third types. As we know, domains of the second type are described in the following manner.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

115

Let p > O. We will consider p xp skew symmetric matrices Z as points in a tp(p-l)-dimensional complex space. The set of Z such that (1)

forms a bounded domain {0. The analytic automorphisms of domain {0 are described in the following manner. Consider the set G of all square matrices M of order 111. = 2p that satisfy the conditions M*HM = H,

where

H

= (

M'KM = K,

-0E pO) Ep

K = (0 ,

Ep

EOp ).

(2)

(3)

An analytic automorphism of the domain {0 Z

-7

(AZ+B)(CZ+D)-1

corresponds to each square matrix l1f

M=G :} The boundary F of this domain clearly consists of all Z such that

det(E-Z*Z)

= 0,

E-Z*Z ~ O.

(4)

We will begin by enumerating all boundary components of the domain {0. Theorem 1. The set ofpoints oftheform

o ) 2r Z 2r

, Z*Z < E p -

2n

Z' = -Z,

p-2r

p-2r

in boundary F forms a component :F,. analytically equivalent to a classical domain of the second type with parameter P1 = P - 2,.. Any boundary component may be mapped into a component :Fr(1 ~ r ~ tp) by som.e analytic automorphism of the domain {0. Every point in F belongs to some component. The proof of this theorem proceeds in exactly the same manner as the proof of the corresponding theorem in Section 10. As in Section 6, we assume that (6)

116

THE GEOMETRY OF CLASSICAL DOMAINS

It is easy to verify that the rank r(Z1' Z2) of the matrix W 12 is an invariant of the pair of points (ZbZ2) under the analytic automor-

phisms of the domain !:0. We will now state a criterion for membership of two points in one component. Lemma 1. Two points Z 1, Z 2 E F belong to one component if and only if (7)

The proof of this lemma is exactly the same as the proof of the corresponding lemma, Lemma 2, in Section 6. We now turn to describing the canonical realizations of the upper halfplane type for the domain!:0. Let n denote the set of all complex rectangular 2p x p matrices U of maximum rank that are such that U' KU = 0, where K is a non degenerate symmetric matrix. If U En, then UR En, where R is any square non degenerate p x p matrix. We agree to say that the two matrices U and UR are equivalent. The set of all equivalent matrices forms a class. The set of classes is a compact complex symmetric manifold D. The dual manifolds D (defined in Section 6), of the domain!:0 which correspond to different K, are analytically equivalent. As a result, Kmay be chosen in any convenient manner. We will assume that K is of the form (3). Let H be the Hermitian matrix defined in (3). Consider the set nH of matrices U En, such that U*HU>

o.

(8)

It is easy to verify that every class of equivalent matrices is either

entirely contained in nH , or its intersection with nH is empty. The set of classes contained in nH is a domain in D. We set U=(U 1 )" p. U2 P

(9)

P It is easy to use the techniques we used in Section 10 to prove that !:0 may also be realized in the form of a domain in D. It is not difficult to see that the boundary F of domain !:0 consists of those U En, for which the determinant of the matrix U* HU is equal to zero and U* HU ~ O. In other words, F consists of those Z satisfying

THE GEOMETRY OF HOMOGENEOUS DOMAINS

117

the following conditions: (1) the determinant of the matrix Ep-Z*Z is equal to zero, (2) Ep-Z*Z ~ 0, (3) Z' = -Z. We will now give a general method for finding the canonical realizations. Let Hbe a Hermitian matrix of order 2p withp positive roots and p negative characteristic roots, and let K be a nondegenerate 2p x 2p symmetric matrix, where HK- 1 HK- 1 = -E.

(10)

Consider the set OR of 2p x p matrices U such that H*HU > 0,

U'KU = 0.

(11)

We agree to say that the matrices U and UR, where R is a p x p nondegenerate matrix, belong to the same class of OIl' The set DR of classes belonging to Q ii is a domain in D (see Klingen [1]). That is analytically equivalent to our domain £0. The analytic automorphisms of this domain may be described in the following manner. Let G be the set of all 2p x 2p matrices A such that A* HJ1 = Hand A'KA = K. With each A E G we associate the analytic automorphism U -7 A U in O. It is easy to see that an analytic automorphism in the set of classes corresponds to it. The boundary of the domain £0 consists of the classes such that

U'KU = 0,

U*HU ~ 0,

det(U*HU) = 0.

First we will describe the realization S[tp] corresponding to the zerodimensional components of the boundary of the domain £0. We will separately discuss the two cases depending on the parity of p. Case I. p even. We set

H=C~£p i:p). K=(~J J

= (_0£,

:'). where s =!P.

Jr

(12)

j

It is easy to see that (10) is satisfied when Hand J( are chosen in this way· We partition Uinto blocks in the following manner:

u=(U )p. 1

U2 P P

118

THE GEOMETRY OF CLASSICAL DOMAINS

Conditions (11) imply that

~(UI u1-ui U 2) > 0, U~JU2 = U~JU1'

(13)

1

As we can easily verify, U2 is nondegenerate, and, consequently, there is a unique representative of the form

(~) in each class. It follows from (13) that

~(Z-Z*) > .

0, JZ = Z'J ,

l

i.e., S[tP] is a Siegel domain of genus 1. Case II. p odd: p = 2s+ 1. We set

H=

o o

o

-1

o

0

K=

010

o

0

0

o o -J s

0

1

o .

1 0

0

0 0

0

(14)

We partition U into blocks:

u=

V 11

U 12

2s

U 21

U 22

1

U 31

U 32

U41

U 42

2s-

1

2s

It is easy to prove (see the analogous situation in Section 10) that the matrix (

U31 U41

U32) U42

is nondegenerate. As a result, we can normalize U with the conditions

119

THE GEOMETRY OF HOMOGENEOUS DOMAINS

Relationships (11) imply that

U'11J = JU 11 , W = (W11 W 21

u*21 U 21,

*) Jt 11 = -:-1 ( U 11 - .U 11 T

l

W12 = Wi\

= -iU 12 ,

W 22 = l.

We have been led to the following domain in a -!-p(p-I)-dimensional complex space (whose coordinates are the entries of the matrix U 12 and are independent of the entries of the matrix U 11 ):

~(U11-U~I)-U12Ui2-JUI2U'J>O,

U 11 J=JU 11 . }

(15)

Thus, in this case, the domain S[tP] is a Siegel domain of genus 2. We will now describe the remaining realizations. From now on we will not distinguish between even and odd p. We set 0

H=

0

iE 2s

0

0

0

Jr

0

0

Er

0

0

-Er

0

0

0

0

Er

0

0

Er

0

0

0

0

0

-Js

0

0

0

- i E 2s

where

0

J, = (

0 -Es

K=

:}

2s+r.

p=

We partition Uinto blocks in the same way we did in Section 10:

U=

U 11

U 12

2s

U 21

U 22

r

U 31

U 32

r

U 41

U 42

2s

2s

r

(16)

120

THE GEOMETRY OF CLASSICAL DOMAINS

and we prove as we did there, that if U satisfies conditions (11), then the matrix

is nondegenerate and, consequently, each class U E OH contains a unique representative such that U 31 = 0, U 32 = En U41 = E 2s ' U42 = 0. Relationships (11) imply that U'11 J s = J s U 11, W 11

U;2 = - U 22 '

U;l = JU 12 ,

1 11 - U 11 * )- U 21 * U 2b = :-(U I

W 12 = wil =

~I U12 -

uil U 22 '

(17)

W 22 = E- Ui2 U 22 ' fV12) > 0.

J

W22

The inequalities (17) define an unbounded domain S in a tp(p-I)dimensional complex space (whose coordinates are the independent entries of the matrices Ul1 , U 22 and U 12 ). , As in Section 6, we may write the inequalities defining the domain in the following form:

l(V 11 - U*) U~·21 W-221U 21 - UT 12 W-221 U~'12 11 -

-:1

- iU 12

}V;l ui2 u21 + iUil V 22 w;l Uj2 > 0,

)

(18)

E- vi2 U 22 > 0.

In order to make it clear that the domain obtained coincides with some Siegel domain of genus 3, we introduce some new notation characteristic of Siegel domains. We set

t = U 22 '

Z

= 2U 11 ,

U

= U 12 '

V

= V 12 .

It follows from (18) that t belongs to a classical domain of type 2 with parameter r = p-2s.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

121

Let the cone V consist of all 2s x 2s Hermitian positive definite matrices Y such that YJs = Js Y. We set

L/u, v) =

u12 Wi} vfz +J V12 W;l u 12 J'

It is not difficult to see that our domain is the Siegel domain corresponding to the cone Vand the form Lt(u, v). The boundary F of this domain consists of all matrices U such that

U*HU ~ 0

det U*HU = 0,

(20)

(compare this with formula (38) of Section 10). Consider the matrices U E F of the following form: E zs

U=

0 0 0

~1

~' j,

Z'= -Z,

Z*Z < Er •

(21)

It is easy to prove the following results, which are analogous to the results of Section 10: There is no more than one matrix of the form (21) in a class of equivalent matrices. The matrices U form a component analytically equivalent to the component!Fn which we naturally treat as "infinitely distant". The analytic automorphisms of our domain that map component (21) into itself are quasilinear transformations (defined in Section 3 of Chapter 1), while the analytic automorphisms leaving the point

rE 2s o o o

0

0

(22)

E 0

fixed are linear transformations. In order to prove the last assertion, it is first necessary to prove that the matrices A corresponding to the analytic automorphisms of the

122

THE GEOMETRY OF CLASSICAL DOMAINS

domain S that map an "infinitely distant" component into itself have the form

Au

A=

°

A12

A13

A141

A22

A 23

A24

o

A32

A33

A 34

° ° °

A*HA = H,

so that

j'

A44 A'KA = K;

(23)

(24)

Hand K are defined in (16). As we did in Section 10, we will prove that the transformations corresponding to these matrices are quasilinear transformations of our domain. The matrices A corresponding to the analytic automorphisms leaving point (22) fixed are separated by the supplementary conditions

A 23 = 0,

A32 = 0.

(25)

Linear transformations of our domain (see Section 10) correspond to such matrices. The proofs of the second fundamental theorem and Lemma 4 of Section 10 for classical domains of type 2 proceed in exactly the same way as they did above. We now give, without proof, formulas for the Riemann distance and geodesics (Klingen [1]). We will assume that the domain {0 is realized in the same way as at the beginning of the section. It is possible to prove that a Riemann metric invariant with respect to the analytic automorphisms is given by the formula. ds 2 = a{(Ep-Z*Z)-1 dZ*(E p-ZZ*)-1 dZ); (26) a(A) denotes the trace of the matrix A. The distance between two points Z1 and Z2 is given by the following formula: 2

1

p (Z1' Z2) = -

8

Ik p

==1

2 1 +..jrk In ----=.,

l-.Jr k

Ak-l rk = - - , Ak

(27)

where A1 , ••• , Ap are the characteristic roots of the matrix

R(Z1' Z2) = (Ep-Zi Z1)-1(Ep-Zi Z2)(Ep-Z! Z2)-1(Ep-Z! Z1).

THE GEOMETRY OF HOMOGENEOUS DOMAINS

123

Any geodesic in !:0 is analytically equivalent to a geodesic of the form

th (Xl s.j

0

0

0

ih (Xl s.j

0

0

0

th (Xr s.j

0

0

0

0

0

Z(s) =

r = [j-p], j

= (

0 -1

0

~}

where the last row is added if p is odd, (Xi + ... + (X; = 1, and s is the arc length. We will omit that statement and proof of Lemma 5, since they are completely analogous to those given in Section 10. The fiber !:0 o corresponding to U22 = 0 is given by the inequality

~(Ul1 l

Url)- U 12 Url -JU 12

U~l J' > O.

(28)

As we should expect, it is a Siegel domain of genus 2. We now turn to a discussion of classical domains of type 3. These domains are described in the following manner. Let p > O. We will treat p x p symmetric matrices Z as points in a j-p(p + 1)-dimensional complex space. The set of Z such that (29) forms a bounded domain!:0. The analytic automorphisms of the domain !:0 are described in the following manner. Let G be the set of all 2p x 2p matrices A such that A*HA = H,

A'JA = J,

where

(30) (31)

An analytic automorphism of the domain !:0 (32) corresponds to each square matrix A.

124

THE GEOMETRY OF CLASSICAL DOMAINS

The formulas for a linear element of a metric invariant with respect to the analytic automorphisms of the domain ~ and for the distance between two points Zl and Z2 are similar to the formulas given in Section 10 (p. 91). Any geodesic in ~ is analytically equivalent to a geodesic of the form

o

o o

Z(s) =

(33)

o where exi + ... + ex; = 1 and s is the arc length. Let a be the set of all complex rectangular

°

2p x p matrices U of maximal rank and such that U'JU = 0, where J is defined in (30). If U E a, then UR E a, where R is a p x p nondegenerate matrix. We agree to say that the matrices U and UR are equivalent. The set of all equivalent matrices forms a class. The set of classes is a compact complex symmetric manifold D. We will prove that ~ may be realized in the form of a domain in D. Let H be the Hermitian matrix defined in (30). Consider the set aH of matrices U E a, such that U*HU > 0.

(34)

It is easy to verify that every class of equivalent matrices is either completely contained in aH or does not intersect aH • The set of classes contained in aH is a domain in D. We set 1

U=(U )P. U2 P

(35)

As we did in Section 10, we can prove that ~ is realized in the form of a domain in D. The boundary F of domain ~ consists of the U E a, such that the determinant of the matrix U* HU is equal to zero and U* HU ~ o. In other words, F consists of the Z satisfying the following conditions:

(1) the determinant of matrix Ep-Z*Z is equal to zero, (2) Ep-Z*Z ~ 0, (3) Z' =Z.

THE GEOMETRY OF HOMOGENEOUS DOMAINS

125

The proofs of the Theorems 4, 5, and 6 for domains of type 3 proceed in exactly the same manner as the proofs for the cases of domains of types 1 and 2. As a result, we will state only the necessary changes in the statements and formulas.

Lemma 1. The set of points of the form

z=

z(p-" P-'J,

1 (36)

Z/=Z,

z*z < Ep-r

J

in the boundary F forms a component that we denote by ,~il ~ r ~ p). Any boundary component is analytically equivalent to a component :Fr. Every point in F is contained in some component. Lemma 2. Two points Zl' Z2 E F belong to one component if and only if (37)

where r(Zi'Z) denotes the rank of the matrix Wij = Ep-Zt Z). We now turn to describing the canonical realizations of domains of the third type. A general method for obtaining such realizations consists in the following. Let H be a Hermitian matrix of order 2p with p positive and p negative characteristic roots, and let J be a nondegenerate asymmetric matrix of order 2p, so that (38) Consider the set OH of 2p x p matrices U such that U*HU> 0,

U'JU = 0.

(39)

We agree to say that the matrices U and UR, where R is a p x p nondegenerate matrix belong to the same class. The set of classes contained in OH is a domain in D (see Klingen [1]). That is analytically equivalent to our domain !:0. The analytic automorphisms of this domain are described in the following manner. Let G be the set of a1l2p x 2p matrices A such that A*HA = H,

A'JA = J.

We associate an analytic automorphism U ~ A U in OH with each A E G. It is easy to see that there is a corresponding analytic automorphism in the set of classes.

126

THE GEOMETRY OF CLASSICAL DOMAINS

The boundary of the domain f» consists of the classes that satisfy the conditions

U'JU = 0,

U*HU ~ 0,

det U*HU = 0.

It is possible to prove that two boundary points U1 and U2 are mapped onto each other under an analytic automorphism of the domain f» if and only if the ranks of the matrices W l = ut HU1 and W 2 = ui HU2 are the same. We may deduce from this that the boundary decomposes into p transitive parts. Thus, there are exactly p canonical realizations. We will first describe the realization Sp corresponding to a zero-dimensional component of the boundary of domain f». We set

H=

(-iEp°

(40)

It is easy to verify that (38) is satisfied when Hand J are selected in this way. We partition Uinto blocks in the following manner:

Conditions (39) imply that l(u*2 -:1

°

(41)

U 1 - U*1 U) 2 > ,

It is easy to verify that U 2 is nondegenerate and, consequently, every

class contains a unique representative of the form

It then follows that from conditions (41)

~(Z-Z*) > 0, 1

i.e., S is a Siegel domain of genus 1.

Z' = Z,

(42)

127

THE GEOMETRY OF HOMOGENEOUS DOMAINS

This domain was introduced by Siegel, as a result of which it is frequently called Siegel's generalized upper halfplane of degree p. We now turn to descriptions of the remaining realizations. We set

H=

0

0

0

iEs

0

-Er

0

0

0

0

Er

0

0

0

0

-iEs

0

0

0

Es

0

0

Er

0

0

-Er

0

0

-Es

0

0

0

J=

(43)

It is easy to verify that condition (38) is satisfied. We partition U into blocks just as we did in Section 10, i.e.,

u=

u 11

V 12

S

U 21

V 22

r

U 31

V32

r

U 41

U 42

S

(44)

r

S

As we did in Section 10, we prove that if U satisfies (39), then the matrix

(

U U32) 31

U 41

U 42

is nondegenerate and, consequently, each class U E nH contains a unique representative for which U31 = 0, U 32 = E r , U l l = E s ' U42 = o. From (39) we obtain U~1

= U 11 ,

U~2

=

U 22,

U~1

=

U

12,

W12) >0, H/22

where

(45)

128

THE GEOMETRY OF CLASSICAL DOMAINS

Inequalities (45) define an unbounded domain S in a -!p(p+ 1)dimensional complex space (whose coordinates are the independent entries of the matrices U11 , U 12 , and U22 ). An "infinitely distant" component consists of matrices of the form E

0

0

Z

0

E

0

0

, Z'=Z, Z*Z<E.

(46)

We now turn to describing the groups Gig;), k = 1,2,3. As we did in Section 10, it is possible to show that group G 1 (ff) consists of all transformations with matrices A of the following form:

A=

Au

A12

A13

S

0

A22

A 23

2r,

0

0

A33

S

21'

S

A*HA = H, A22 =

(47)

S

A'iA = J,

(Bl B2} B3

B4

where Hand J are defined by (43). The following automorphism of a component corresponds to each transformation with such a matrix A:

E

0

E

0

o Z o E o 0

0

Z

0

E

0

0

whence it follows that matrix A in G 1 (g;) belongs to GzCg;) if and only if

THE GEOMETRY OF HOMOGENEOUS DOMAINS

129

In addition, it is possible to prove that the fiber f» 0 corresponding to 0 is the following domain:

U22

1 * *_ -:-(U 11 -U 11 )-U 12 U 12 - U 12 U 12 > 0, 1

i.e., some Siegel domain of genus 2.

E

CHAPTER 3

Discrete groups of analytic automorphisms of bounded domains Section 1. Introduction

Let r be a discrete group of analytic automorphisms of some bounded homogeneous domain {0. We will call the functions that are meromorphic in {0 and invariant with respect to the group r automorphic functions. It is not difficult to show that the degree of transcendence of a field of automorphic functions is no less than the complex dimension n of the domain {0 (see C. L. Siegel [7] and Lemma 2, Section 4). As we know, the degree of transcendence of a field of merom orphic functions on a compact analytic normal space is no greater than its dimensions (Remmert [1]). Thus, in order to prove that the degree of transcendence of a field of automorphic functions is no larger than n, it is sufficient to imbed {0/r in a compact analytic normal space M in the form of an everywhere dense set and show that any meromorphic function on {0/r extends to all of M. In order to do this, it is sufficient to show that the complex dimension of M' = M - {0/r is no greater than n - 2, where n is the dimension of {0. In this chapter, we will, for any arithmetic group in the sense of A. Borel, give an explicit construction of an extension of the factor space {0/r with all of the necessary properties.t t The unit disk I z I < 1, where

1, is an exception, for the dimension of 2. Here we must impose additional assumptions on the definitions of automorphic functions so that it will be possible to extend them analytically to all of M. 131

M'

=

n

M - flZjr is 0, i.e., larger than

II -

132

THE GEOMETRY OF CLASSICAL DOMAINS

For the case in which r is Siegel's modular group, a construction for such an extension was first given by Satake [2]. Satake's extension coincides with the one obtained from the general construction of the present chapter. A. Andreotti and H. Grauert [1] recently proposed a very elegant general method for proving the theorem on the degree of transcendence of a field of automorphic functions. As we will show in Section 7 of this chapter, the conditions under which this method is applicable hold (with trivial exceptions) for all arithmetic groups of analytic automorphisms of symmetric domains. The construction of the extension M of the factor space {0/r is applicable to discrete groups of analytic automorphisms of arbitrary bounded homogeneous domains. It is stated in this form in the present chapter. It is understandable that, in this case, the space M is, as a rule, not compact, but it is always an analytic normal space. This can be proved by means of a theorem due to H. Cartan. We should also note that our construction of the extension M is applicab'le not only to homogeneous domains, but, in general, to arbitrary Siegel domains. The examples of complex nqrmal spaces arising as a result of this construction will be of value to us. As a rule they are not complex manifold, even if they are compact. U. Christian [2] discussed this in detail for the spaces that arise in the study of Siegel-Hilbert modular groups. When {0 is a symmetric domain and r is an arithmetic group, the space M is always compact. We will use a recent result of A. Borel [3] to prove this proposition. We will now turn to a section-by-section survey of the contents of the present chapter. Section 2 presents the construction of the extension M for the factor space {0/r for the case in which {0 is a bounded homogeneous domain and r is a discrete group of its analytic automorphisms. In Section 3 we will use one paper ofR. Cartan as a basis forintroduction of an analytic structure into the space M. Satisfaction of Cartan's conditions is established by means oflemmas proved in Sections 4 and 5. In Section 6 we will show that if {0 is a symmetric domain and r is an arithmetic group of its automorphisms, then M is compact. In Section 7 we will show that, except for certain trivial exceptions, any arithmetic group is a pseudoconcave group in the sense of Andreotti and Grauert.

133

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

Section 2. Construction of the Extension of the Factor Space

~/r

Before we give this construction in the general form, we will consider it in a very simple, almost trivial case, the case in which {0 is the ordinary upper halfplane, i.e., the set of points of the form z = x+iy, 0< y, and r is an ordinary modular group. We let denote the space obtained by adding all of the rational points on the real axis and the point 00 to {0. The space M is the factor space 9)e/r. It is clearly sufficient to introduce a topology into the space in order to introduce one into the space M. We provide a topology in by means of a neighborhood base. A neighborhood of a point Zo E {0 is defined as usual. A neighborhood of the point 00 consists of those points of the domain {0 that have the form y = Imz> c > 0 and, naturally, the point 00 itself. A neighborhood of point z = r (r is a rational number) consists of the points of domain {0 that lie inside some circle tangent to the real axis at the point z = r and the point r itself. It is clear that the space M constructed in this manner is homomorphic to a two-dimensional sphere. It is very important to give the proper definition of the analogs of rational points in the example given above so that we can extend this scheme to the general case. Let {0 be a bounded homogeneous domain, and let r be a discrete group of analytic automorphisms of the domain {0. The most important point of the construction discussed below is the notion of a r-rational homogeneous fibering. Consider some homogeneous fibering of the domain {0. Let ® be the group of all analytic automorphisms of the domain {0 that preserve the given fibering. By ¢ we will denote the natural homomorphism of the group ® onto the group ®' of analytic automorphisms of the base, and by 3 we will denote the group of parallel translations of this fibering (see Chapter 2, Section 7). We will say that a homogeneous fibering is r-rational if: (1) the factor space 3/11 where 11 = r n 3, is compact, and (2) the group r' = ¢(r n ®) is a discrete subgroup of the group ®'. In what follows we will call r' the induced group. It is not difficult to see that there is a single family of rational fiberings for any two commensurable subgroups r 1 and r 2. Let us consider an example. Let G be an algebraic group defined over the rational numbers Q. Assume that the group GR is transitive in the

we

we

we

134

THE GEOMETRY OF CLASSICAL DOMAINS

domain £0, and let r = Gz • Now, a homogeneous fibering is rational if and only if the group of its parallel translations is an algebraic subgroup of the group G and defined over Q. Indeed, if the fibering is rational, the factor space 3//)., where /). = 3 n Gz is compact. This implies that 3 is a subgroup of the group GR. Moreover, we know (Chapter 2, Section 7) that 3 is an algebraic subgroup of the group of all analytic automorphisms of the domain £0. As a result, the subgroup 3 is an algebraic subgroup of the group GR. Compactness of the factor space 3//). implies that the subgroup 3 is defined over Q. We will now prove the converse, i.e., that if the subgroup 3 of some homogeneous fibering is an algebraic sub-group of a group G and defined over Q, this fibering is rational with respect to the group r = Gz • The group 3 is unipotent, and, consequently, the factor space 3/3z is compact (A. Borel, Harish-Chandra [1]). Consider its center 30' and let Go be the normalizer of the group 30. The group r in the definition of a rational fibering is, as we can easily show, commensurable with the group Gz, where G = G o/3. As a result, the group r' is automatically discrete in this case, i.e., the second condition of the definition of a r-rational fibering is a consequence of the first. This fact is clearly related to the fact that algebraic Lie algebras contain, along with any given element, all of its replicas. Let M denote the set-theoretic union of the domain £0 and the domains £0' that are bases of r-rational homogeneous fiberings of the domain £0. When £0 is the unit disk and r is an ordinary modular group, the space coincides with the space introduced in the example discussed at the beginning of the present section. In what follows, the domains £0' will sometimes be called r-rational components. The group r is naturally defined in the space IDe. As in the simple example considered at the beginning of this section, we define M to be a factor space IDe/r. Now our problem is to introduce a topology into the space M. Toward this end, we now construct a fundamental system T(rno) of open subsets M o =£0/r for each point rnoEM; henceforth, we will define topologies by means of these subsets so that the subsets of T(rno) will be the intersection of the neighborhoods of the point rno and Mo· Assume that the point rno is contained in Mo. Let Zo denote one of its preimages in £0. The system of sets that we seek is the image of a

me

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

135

fundamental system of neighborhoods of the point Zo under projection onto Mo. We will now consider the case in which m o E£0'jr', where £0' is the base of some r-rational homogeneous fibering and r' is the discrete group induced on £0'. Let Zo be some preimage (in £0'), of the point rno and let U denote some neighborhood of the point Zo in £0'. Moreover, let A(U) be some section in £0 over U, i.e., a subset of £0 for which the projection onto £0' induces a homomorphism with U, and J1(U) is the union of all orbits of the group 3 of parallel translations of this fibering that pass through points of the set A(U). The bounded holomorphic hull O(J1(U)) of the set J1(U) is an open subset in £0 (see Chapter 1, Section 4). T(mo) consists of the images of sets of the form O(J1(U)) under projection onto £0 jr, where U is an arbitrary neighborhood of the point Zo in £0' and A(U) is an arbitrary section over U. In what follows, we agree to call sets of the form 0(J1(U)) cylindrical sets. It is not difficult to show that our construction is independent of the selection of the point Zo in £0'. It is clear that if 0lET(mo) and OoET(mo), then there exists an 0 3 ET(mo) such that 0 3 E 0 1 n O2 , Now we can introduce a topology into M. Let mo EM, and associate the set W of all lnEM for which there exists an OET(m) such that Om E 0 with each set 0 E T(mo)' The family of sets W obtained in this manner is a base for the neighborhoods at the point mo. In addition, we will assume that the following assumptions are true: (A) for any point Zo E £0', where £0' is the base of some r-rational fibering, there exists a cylindrical set of the form O(J1(U)) (U is some neighborhood of the point zo) with the following property: if Zl E 0(J1(U)), yZl E 0(J1(U)), where y Er, then ypreservesthegivenfibering; (B) For any two differentpointsm 1 ,m2 EM, there exist sets 0 1 and O2 , OkET(mk), k= 1,2, whose intersection is empty. It follows from (B) that the topology in M is Hausdorff. Indeed, let ml and m 2 be two distinct points of M. Let 0 1 ET(md, O2 ET(m 2 ) be such that their intersection is empty. By W k , k = 1,2 we denote the set of all mEM for which there exists an OmET(m) such that Om C Ok for k = 1,2. It is clear that W 1 n W2 = 0. It is clear that conditions A and B are always satisfied. For spaces M obtained from arithmetic groups defined on symmetric domains, satisfaction of these conditions is a corollary of the results of A. Borel [3].

136

THE GEOMETRY OF CLASSICAL DOMAINS

Section 3. Analytic Normal Spaces

In this section we will introduce an analytic structure into space M. In the case of one complex variable, we can prove that M is a complex manifold. This is not so in the general case. As we will prove below, however, M is an analytic normal space. We now give the definition of analytic normal spaces. First we will present the notion of a ringed space (H. Cartan [3]) due to Serre. A Hausdorff space X such that a subring Ax of the ring of all germs of continuous complex-valued functions is defined at every point x EX, is called a ringed space. We denote the set of rings Ax by A. Homomorphisms and isomorphisms of such spaces are defined in the usual way. Let U be a domain in the complex space eN. An analytic subset in U is a closed subset V c U such that in a sufficiently small neighborhood of each of its points, it is the set of common zeros of some finite number of functions analytic in this neighborhood. We should note that every analytic set is a ringed space if we take the set of functions induced by functions analytic in some neighborhood of a point x for Ax. A ringed space is said to be an analytic space if for each of its points there is a neighborhood isomorphic as a ringed space to some analytic subset in eN' A ringed space is said to be normal if every local ring Axis an integrally closed integral domain. Recall that an integral domain is a commutative ring with no zero divisors. An integral domain 0 is said to be integrally closed if every solution of the equation yll+a i yll-i + ... + all

= 0,

(where ai' ... , all EO) that belongs to the quotient field of integral domain o belongs to O. For example, the ring of integers is an integrally closed integral domain. A point x E X is said to be regular if it has a neighborhood isomorphic to a domain in eN. We will prove that it is possible to introduce the structure of an analytic normal space into the space M defined in Section 2 in such a manner that on each M j = !?)r'($P), where $Pj is some r-rational

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

137

component, the structure will coincide with the natural analytic structure already there. We will use H. Cartan's theorem on extensions of analytic normal spaces (H. Cartan [1]) to prove this statement. Let X be some locally compact space. V is an open set that is everywhere dense in X and W = X-V. We will assume that the structure of an analytic normal space of dimension 111 is defined on V. H. Cartan posed the following question: Does X have the structure of an analytic normal space with the following properties?

(0:) it induces the indicated structure in V; (fJ) W is an analytic subspace of X with dimension less than m.

As H. Cartan remarked, if such a structure were possible, it lVould be unique.

Indeed, let x EX. The ring Ax of germs of continuous complexvalued functions at the point x is uniquely defined by the following condition: a function f belongs to Ax if and only if it is continuous in some neighborhood U of point x and at each point y E Un V it belongs to By (By is the ring of germs of analytic functions at the point y). As Cartan proved, satisfaction of the following three conditions is sufficient for existence of the required structure: (1) Any point Xo E W has a fundamental neighborhood system whose intersection with V is connected; (2) Every point Xo E W has a neighborhood U in which the functions continuous in U and analytic at every point of V n U separatet all points

ofVn U; (3) The structure A of the rings of germs of continuous functions that naturally appears in W induces the structure of an analytic normal space of dimension less than 111 in W.

We will prove the following proposition, which we will use later on, as an example of an application of Cartan's theorem: Let fifi be some complex manifold, and let r be a discrete group of its analytic automorphisl11s. It is always possible to introduce the structure of an analytic normal space into thefactor space fifi/r. Proof First of all, note that if the group r contains no nontrivial transformations with fixed points, then the natural mapping fifi -+ fifi /r

t We say that a function/(z) separates points Zl and Z2 if/(Zl)

=I=-

/(Z2).

138

THE GEOMETRY OF CLASSICAL DOMAINS

is locally one-to-one and, therefore, defines the structure of a complex manifold in £0 /r. Now for the proof of our assertion. We denote the set of all points in £0 that are not fixed points for elements of the group r by £0 0 , Now we set X = £0/r and V £0 o/r. It is clear that £0 0 , and, therefore, Vas well, are complex manifolds. We will show that the conditions of CaI'tan's theorem are satisfied for X and V. Let Xo E W = X-V. We denote any preimage of this point in £0 by Yo. lt is not difficult to see that any fundamental neighborhood system of the points Yo is mapped onto a fundamental neighborhood system of the point Xo by the natural mapping £0 -+ £0 /r. lt immediately follows that condition 1 of Cartan's theorem is satisfied. In order to prove that condition 2 is satisfied, it is sufficient to find for any point YoE£0-£0 o a neighborhood U in which analytic and r-invariantt functions separate all r-nonequivalentt points. We denote the set of all l' E r such that Y(Yo) = Yo by r o. We choose a neighborhood U of the point Yo so small that l' E r 0 is a consequence of nonemptiness of Un y(U). Furthermore, we may assume that the neighborhood U is r o-invariant. Let Yb Y2 E U. If Yl i= YY2 for any l' E r 0' then there exists a functionf(z) that is analytic in U and such that !(YYl) = 0

and !(YY2) = 1

for all l' Er o' The function

cp(z) = If(yz) )'ErO

is r-lnvariant and separates the points Yl and Y2' We have proved that condition 2 is also satisfied. We now note that in a sufficiently small neighborhood U of point Yo, it is possible to choose a coordinate system in which all transformations in r 0 are linear. The set of fixed points for r 0 in this coordinate system will be a linear subspace, and, therefore, a submanifold of the manifold £0. The mapping of £0 - £0 0 onto W is locally one-to-one and, consequently, the structure induced in W is that of a complex manifold. Thus, condition 3 is also satisfied. Our proposition is proved. t Functions such that the functional equation fez) = f(yz) holds for all y Erare said to be r-invariant. t Two points Zl and Z2 are said to be r-equivalent if Z2 = yZl for some yEr.

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

139

We will now prove that the space constructed in the preceding section has the structure of an analytic normal space. We will do this by applying Cartan's theorem in the case X = M, V = Mo. Verification of condition 1. Let Xo E ff' jo' and consider the sequence of cylindrical domains QIl with bases BII E JiP jo' Xo E BII" We assume that

Then, as we can easily see, there is a fundamental neighborhood system Un of the point Xo such that Un n M 0 is the image of QIl under the natural mapping flfi -+ flfi /r. Every cylindrical domain Qn is connected. As a result, Un n Mo, as a continuous image of Qn, is also connected. Verification of condition 2. Again let Xo E JiP jo • Lemma 2 of Section 9 implies the existence of a cylindrical domain Q with base Be JiP jo ' Xo E B, and the property that if Z E Q and yz E Q, where l' E r, then yXo = Xo. Let U denote the neighborhood of the point Xo such that Un Mo is the image of Q under the natural mapping flfi -+ flfi/r. Every function that is continuous in U and analytic in Mo n U induces some r o-invariant analytic function in Q (r 0 is the set of all l' E r such that y(xo) = xo). We map the domain flfi onto some Siegel domain S so that all transformations leaving the point Xo fixed become linear transformations, while all transformations mapping JiP jo onto itself become quasilinear. Note that the Jacobian of any transformation Yo E r 0 in the domain S is equal to 1. As a result, the concepts of r o-invariant functions and r o-automorphic forms coincide (see the definition in Section 4). Continuity of a function in U implies that the function induced by it in Q is "analytic at infinity" in the sense of Section 5. The converse will follow immediately from Lemma 1 of Section 5, namely that any r o-invariant function that is analytic in Q and "analytic at infinity" is induced by some function that is continuous in U and analytic in UnMo· It will follow from Lemma 4 of Section 5 that any two r o-nonequivalent points of the domain Q can be separated by functions that are analytic in Q, ro-invariant, and "analytic at infinity". Verification of condition 3. We will now prove that the structure of the rings of germs of continuous functions that naturally appears in W induces the structure of an analytic normal space of dimension less than

140

THE GEOMETRY OF CLASSICAL DOMAINS

111 in W. It follows from Lemma 1 of Section 5 that the ring structure induced on each Mk = flfiur£ coincides with the natural analytic structure already there. Let Wo denote the set of all Mk c M that are closed in M. In addition, let W 1 be the set of all Mk c M whose boundaries, i.e., Mk-Mk, belong to WOo We define W 2 , W 3 , etc., similarly. It is clear that W k = W for some k. We will now show that Wj is an analytic normal space whose dimension is equal to the maximal dimension of the Mk contained in it. This is clear for i = 0, because Wo is the union of no more than a countable number of closed disjoint sets M i , and the structure induced on each Mi coincides with the structure of an analytic normal space that is already there. Assume that we have already proved that Wj has the structure of an analytic normal space; we will now prove that Wj + 1 also has the structure of an analytic normal space. Wj + 1 is the union of no more than a countable number of closed disjoint sets. It is clear that it is sufficient to prove our assertion for any of these closed subsets. We again apply Cartan's theorem for the proof. For the same reasons as above, conditions I and 2 of this theorem are satisfied. Condition 3 is satisfied by virtue of the induction hypothesis.

Section 4. Poincare Series

In this section we will study Poincare series. The results of this section and the next will be used to prove Lemma 4, from which follows satisfaction of one of the conditions of Cartan's theorem. First we define automorphic forms. Let flfi be some domain, and let r be a discrete group of analytic automorphisms of this domain. Definition l.t A function fez) that is analytic in flfi is said to be a r -automorphic form of weight /11, if it satisfies the functional equation J(Y(Z))j~I(Z)

= fez) for all

"I E r,

(1)

where irCz) denotes the Jacobian of the transformation z -+ y(z) .. We should note that r-automorphic forms are transformed in the t Thi'J definition departs somewhat from the usual terminology. Properly speaking, automorphic forms are solutiqns of the functional equation (1), where additional hypotheses concerning the behavior close to certain boundary points are imposed on the solutions (see the example, Section 2). All the same, we will use this definition.

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

following manner under an analytic mapping z -+ Z1 domain £0 onto some domain £0 1 : 1(z) -+ 1(cp -1(Z 1»)j;-1(Z1)'

141

= cp(z) of the (2)

When the domain £0 is bounded, there is a very convenient method for constructing r-automorphic forms by means of Poincare series. Let r be an arbitrary discrete group of analytic automorphisms of a bounded domain £0. The series (3) is uniformly convergent in any compact subdomain of the domain £0 (Siegel [1], p. 103). As a result, the series

I

h(y(z»j~'(z),

111

~ 2,

(4)

YEr

where h(z) is a bounded function analytic in £0, is a function analytic in £0. It is easy to verify that equation (1) is satisfied by this function and, therefore, it is an automorphic form of weight m. Preliminary discussion of the construction of some special fundamental domain will be useful in the study of Poincare series. Definition 2. By afundamental domain, we will mean a closed domain bounded by a finite or countable number of real analytic manifolds, where every point in the domain £0 must have at least one point r-equivalent to it in the fundamental domain, and any two interior points of the fundamental domain must be r-nonequivalent. Let F denote the set of points in £0 for which Ii/z) I ;;; 1 for all y E r. We will prove that ifr 0 contains no nontrivial mappings with lacobians having modulus identically equal to 1, then F is a fundamental domain in the above sense. Indeed, it is clear that F is closed. Let Fo denote the set of points z E F such that Il/z)1 < 1 for all y except y = 8. It is clear that Fo is an open set. The points contained in Fbut not in Fo are clearly contained in one of a countable number of hypersurfaces Iliz) I = 1. Every point Zo E D has an equivalent point z' E F. In view of the convergence of series (3), in fact, among the numbersl/zo), YEr, there is one with maximum modulus. We denote it by lyo(zo). Set z' Yo z. Then (z 0) I ~ 1 IJ. ( ') I = IJ. (Yo (Zo) I = IIj jiz) I- , y z

and, therefore, z' E F.

y

)')'0

(5)

142

THE GEOMETRY OF CLASSICAL DOMAINS

Let Zl EFo and Z2 = Y(Zl) EFo; then

jy-1(Z2)j/Z1) = jy-1(Y(Zl»)j/Z1) = je(Zl) = 1. This is impossible (see the definition of Fo). Consequently, there are no r-equivalent points in Fo. We will now prove a property of the fundamental domain we have constructed that will be useful in what follows. We will prove that any closed set Do c D can be covered by a finite number of images of F. Indeed, if Zo E Do and Zl = 1'1 Zo E F, then (6)

Because series (3) converges uniformly, Do may contain only a finite number of different I' for which the last inequality is true. Now consider the space in which there are nontrivial mappings I' E r with lacobians identically equal to one. It is clear from (3) that there are only a finite number of such mappings. It is also clear that they form a group, which we will denote by r o' F is defined as above, and Fo is defined as the set of all Z such that Ii/Z) I < 1 for all I' Er except Y E r o. As we did above, we will prove that any point in f0 is equivalent to some point in F and that if two points Zl and Z2 in Fo are equivalent, then Z2 = YOZb where YoEro. The following lemma proves that it is possible to separate r-nonequivalent points in f0 by means of Poincare series. Lemma 1. For any two points Zl and Z2 that are not equivalent with respect to r, there exist Poincare series ¢l(Z) and ¢2(Z) of the same weight such that (7) Proof Without loss in generality, we may assume that Zl and Z2 belong to F. Note that when m ~ CIJ

(8) We choose hi(z), i = 1,2, so that the following conditions are satisfied:

h 1(Zl)hi z 2)-h 1(Z2)hi z 1) h/yzj)

= 0, if

Ij/z j)1 = 1,

0,

(9)

yZj =1= Zj'

(10)

=1=

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

143

Conditions (9) and (10) are consistent because the points Zl and Z2 are not equivalent. We denote the number ofy such that yZj = Zj by Sj. Itis clear that the y having a common fixed point form a group. By Lagrange's theorem, therefore, it follows from yZj = Zj that ySj = 8 and, therefore,

(11)

the lemma follows trivially from (9) and (11). We should note that a similar method may be used to prove a somewhat more general assertion, namely: let Zl' •.. , zp be some system of pairwise distinct points. Then there exist Poincare series CPl' ... , CPP of the same weight such that

CPl(Zl) ... CPi Z l) CPl(Z2) ... CPi Z2)

0

i= .

(12)

Expression (12) clearly implies the existence of a Poincare series with sufficiently large weight and any preassigned values atthe points Z 1, ••• , Z po Lemma 2. Let Zo E F be a point that is notfixed under any transformation in the group r except the identity transformation. Then there exist Poincare series CPo, ... , CPIl such that

CPo ... CPIl oCPo oCPn OZl ···OZl

OCPo OCPIl iJz 1 ••• iJzll

i=

o.

Z=Zo

Proof As above, we seek them in the form

cplz) =

L hlyz)j~(z). )'Er

(13)

144

THE GEOMETRY OF CLASSICAL DOMAINS

For sufficiently large m, the behavior of the functions CPi(Z) depends only on the terms in series (13) for which IJ/zo)! = 1. We set

/z) =

L

h/yz)j~J(z).

liy(zo) I = 1

It is clear that it is sufficient to select hi(z) so that

<1>0 '" n 0<1>0

0<1>11

OZl '" OZl

0<1>0

:f 0.

8<1>n

OZII '" oZn

z=zo

This is not difficult to do, if we use the fact that the points yZo are different for different y. We should note that without the restriction that Zo be a fixed point of none of the transformations, our assertion is false. Indeed, let the point Zo be a fixed point of transformations in r. These transformations form a finite group K. We expand the Poincare series cp(z) in a power series about z. For simplicity in the calculations we will assume that Zo is the origin. By a well-known theorem of Cartan (see Bochner S. and Martin U. T. [1]), we may assume that the l' E K are linear or even unitary, transformations. Thus,

cp(yz) = 8cp(Z), where 8 is a root of unity and l' E K. As a result, either cP or some of its partial derivatives are equal to 0. The following lemma may be proved by arguments similar to those given above. Lemma 3. Let Zo be a point that is fixed under none of the transformations in the group r except the identity transformation. Then for all sufficiently large m, for some a > 0, and for some any ak t ... k" (s = kl + ... +kll ~ am) of complex numbers, there exists a Poincare series cp(z) such that

(14)

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

145

This lemma generalizes to the case in which Zo is a fixed point. That is, it is possible to prove that the only restrictions on the possible selection of the ak( ... k" are consequences of the functional equations for the automorphic form for the 'Y that leave the point Zl fixed. Further on it will be important to be able to estimate Poincare series close to the boundary of a domain. We agree to adopt the following notation: Let £0 be some bounded domain; z -4 w = ¢(z) is some one-to-one analytic mapping of £0 onto some domain £0'; r is a discrete group of analytic automorphisms of the domain £0; and r' = ¢r¢ -1 is the corresponding group of automorphisms of the domain £0'. Note that the domain £0' may be unbounded. A subset T c £0' is said to be a proper subset if there exist 8 and N such that in any polycylinder C( wo, 8), t where Wo E T, has no more than N r'-equivalentt points. Lemma 4. Letfo(z) be some Poincare series. Thefunction few) =fO(¢-l(W»)j;-l(w) is bounded on any proper subset of the domain £0'. Proof It is sufficient to prove that the function

k(~) =

L: !j/¢-1(W))!2!jljJ_l(W)!2 )'Er

is bounded on any proper subset. Indeed, the inequality !f(w)! ~ c(!c(w))1Il/2, where c is the maximum modulus of the function h(z), follows immediately from (4). Let T be a proper subset of the domain £0'. We will prove that the function k(w) is bounded on T. The general properties of analytic functions can easily be used to derive the inequality

f

d(J,

C(wo, r)

where d(J is a Euclidean volume element of an affine complex space. t C(wo, e) is the set of points W (of an affine complex space) such that the modulus of any difference of coordinates W - Wo does not exceed e. t We assume that C(wo, e) C {J2' for any WOE T.

146

THE GEOMETRY OF CLASSICAL DOMAINS

We now apply this inequality, assuming that

few) =

(
jy

X

jlji-l (w),

Wo E

T, r

=

8,

where 8 is the same as in the definition of T:

Furthermore, we have

j/
Carrying out the change of variables v = y'(w), we find that

k(w o)

~ .Ie I -1

'I'

f

1jlji - 1(v)12 dO"

~ .Ie

-1

N

r Ijlji-l(V)12 dO",

J

ffi'

Indeed, because the set T is proper, each point WE £0 is contained in no more than N domains of type "1'( C(wo, 8»). The last integral is clearly equal to the Euclidean volume of the domain £0 and, therefore, is finite, because £0 is bounded. The lemma is proved. Section 5. Lemmas Let S be some Siegel domain of genus 3 with base $7, and assume that it is, as usual, given in the form (see Chapter 1, Section 3) Imz-ReLtCu, U)E V,

tE$7.

Let Q be some domain in $7, rEV. Definition 1. A cylindrical domain SeQ, r) in the domain S is the set of all w = (z, u, t) such that Imz-ReLtCu, u)-rE V,

tEQ.

This definition coincides with the definition given in Section 2 for cylindrical domains as the bounded holomorphic hulls of certain sets of orbits of groups of parallel translations. Let r be a discrete group of analytic automorphisms of the domain S,

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

147

3 be the group of parallel translations of the domain S, and let 30 be the center of 3. We set ~ = 3 n r, ~o 30 n r o. In this section we will only consider groups r such that the factor space 3/ ~ is compact. The most important automorphic forms are those which are said to be "analytic at infinity". Definition 2. A r -automorphic form is said to be "analytic at infinity" if it is bounded in any cylindrical subdomain of the domain S. Later on we will show that any automorphic form is "analytic at infinity" for a very general class of groups r. The following definition makes it possible to make the meaning of "analytic at infinity" more precise. Definition 3. Let Wl, W2, ..• , be a sequence ofpoints in S. We agree that lim

Wv

= to,

where

to E £0,

v-> 00

iffor any cylindrical domain seQ, r) (to E Q) there is a Vo such that for v > Vo the point Wv belongs to SeQ, r). Definition 4. Let few) be some function in S. We agree to say that limf(w) = A, w->to

iffor any sequence

Wv E

S such that

the limitf(w v) exists for v --+ Lemma 1. The limit

00

and is equal to A.

limf(w) = ljJ(t) w->t

exists for any r-automorphic few) of weight J.1 that is "analytic at infinity". The limit function ljJ(t) is analytic in $7. Proof We denote by 30 the subgroup of 3 consisting of transformations of the form

(1)

where a is an arbitrary real vector.

148

THE GEOMETRY OF CLASSICAL DOMAINS

We set ~o r n 30. It is clear that ~o is a commutative group with the same number of generators as the dimension of group 30' i.e., ~o is a lattice for 30. Let the function fez, u, t) be a r-automorphic form of weight /1. The invariance of/with respect to the transformations of the group ~o implies that it is possible to expand/in a Fourier series of the following form:

f(z,u,t) = It/Jp(u,t)e 2ni (P,z),

11

(p,z) =

p

L

PkZk,

(2)

k=l

where P runs through the dual lattice of ~o, i.e., the lattice consisting of all vectors P such that (p, a) is an integer for any a E ~o. It is clear that each of the functions If p(u, t) is analytic in the domain

c m x $7. We denote the dual cone by V'. (The set of P such that (p,y) > 0 for all y E V, y =1= 0 is called the dual cone.) We will prove that if the r-automorphic form/(w) is bounded in any cylindrical domain, then t/J p(u, t) == 0 for all p E V'. (V'is the closure of V'.) We use Parseval's equality

4n (p,y), ~Lflf(x+iY, U, t)12 dx = L It/Jiu, t)12 eL p

mes

(3)

where L is the parallelepipedt of the lattice ~o. If t/J ilio, to) =1= 0, then for all sufficiently large y

e-41t(p,y)~

1 2f mes Llt/J p(uo, t o)/ L

If(x+iy,uo,to)12dx~C,

(4)

where C is some constant.t It immediately follows from inequality (4) that

(p, y) > C 1,

if y - rEV,

(5)

where C 1 is some constant.

t The fundamental parallelepiped is the parallelepiped constructed of the generators of the lattice Llo. :j: This means that there is a vector rE V such that if y - rE V, then inequality (4) is satisfied.

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

149

Substituting AY for y in (5), where A is a real number, and letting A go to 00, we obtain, from (5), the inequality which is true for any y E V;

(p, y)

~

0

implies that p E V'. We will now prove that the function I/Jo(u, t) is independent of u. Indeed, the factor space 3/(r n 3) is compact, and, therefore, the function I/J o(u, t) has 2m, periods with respect to u, where m is the complex dimension of u. Hence we conclude, by means of Liouville's theorem, that I/J o(u, t) is independent of u. Now let Wv = (zv' uv, tv) be a sequence of points in S that converges to the point to E D. We will now show that a limit, which is equal to I/Jo(to) , exists for few) as w ~ to. Without loss of generality, we may assume that U v is bounded and tvE Q, where Q is a fixed neighborhood of the point to' The Fourier series of an analytic function converges absolutely; Thus,

I

II/Jp(u, t)1 e- 2rr (p,y) <

00.

(6)

p

It is clear from the general properties of analytic functions that series (6) is uniformly convergent with respect to the set of u, t such that t E Q and u is bounded. It is clear that if Wv = (zv, uv, tv) ~ t, then Yv ~ 00, and

If(wy} -l/Jo(tv)1 ~ I p

as v ~

00.

II/J p(U v, tv)1 e- 2rr (p,yv) ~ 0

As a result limf(w)

= I/Jo(t).

w->t

The lemma is proved. Later on we will need a criterion for "analyticity at infinity". Lemma 2. A r-automorphic formf(w) is "analytic at infinity" if and only ifin series (2) the Fourier coefficients I/J p(u, t) == 0 when p E V'. Proof Necessity of the conditions of the lemma follows from the proof of Lemma 1. Sufficiency is proved in the following manner.

150

THE GEOMETRY OF CLASSICAL DOMAINS

Let SeQ, r) be some cylindrical domain in S. Consider the subset in r) that consists of the points of the form

SeQ,

1m z - Re LtC u, u) - rEV, } lul
(7)

tEQ,

where Ko is some constant. t Any point in SeQ, r) is ~-equivalent to some point of the form (7) if Ko is sufficiently large. As a result, we must prove thatf(w) is bounded in domain (7). Note that the Fourier series of an analytic function converges absolutely, and, consequently,

I lifi u, t)1 e- 2n (p,y) <

00

p

for any point w = (z, U, t) E S. Since the convergence is uniform on any compact subset of domain S, we have

I lifi u, t)1 e- 2n (p,r+ReL

t (II,II))

< Kl

p

for any tE Q, lui < Ko. Furthermore, for any point of the form (7) we have

I~ if peLt, t) e2ni(P,Z)! ~ I lifill, t)1 e- 2n (p,y-ReL (lI, 1I)-,.)-2n(p,r+L (lI, II)) t

~

I lifiu, t)1 e- 2n (p,r+L

t

t (II,II)).

We now use the fact that p E V', which implies that

(p,y-ReLt(u,u)-r)

~

o.

Our problem consists of proving that it is possible to separate points that are not equivalent by means of automorphic forms that are "analytic at infinity" and are of sufficiently large weight. In the preceding section we proved that automorphic forms representable in the form of Poincare series can be used to separate any two points that are not equivalent. Below we will prove that automorphic forms representable in the form of Poincare series are "analytic at infinity". More accurately, we will prove the following. Let S be a Siegel domain and r be some discrete group of its quasilinear mappings onto itself. We map S onto some bounded domain f0 and then prove that the

t Remember that the modulus of any coordinate does not exceed Ko.

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

151

functions in S that correspond to Poincare series in £0 are automorphic forms that are "analytic at infinity". First we note the following. We denote the factor group of group G (consisting of all quasilinear mappings) by a subgroup 3 of parallel translations by G 1 • We denote the factor group r/(r n 3) by r l' We will assume that the group r 1 is a discrete subgroup of group G 1 • This occurs, for example, if the centralizer of the group 3 in the group G consists of the identity element. The last condition is satisfied for all Siegel domains that are canonical realizations of irreducible classical domains. Let z~Az+a(u,t),

u~B(t)u+b(t),

t~g(t)

(8)

be some quasilinear transformation of the domain S. With it we associate the following transformation in the space of y, U, t: y

~

Ay,

u

~

B(t)u,

t ~ get).

(9)

We will call this transformation the principal part of transformation (8). It is clear that there is a one-to-one correspondence between the principal parts and the cosets of the subgroup 3 in the group G (see Section 6). The following lemma contains examples of proper subsets of the domain S, which subsets will be used later on (proper subsets are defined in Section 4). Lemma 3. Let Wi = (iYl' Ul, t 1), where the point (Yl' Ul' t 1) is not a

fixed point of the principal part of any transformation in r. The set T of points of the form (X+iAY1' u 1, t 1), where A;?; 1, Ixl ~ c and c is an arbitrary constant, is a proper subset. Proof We choose 8 so that C(Wo, 8) is contained in S for any WET. We denote by R(w o) the set of all quasilinear mappings w ~ ljJ(w) that map S onto itself and are such that there is aWE C(wo, 8), such that IjJ(W) E C(wo, 8), for each of them. Let R be the union of all R(wo). We will first prove that the image of R under the natural homomorphism G ~ G1 is a compact set. Let IjJ E R(w o), where Wo = (xo + iA O Y1' Ul' tl)' We denote the following analytic automorphism of the domain S by (j).:

z ~ AZ,

u~ u.J A,

t ~ t.

152

THE GEOMETRY OF CLASSICAL DOMAINS

The mapping 6;'01/16 }~o 1 maps some point of the set IImz-ImYll <

"'01 8, lu-"'ot u1 1 < Aot8, It-t 1 < 8 1

(10)

onto some point in the same set. The set (10) belongs to S for sufficiently sma118. It is possible to verify directly that the set of all quasilinear transformations that map any point of the form (10) into a point of the same form under the natural homomorphism G -+ G1 map into a compact set. Note that 6;. belongs to the center ofG 1 and, therefore, 1/1 and 6;. 1/16-;1 are mapped onto the same element of the group G1 under the homomorphism G -+ G1 (see Chapter 1, Section 3). We now proceed directly to the proof of the lemma. Let Wo E T. We must prove that the set r(wo) = r n R(w o) contains no more than N elements, where N is some constant independent of the selection of woET. Note that under the mapping G -+ Gb the elements of r(wo) may be mapped onto only a finite number of different elements, because r 1 is a discrete subgroup of the group Gl' In addition, the set r(wo) consists of elements mapped onto the identity of the group G1 when 8 is sufficiently small. We will show that r(wo) contains only the identity element when 8 is sufficiently small. Indeed, let the transformation

z -+ z+ a + 2iLtCu, e(t» + iLtCe(t), e(t»), u -+ u +c(t),

t -+

i,

belong to r(wo). Then Ie(tl)/ < 28 whence, by virtue of the fact that 8 is arbitrarily small, it follows that c = O. Furthermore, /a/ < 28, i.e., a = O. We have thus proved that Tis a proper subset. Now we will prove the following basic lemma. Lemma 4. Let r be a discrete group of quasilinear mappings of the domain S, and let Wl and W 2 be two points of the domain S that are not r-equivalent. It is possible to separate the points W 1 and W2 with automorphic forms that are of sufficiently large weight and "analytic at infinity" . Proof. In view of Lemma 1 in Section 4, it is sufficient to prove that the function corresponding to the Poincare series will be "analytic at infinity", i.e., bounded in any cylindrical subdomain. It follows from

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

153

Lemma 3 that the function few) is bounded on the set of points w of the form

where Ixl < c, c is an arbitrary constant, Yo, uo, to are fixed, and A > 1 so that (Yo, uo, to) is not a fixed point for the principal part of any transformation. Hence, by the argument used in Lemma 1 of this section, it follows that the Fourier coefficients !/J /u, t) for the expansion of the function few) in a series of the form (2) are equal to zero for almost all U o and to if P E V'. It remains to note that the functions !/J p(u o, to) are analytic with respect to the set of u, t and, consequently, their being equal to zero on an everywhere dense set implies that they are equal to zero everywhere. By Lemma 2, we conclude that the function few) is bounded in any cylindrical domain. This completes the proof of the lemma.

Section 6. Arithmetic Groups in Symmetric Domains In this section we will show that the space M for an arithmetic group which acts on a symmetric domain is compact. In order to prove this we must use A. Borel's work on the fundamental domains of arithmetic groups (A. Borel [3]). We will now present his results: Let G be a semi simple linear algebraic group defined over the field of rational numbers Q. By a Satake subgroup we agree to mean an algebraic subgroup B £ G defined over Q with the following properties:

r

(1) B is solvable and split over Q; (2) a maximal solvable and split (over Q) normal subgroup of the normalizer S(B) of the group B coincides with B. Let !£ be the Lie algebra of some Satake subgroup, and represent it in the form W+ S)(:, where W is a subalgebra consisting of semisimple elements and S)(: is a nilpotent ideal. Since the sUbjective representation of Win S)(: is completely reducible, S)(: can be represented in the form of the sum of spaces 97:0; consisting of n such that

[a,nJ = a(a)n,

aEW,

nES)(:o;'

(1)

154

THE GEOMETRY OF CLASSICAL DOMAINS

Here o:(a) is some linear form on Sll. The linear forms o:(a) for which ~)(a is nontrivial are called roots of the algebra !l'. It can be shown that Sll contains a vector a o such that o:(a) > 0 for all roots 0:. We will denote the set of all a E Sll such that o:(a) > 0 for all roots 0: by V(Sll). We now introduce a partial ordering into Sll by setting a 1 > a2 if a1 -a2 E V(Sll). We denote the maximal compact subgroup of the group GR by K. Let X = GR/K be the corresponding symmetric space. Consider the set L of all orbits of the form {BRx}, where XE X and B is some Satake subgroup of the group G. Note that Xc L, because the identity subgroup is a Satake subgroup. The group GQ is naturally defined in the space L, namely, a transformation . BRX~gBRX=gBRg-1gx corresponds to each gEG Q • r,which will remain fixed to the end of the argument, is a group commensurable with Gz . We set (2)

It is clear that

r\xc s.

(3)

We now introduce a topology into S using the same method as that used to introduce a topology into the space M in Section 2 of the present chapter. Namely, with each point S E S we associate a fundamental system reS) of open sets r\x. A topology will be introduced into S in such a manner that the sets of the system reS) prove to be the intersections of the neighborhoods in a fundamental neighborhood system of the point y and r\x. We will now define the system of sets r(so)' Let So E r\X; then, for the system r(s) , we take a fundamental neighborhood system of the point s. Now let So = B R xo, where B is some nontrivial Satake subgroup of the group G. We denote the maximal unipotent normal subgroup of the group B by N. The orbit BR Xo fibers naturally into orbits of the group N R • There is a natural isomorphism between the orbits of the groups N into which the orbit B R x fibers and the Lie algebra SllR (recall that Sll is the maximal commutative subalgebra consisting of semisimple elements of the Lie algebra of the group B). We can use this isomorphism to carry this partial ordering in Sll over to the set of orbits. Let N Rx 1 be some orbit contained in BRx O, and let U(e, N Rx 1 ), where e > 0, be the set of all x E X such that (4)

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

155

where p is an invariant distance in X and Q X1 denotes the union of all orbits of the group NR that are larger than the orbit NR Xl in the sense of the partial ordering in the set of orbits. The projection of all sets of the form U(e, NR Xl) under the natural mapping of X onto r\X forms the system T(so). It is not difficult to verify that the family of sets obtained in this manner has the following properties: (1) if Ul E T(so) , U2 E T(so) , then there exists a U3 E T(so) such that

(2) if So E r\X, the intersection of aJI sets in the family T(so) coincides with the point So, (3) if So ¢ r\X, the intersection of all of the sets in the family T(so) is empty. We now define a topology in S in the following manner. Let So E S and Uo E T(so); denote by 0 0 the set of all s E S such that for each there exists a set Us E T(s) contained in Uo. It is not difficult to use properties 1, 2, and 3 to show that the family of sets 0 0 forms a fundamental neighborhood system for the point So. Borel's remarkable result consists of the fact that the space S is compact and Hausdorff. Even when Xhas a complex structure, however, the space S, as a rule; does not. We will now change the construction of the space S so that the space obtained does have a complex structure. Assume that Xis a complex symmetric space; then, as we know, Xis a bounded homogeneous domain in CII • Denote the set of bounded holomorphic hulls of the orbits {B R x} of the subgroups of the Satake group G by 1:. Moreover, set (5)

The natural mapping of L onto 1: induces a mapping of S onto S. Henceforth we will discuss S with the topology induced by the mapping S -+ S. In this topology S is compact. Below we will prove a lemma from which it follows that S coincides with the space M introduced in Section 2 of the present chapter and, therefore, is a complex compact sPFlce. Let ~ be a bounded homogeneous domain. A subalgebra R of the

156

THE GEOMETRY OF CLASSICAL DOMAINS

Lie algebra of all analytic automorphisms of the domain!» is called a Satake subalgebra if (1) R is solvable and split over R.

(2) a maximal solvable and split (over R) ideal of the normalizer of the algebra R coincides with R. We also agree to call a subgroup of the group of analytic automorphisms of the domain!» whose Lie algebra is a Satake sub algebra a Satake subgroup. We have the following lemma.

Lenuna. Let!» be a bounded homogeneous domain. A fibering into bounded holomorphic hulls of the orbits of the Satake subgroup B is a homogeneous analytic fibering of the domain!». We denote the normalizer of the group B in the group of all analytic automorphisms of the domain!» by sJC(B), while 'we denote a maximal commutative normal subgroup by A. The group ,3 of parallel translations of the given fibering is a maxim.al unipotent normal subgroup in the centralizer of the group A. We will first show how this lemma implies our statement about the space S and then we will prove it. lt is sufficient to show that the space we for the group r coincides with the space 1:. In order to do so, it is clearly necessary to show that: (1) a fibering into bounded holomorphic hulls of the orbits of a given Satake subgroup B of the group G is a r-rational fibering in the sense of Section 2 of the present chapter; (2) the maximal solvable and split over Q normal subgroup of the group of aut om orph isms of the domain !» that preserve the given r -rational fibering is a Satake subgroup B of the group G, and the fibering into bounded holomorphic hulls of the orbits of the group B coincides with the initially adopted fibering. We will first prove the first statement. Let B be some Satake subgroup of the group G. lt follows from the lemma that the fibering into bounded holomorphic hulls of the orbits of the group B R is a homogeneous analytic fibering. In order to show that this fibering is rrational in the sense of Section 2, it is sufficient to show that the group ,3 of parallel translations of this fibering is defined over the field of rational numbers Q. First note that the normalizer SJC(B) of the group B is an algebraic subgroup of the group G and is defined over the same field as B, i.e., over Q. Moreover, the maximal commutative normal subgroup A of the group SJC(B) and its centralizer ,3(A) is clearly an algebraic subgroup of the group G. As a result, the maximal unipotent normal subgroup of the group ,3(A) is an algebraic subgroup of the group G and

157

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

is defined over Q. It remains to note that this normal divisor, by the lemma, coincides with.8. We will now prove the second statement. Consider some r-rational fibering. As we noted in Section 2, the subgroup .8 of this fibering is an algebraic subgroup of the group G and is defined over Q. As a result, the normalizer 9((.8) of this subgroup in G is also defined over Q. The group ~(R(.8) coincides with the group of all analytic automorphisms of the domain !!fl that preserve the given r-rational fibering. Its maximal solvable and split (over R) normal subgroup B is generated by the group .8 and the directing subgroup of the given fibering (see Theorem 4, Section 8, Chapter 2). As a result, by what we proved in Section 4 of Chapter 1, the bounded holomorphic hull of the orbits of the group BR coincides with the fiber of the given fibering. Thus, we have proved, by means of the lemma, that 1: = and, therefore, S = M. It now remains to prove the lemma. Proof of the lemma. The normalizer m(B) contains a maximal solvable subgroup of the group of all analytic automorphisms of the domain !!fl and, consequently, is transitive in the domain !!fl. Let W denote the Lie algebra of the group ~(B). We can thus introduce the structure of a j-algebra into W in a natural manner (see Section 2, Chapter 2). As was proved in a paper contained in the appendix, any algebraic j-algebra can be repreEented in the form

we

(6)

vV=2'+j2'+K+U+W' ,

where 2' is a commutative ideal, J( is a compact algebra, j(K) = 0, W' is a semisimple algebra that is invariant with respect to j, and [u, U]

c

2',

[2'+j2'+K, W'] = 0,

[U, W']

c

U.

(7)

In addition, 2' contains an element 1o such that Ulo, I] = 1 for all = tu,

IE 2', Ulo, u)

(8) Consider the following subalgebra of the algebra W: L

2'+U+{jlo},

(9)

where {}lo} denotes the one-dimensional linear space generated by the vector }lo. It is not difficult to see that L is a solvable ideal of the algebra G and, therefore, is contained in the Lie algebra of the group B. With the

158

THE GEOMETRY OF CLASSICAL DOMAINS

representation of the Lie algebra W in the form (6) we can naturally associate a homogeneous fibering n of the domain ~ with the base ~', in which the Lie algebra W' is transitive, and fiber a Siegel domain of genus 2 in which a group whose Lie algebra is

is linear. It is not difficult to show (see Section 4, Chapter 1) that the bounded holomorphic hulls of the orbits of the groups expSP

and

exp(SP+jSP+K+U)

(10)

coincide with the fiber of the fibering n. The obvious relationship expSP c B c exp(SP+jSP+K+U)

(11)

implies that the bounded holomorphic hulls of the orbits of the group B are also a fiber of the given fibering. We will now show that the maximal commutative ideal of the algebra W coincides with SP. Let SP m be a maximal commutative ideal of the algebra W. It is clear that

SPlllcSP+jSP+K+U. It follows from (8) that SP m = SP n SP m+ U n SPm+(jSP+K) n SP m.

(12)

We will first show that (jSP+K)n SPill =0. Let gl =jll+klE(jSP+ K) n SP m • It follows from the definition of j-algebras (see Section 2, Chapter 2) that

[g 1,1 0] +j([jg l' 10J) +j([g 1,j 10]) - [jg 1,j 10] E K.

(13)

It follows from (8) that [jgl,loJ

= [gl,j IO] = 0,

[jlo,jgl] = [jlo, -1 1] = -1 1 , As a result, [gi' 10J = 11, Thus, if gl = jli +kl E SPill' we have 11 E SP m • Moreover,

(14)

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

159

But (14) contradicts the fact that the Lie algebra .fi'1II is commutative. The proof that .fi'1II n U = 0 is similar, but somewhat more complex. In order to prove this statement, we construct a differentiation ljJ for the j-algebra W. We set

(15)

W',

joEW'

where j is an endomorphism of the complex structure in Wand adujo

j

on

U o = [fli', UJ.

(16)

The existence of such an element jo in semisimple j-algebras is a well-known fact. It is not difficult to show that ljJ is a differentiation for thej-algebra W and, therefore, ljJ carries a maximal solvable ideal of the algebra W into itself. As a result, ljJ E W. We will now show that Un.fi'1II = 0. If U E Un .fi'IlP then ljJ(u) c Un .fi'1II. It follows from the definition of j-algebras that [ljJ(u), u] = [ju, u] :j:. 0, if u :j:. o. As a result, u = 0 and, therefore, Un .fi'1II = 0. We have proved that .fi'1II c .fi' and, therefore .fi'1II = .fi'. The centralizer of the ideal .fi' is clearly equal to .fi'+ U + W'.

(17)

A maximal nilpotent ideal of the algebra (17) is equal to and, therefore, coincident with, the Lie algebra of the group of parallel translations ,8 of our fibering. This completes the proof of the lemma. Section 7. The Andreotti-Grauert Method

Andreotti and Grauert [1] proposed a very elegant method for proving that the degree of transcendence of a field of automorphic functions is no greater than the complex dimension of their domain of existence. We will now present a brief outline of this method and show that it applies to our case. The ingenious notion of pseudoconcave boundary points plays a central role in this method. Let X be a strict subdomain of a domain ~, i.e., contained in X together with its closure. We denote the boundary of the domain ~ by X'. We agree to say that the domain X is pseudoconcave at a point

160

THE GEOMETRY OF CLASSICAL DOMAINS

X' if for any neighborhood U of the point Zo and any function ¢(z) that is regular in U there exists a point Zl E Un X such that

Zo E

(1)

Andreotti and Grauert proposed the following simple criterion for verification of pseudoconcavity. If there exists a two-dimensional complex plane E = E(tl' t2 ) passing through the point zo, an infinitely differentiable function q(z) defined on E, and a neighborhood of the point Zo in E such that LgaP~a~p > 0, q(z)
(}2q

gap

for

I

=-;-;::-\ uta utp z=zo

(2)

zEWnX,

then the point Zo is pseudoconcave. Now let r be a discrete group of analytic automorphisms of the domain~. We agree to say that the group r is pseudoconcave if there exists a subdomain X c ~ that is contained in ~ together with its closure and possesses the following property. For each point Zo of the boundary X' of the domain X, the orbit rzo of the point Zo contains a Zl that is either contained in X or is a pseudoconcave point of the boundary of X. The fundamental result of Andreotti and Grauert consists of the following. If the group r is pseudoconcave, then the degree of transcendence of its field of automorphic functions is no greater than the complex dimension of the domain ~. We will not prove this theorem here-we refer the reader to the appropriate paper. We should note that this proof is a modification of C. L. Siegel's proof that the degree of transcendence of a field of meromorphic functions on a compact complex manifold is no greater than its dimension (C. L. Siegel [8]). We will now show that the Andreotti-Grauert conditions hold true for arbitrary arithmetic groups.t Let r be an arithmetic group defined on a symmetric domain ~. As we showed in Section 6, the space M for it is compact. For each t The case in which r is commensurable with a group containing, as a factor, an arithmetic group defined on the Lobachevskian plane is an exception.

DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS

161

point mE M, we denote by UI/I the neighborhood with the following property: There exists a cylindrical set Om C !!fi whose projection onto !!fi/r coincides with Um n (!!fi/r) and is such that if y E r and the intersection ofy(Om) and Om is nonempty, we have y(Om) = 01/1' Compactness of the space M implies existence of a finite set n of points ml' ... , mk E M such that k

M=

L

U mi •

(3)

i=l

Assume that m 1 , ... , ms are all of the points in n that are not contained in !!fi/r and that 0 1 , ... , Os are the corresponding cylindrical sets. It follows from (3) that the fundamental domain F = !!fi /r is contained in the sum

(4) where Fo is a compact subset in !!fi. We now consider a mapping of the domain!!fi onto a Siegel domain St, 1 ~ t ~ S, such that all automorphisms of the domain !!fi that map Ot into itself are linear transformations of St. Moreover, let 3t be a subgroup of parallel translations of the domain St and let ~t = r n 3t. We represent an invariant volume du in St in the form du = (PrCz)) -1 dUe, where dUe is an affine volume. Then the function PrCz) will be invariant under all unimodular affine transformations of the domain St into itself, and, in particular, under the transformations in the group 3t. It is clear that Ot contains an open subset P t such that: (1) For any point z E 0 t there exists abE ~t such that b z EP t, and (2) The closure of the set PrCc) = {ZEPt,PrCZ) < c} is contained in !!fi for any c > O. It follows from (4) that we can select c so large that the set Fc = Fo +P 1 ( c) + ... + Pic) will have the following properties: (1) For any point Zo EFo there exists a Yo E r such that Yo Zo E Fc ' and (2) For each point Zo E!!fi contained in both boundary of P t and the boundary of 0 t there exists a Yo E r such that Yo Zo E Fc. We will now show that the set Fc selected in this manner has the properties of the set X that we discussed in the definition of pseudo·· concave groups. We split the boundary of the set Fc into two components !F 1 and !F 2' The component !F 1 consists of all points Z E Ot, 1 ~ t ~ S, such that Pt(z) = c. On the other hand,!F 2 consists of all remaining boundary points. It is clear that if Zo E!F 2' then the orbit F

162

THE GEOMETRY OF CLASSICAL DOMAINS

rZo of the point Zo is an interior point of Fc. Let Zo E!F l' As we know, the form

is positive definite at any point in the domain!». As a result, Zo is a pseudoconcave boundary point of the domain Fc. Thus, we have shown that any arithmetic group is a pseudoconcave group in the sense of Andreotti and Grauert.

CHAPTER 4

Automorphic forrl1s Introduction

In this chapter we will study automorphic forms for discrete groups of analytic automorphisms of symmetric domains. Certain results, e.g., the construction given in Section 1 for Fourier-Jacobi series, hold true for arbitrary Siegel domains of genus 3. In Section 3 we will use Fourier-Jacobi series to prove the theorem on algebraic relations for arbitrary arithmetic groups in symmetric domains. This proof is the third in this book. Two others, one based on a construction of a compactification, the other based on the Andreotti-Grauert method, were given in Chapter 3. In itself, the proof we will give for the theorem on algebraic relations is not, at the present time, particularly interesting. As far as we are concerned, however, its use of the apparatus provided by Fourier-Jacobi series is of value. In Section 2 we will study automorphic forms. At the end of Section 2 we will give an outline of Selberg's method for computing the dimension of spaces of automorphic forms of a given weight. Section 1. Fourier-Jacobi Series

Let S be a Siegel domain of genus 3 given, as usual, in the form (see Section 3, Chapter 1)

1m z- ReLtCu, u) E V,

tEf»

where V is some cone and f» is the base of the domain S. We will assume that the domain S was obtained by means of the construction given at the end of Section 3 of Chapter 1. Let 3 be the group of parallel translations of the domain S, and let 30 be the normal divisor of 3 consisting of all transformations of the form z-+z+a,

U-+U, 163

t-+t.

(1)

164

THE GEOMETRY OF CLASSICAL DOMAINS

Let r be a discrete group of analytic automorphisms of the domain S. We set r 0 = r n 20' r 1 = r n 2. In the present section we will assume that the factor space 2/r 1 is compact. As we showed in Section 3 of Chapter 3, in this case the numbers of generators of the groups r 0 and r dr0 are, respectively, n and 2m, where n is the dimension of 20 and 2m is the dimension of 2120' Since ji w) = 1 if Y E 2, each r -automorphic form is invariant under the transformations Y E r l' As a result, it can be expanded in a Fourier series (w) =

L l/Jp(u, t) e

21ti

(p.z)

(2)

p

where p runs through the set of linear functionals on the group 20 that are integer-valued on r o. Further study of automorphic forms in Siegel domains is based on a detailed investigation of the Fourier coefficients l/Jiu, t) in series (2). We will show that the functions (u, t) as functions of u are Jacobian functions. In connection with this, we will call series (2) a FourierJacobi series. Note the following relationship, which is necessary for what follows. Let (c l , a1) and (c 2, a2) belong to ~; then 2Q(e 1,c 2)Ero,

(3)

where Q(c 1 , C2) is defined in Section 3 of Chapter 1. Indeed,

The following relation is a consequence of the functional equation for an automorphic form:

fez + a +2iLtCu, e(t)) + iLtCe(t), e(t)), U + e(t), t)

fez, u, t)

(4)

= l/Jp(u, t)exp[ -2ni{p, a +2iLtCu, e(t))+iLtCe(t),e(t))}].

(5)

=

for all (c, a) E~. As a result, we find that

l/Jp(u + e(t), t) for the functions l/Jiu, t). The expression in the exponent is a linear function of u. We write it in the form (6)

165

AUTOMORPHIC FORMS

where bp(t) and [3p(t) are defined by the following relationship: (b p([3t),(U) = 4 n((p, Lt((U(, C(t)()),) ())}. p t) = 2 n p, L t c t), c t) - 2ni p, a

(7)

It is clear that bp(t) and [3it) depend on c. Let C(l), ... , c(2m) be a basis for the lattice of r' = fl.jrl. For the basis, relationship (5) may be written in the form

ljJ p( u + C(k) (t), t) = ljJ p( u, t) exp [{b~) (t), u} + [3~k)(t)],

(8)

where b~k)(t) and [3~k\t) are given by (7) upon substituting C(k)(t). Recall that Jacobian functions c(u) are functions that are analytic in an m-dimensional complex space and satisfy the relationships ljJ(U+Ck)

= ljJ(u)e(bk.u)+P\

k

= 1, ... ,2m,

(9)

where C l , ... , C2m is some set of vectors linearly independent over the reals. The matrix C whose columns are the vectors c 1 , ... , C2111 , is called the period matrix. It is known that functions satisfying relationships (9) need not exist for an arbitrary selection of vectors Ck and bk • Recall (see Siegel [1], pp. 50-4) that the following conditions are necessary for existence of functions satisfying (9) : (a) every element of the matrix R=

~(B'C-C'B)

2m

is an integer (here B is the matrix with columns b l , ... , b 2m ); ([3) the Hermitian matrix 1 _ H=-G'RG i

is negative definite (G is found from the conditions CG = E III , CG = 0). Furthermore, the dimension of the space of Jacobian functions with given Ck, bk, and [3k is always finite and no greater than 2111 ..jdR (dR is the greatest common divisor of the minors of a matrix R of maximal possible order for which the minors are not all equal to zero; in particular, if R is nondegenerate, then dR is the determinant of the matrix R). If the matrix H = i-1G'RG is positive definite, then the dimension of the space of Jacobian functions is equal to 2111 ..jdR (it is easy to verify that here the matrix R is nondegenerate).

166

THE GEOMETRY OF CLASSICAL DOMAINS

We will now prove that condition (0:) is satisfied. It follows from (7) that 2nirsk = (b s' ck) - (b k, cs) = 4n(p, L t ( C(k)(t), c(s)(t») - 4n(p, Lt(c(s)(t), C(k\t»)

= 4ni(p, Q(c s' ck» (rsk is the element of the matrix R that is located at the intersection of the sth row and the kth column). The fact that rsk is an integer follows from (3). We now turn to verification of condition ([3.) We write Llu, v) in the form L~l)(U, V)+L~2)(U, v), where L~l)(U, v) is the symmetric part of the form Llu, v) and L~2)(U, v) is the Hermitian part. As we know, such a representation is unique. Let wtCt E flJ) denote the set of all vectors p in an n-dimensional real space that are such that (p, LF)(u, u» ;;:;; 0 for all u.

(10)

We will prove that ([3) holds if and only if p E Wt. Note that (7) associates some vector function bpet) that is analytic on flJ with each vector function e(t) of our set so that if

c'(t) -+ then

b~(t),

c"(t) -+

b~(t),

(111 c' + 112 c")( t) -+ 111 b~( t) + 112b~( t)

for any real 111 and 112' As a result, the relationship between e(t) and bp(t) may be written in the form (11)

where the K~i)(t), t = 1,2 are square complex matrix functions of t that are linearly dependent on p. Generally speaking, the matrix functions K~i)(t) are not analytic. Substituting the expression for bpet) into (7), we can easily see that the following relations are valid: (K~l)(t)C(t), u) = 4n(p, L~l)(U, c(t»), } (K~2)(t)C(t), u) =

4n(p, L~2)(u, c(t»).

It is easy to verify that the matrix K~2)(t) is Hermitian.

(12)

AUTOMORPHIC FORMS

167

It follows from (12) that K~2)(t) is non-negative if and only if p E Wt • It remains for us to show that the matrix H given in ([3) is equal to K~2)(t)

up to a positive real factor. Substituting the expression for R in terms of Band C into the formula defining H, we find that . H = -(2n)-lG'(B'C-C'B)G = (2n)-lBG = (2n)-1(K~1\t)C + K~2)(t)C)G

= (2n)-1 K~2)(t).

Here we have used the relationships CG

= CG = E,m CG = O.

The following lemma follows directly from our discussion: Lemma 1. Let r be a discrete group of quasilinear transformations of the Siegel domain S, where the factor space B/(r n B) is compact. If the convex hull of the vectors LtCu, u) coincides with V' for all t in some open subset of the domain!», then any r-automorphicformf(w) will be bounded in any cylindrical domain. Proof We expand few) in a Fourier-Jacobi series. It follows from

our discussion that when the hypothesis of the lemma is satisfied, the Fourier coefficients are ljJp(u, t) == 0, if pE V'. It follows from this and Lemma 2 of Section 5 of Chapter 3 that the function few) is bounded in any cylindrical domain. The first such type of "effect" was, in another context, discovered, in fact, by the German mathematician Kocher while he was proving the following important theorem (Kocher [2]): Kocher's Theorem. Let H be Siegel's upper halfplane, i.e., the set of all complex symmetric matrices Z = X + i Y, where Y is positive definite. We denote by r the group of transformations of the domain H that have the form

Z-+A'ZA+S, where A is any unimodular integer matrix and S is any symmetric integer matrix. Then a r-invariantfunction analytic in H will be bounded in any domain consisting of points of the form Z + iT, where Z E Hand T is an arbitrary positive definite matrix.

The lemma proved above shows that the Kocher "effect" occurs, as a rule, for Siegel domains of genuses 2 and 3.

168

THE GEOMETRY OF CLASSICAL DOMAINS

We will now prove a lemma that includes Kocher's theorem as a special case. This lemma contains the conditions that must be imposed on a discrete group of affine transformations of a Siegel domain S of genus 1 in order for the Kocher "effect" to occur. As we know, the affine transformations of a Siegel domain of genus 1 S have the form z

-+

Az+a,

(13)

where A is the matrix of an affine transformation of cone V into itself and a is an arbitrary real vector. As a result, the group of all affine transformations of S into itself may be treated as a group of pairs (A, a) with law of composition (14)

We denote the subgroup of group G consisting of elements of the form (E, a) by d. Let G' denote the set of all affine transformations of cone V and let G~ denote the subgroup of G' that consists of the unimodular affine transformations of the cone V. Consider the natural homomorphism G -+ G'. If r A = r n d has as many generators as the dimension of the group d, then r' = r /r A is a subgroup of the group G~. Indeed, an automorphism (A, a)(O, b)(A, a)-l = (0, Ab)

of the lattice of r A corresponds to each (A, a). It is clear that the determinant of the matrix of an automorphism of the lattice equals ± l.

Lemma 2. Let r be a discrete group of linear transformations of the domain S such that the factor space G' /r', where r' = r /r A has finite volume. Then any r-invariant function is bounded on every cylindrical domain. Proof Letf(z) be some analytic r-invariant function. We expand it in a Fourier series: n

fez) = LAp e21ti (p,z), p

(p, z) = L

PkZk'

(15)

k=l

Note that invariance of fez) with respect to the group r entails a relationship between the coefficients Ap. Let (A, a) E r, then f(Az+ a) =fez) and, consequently, AA'p = Ap e21ti (p,a) , (16)

AUTOMORPHIC FORMS

169

where A' is defined by the condition (Ay, p) = (y, A'p). From now on, we agree to call the vectors P and A'p associate vectors. The lemma will be proved if we show that Apo = 0 for any PoE V'. We will prove this fact by contradiction. Let Apo :j:. 0 for some PoE V'. We denote an arbitrary point in V by Yo and fix it. By Parseval's equality,

L IApl2 e- 4n

(p,yo)

<

00.

p

We denote the different P that are associates of Po by Mo. From (16) we find that

L

e- 4n (p,yo) <

00.

(17)

pEMo

We can easily verify that there is a one-parameter subgroup g(-r) in the group G' such that lim (g(-r)po, Yo) = -

00.

t ..... 00

By Selberg [3], the group r ' contains an infinite number of elements of the form Bl g(-r)B2' -r > 0, where Bl and B2 are arbitrarily close to unity. It is not difficult to show that, for sufficiently small Bl and B 2 , lim (Bl g(-r)B2 Po, Yo) = t .....

+ 00

00.

As a result, there is a sequence of elements Ap in the group r that lim (A~po, Po) = -

00.

p ..... oo

This clearly contradicts (17). The lemma is proved. We now turn to the proof of a lemma that we shall use in the next section to find a bound for the dimension of a space of automorphic forms. Lemma 3. Let S(Ql' 1'1) c S(Q2, 1'2) be two cylindrical domains, Q1 C Q2, 1'1 - 1'2 E V. For any r-automorphic form few) that is "analytic at infinity" and whose Fourier coefficients ljJp(u, t) are equal to zero when Ipl < -r, we have the inequality

sup

If(w)1 <

C1

e-

C2t

WES(Q1, "1)

where c 1 and c 2 depend only on r

sup

If(w)l,

WES(Q2, "2)

= r1 -

1'2

and are independent of -r.

(18)

170

THE GEOMETRY OF CLASSICAL DOMAINS

Proof Direct integration may be used to prove the following formulas: !(z,u,t)

= _l_J"!(Z-X-ir)f0(X;r,i)dX,

f0(x;r,i)

=

mesL

I

1

(19)

e21ti (p,x+ir), J

peV', Ipl ~t

where L is the fundamental parallelepiped of the lattice of r o' It follows from (19) that sup weS(Ql,l"l)

1!(w)l<

-J

1 1!(w)IweS(Q2,r2) 111es L sup

1f0(x;r,i)ldx.

(20)

L

It remains for us to estimate the integral in the right side of (20). We have -1mesL

J L

( J

)1/2

( I_

(I

If»(x; r, i)12 dx ~ -1mesL

~

1f0(x; r, i)12 dx

L

e- 41t (p,r»)t =

Ipi ~t,peV'

/

A(k)e- 41tk)1

2 ,

where A(k) is the number of solutions for the equation

(p, r) = k,

pE V',

Ipl ~

i.

The relation rEV implies that there are positive constants C 3 and C4 such that c 3 1pl < (p, r) < c4 lpl. This implies that: (1) A(k) = O(k"), where n is the dimension of the cone V; (2) A(k) = 0 if k < Cs i, where Cs is some constant. As a result, (21)

the lemma follows directly from (21) and (22). The following lemma is the basis of the ordinary method of estimating the dimension of a space of automorphic forms when the fundamental domain is compact.

Lemma 4. Let A be SOlne domain in an n-dimensional complex space, B be a domain strictly contained in A, i.e., 13 c A, and let E be some

AUTOMORPHIC FORMS

171

If there

exists a constant M such

linear space of analytic functions fez). that sup\f(z)\ ~ M sup\f(z)\ zeA

for any fez) EE,

(22)

::eB

then the dimension of E is finite and no greater than y(ln M)N where y depends only on A and B. Proof Denote the distance between B and the boundary of A by 2p. In B we select a finite system P of points Zl' .•. , Zq that has the following property: For any point zEB, there is a point ZkEP such that IZ-Zkl < p. It is clear that in order for the function fez) to have zeros of order no less than In at the points Zl' ... , Zq' it is sufficient to impose qmll linear homogeneous relations on the functionf(z). Thus, if the dimension of E were larger than qnlt, then E would contain functions having zeros of order no less than m at the points Zl' ..• , Zq. Using the hypothesis of the lemma, we will now prove that E may not contain a function fez) that has zeros of order strictly greater than (1n M)/(ln2) at the points Zl, .•. , Zq. We assume the contrary. Let Zo be a point at which the modulus If(z) I attains its maximum in B. There exists a point Zk EP such that Izo -zkl < p. It is clear that for any complex A, 1,1.1 ~ 2, the point Zk+},(ZO-Zk) is contained in A. The function

is regular in the disk 1,1.1 ~ 2 and, consequently, has its maximum on the boundary of this disk. Thus, A contains a point z' such that \f(z')\ ~ 211l max \fez),. zeB

As a result, 11l max \fez')\ ~ 2 max 'fez)', zeA

111

zeB

lnM] = [- +1. In2

(23)

But (22) and (23) contradict each other, whence follows the lemma. We will introduce some definitions before we state Lemma 5. For the Siegel domain S under discussion, let G be the set of all quasilinear transformations of the following form: Z -+

z + a(u, t),

U -+

B(t)u + bet), t

We denote the factor group G/z by G •

t -+ get).

(24)

172

THE GEOMETRY OF CLASSICAL DOMAINS

Lemma 5. Let r be a discrete subgroup of the group G such that the factor spaces G' /r' and z/r 1 are compact, where r 1 = r n L1, r' r /r l' In addition, let E(p, J1) be the set of allfunctions l/Jin, t) that are analytic in !» x em and may have any r -autOlnorphic fonn of weight J1 as the coefficient of e2ni (p, z) in the Fourier series. The dimension of the space E(p, J1) isjinite, and does not exceed

c(lpl + J1)III+\

(25)

where m is the complex dimension of the space of u, while k is the complex dimension of!» and c is a constant depending only on r. Proof Let z -+ z + a(u, t), u -+ B(t)u + bet), t -+ get) (26)

be some quasilinear transformation of domain S that belongs to the discrete group r. It follows from the functional equation of the automorphic form that j-Il(wfil/J p(u, t) e2ni (p,z) = p

L l/J p(B(t)u + bet), get)) e2ni (p,z+a(lI,t)), p

where jew) is the Jacobian of transformation (26). It is immediately clear that jew) depends on t and g, whence we obtain the following functional equation for the functions l/J p(u, t):

l/J p(u, t)

l/J p(B(t)u + bet), get)) e2ni (p, a(lI, t))(xit))Il,

(27)

where xit) is the Jacobianj(w) of transformation (26). In addition, as we proved earlier, the functions l/J p(u, t) satisfy functional equation (8). It follows from (27) and (8) that the functions l/J p(u, t) may be treated as forms that are automorphic with respect to some discrete group K and are analytic in the domain!» x elll • It follows from the hypothesis of the lemma that the factor space!» x em / K is compact. Thus we may apply the ordinary method of estimating the dimension of the space of automorphic forms, which method is based on Lemma 4, which we proved above. Let B be the fundamental domain of the group K and let A be some open set in !» x em that contains the fundamental domain B and its closure. We may obtain the following bound by means of the ordinary method and functional equations (8) and (27) : sup Il/Jiu, t)1 ;;.;;; eC1 (lpl +Il) sup (lI,t)eA (lI,t)eB

Il/J peLt, t)l.

(28)

AUTOMORPHIC FORMS

173

It immediately follows from this bound and Lemma 4 that the dimension of the space E(p, p) is finite and, furthermore, that it does not exceed

where c depends only on the group r. Lemma 5 is completely proved. Section 2. Automorphic Forms

Let!» be a bounded homogeneous domain, and let r be an arbitrary discrete group. In the present section we will describe construction of a scalar product in the space of r-automorphic forms. At the end of the section we will give a brief outline of A. Selberg's method for computing the dimension of automorphic forms of a given weight. We now turn to a description of the scalar product in the space of automorphic forms of weight m. Following Peterson, we define the scalar products as an integral over the fundamental domain B = !»/r. Let p(z) be a continuous positive function satisfying the functional equation p(gz)

= p(z)ljiz)12 for all g E G

(1)

(G is the full group of analytic automorphisms of the domain !»). Such a function p(z) is defined uniquely up to a constant factor. Let /1 and 12 be functions satisfying the functional equation for automorphic forms of weight 111. We set

(f,J2l =

J/l J,

p"(z) dv.

(2)

It is easy to verify that integral (2) is independent of the selection of the . fundamental domain B. As we know, integral (2) does not exist for all automorphic forms. Notation: m(r,111) is the space of forms automorphic with respect to a given discrete group r that satisfy the following additional condition. Consider an arbitrary r-rational homogeneous fibering, and let cx(z) be the Jacobian of the mapping of the domain!» onto the corresponding Siegel domain of genus 3. The additional condition consists in requiring that the automorphic form/ofweight m belongs to mer, m) if and only if

174

THE GEOMETRY OF CLASSICAL DOMAINS

the function/(z)a-m(z) is bounded in any cylindrical domaint associated with the given fibering of the domain ~. A(r, m) is the space of automorphic forms of weight 111 for which (j, I) < + 00. We will prove that the space A(r, 111) is complete as a Hilbert space. Indeed, let the sequence of functions flz) E A(r, m) satisfy Cauchy's criterion, i.e., for any e > 0 there exists an n. nee) such that for all

n1 > nee), n2 > nee), (fill -1,'2,1,'1 -1,,) < e. It is easy to use the functional equation of an automorphic form to show that

where ~1 is any fixed subdomain of the domain ~ such that ~1 c ~. Using the general properties of analytic functions, we can easily use this fact to show that the sequence of functionsf,(z) uniformly converges in any subdomain ~1 of the domain ~ to some function lo(z). The functional equation of an automorphic form of weight 111 will clearly be satisfied by the limit function/o(z). We denote the upper bound of the numbers (f" Ill) by c. It is clear that

f

I

lJo(z) 2plll(Z) dv

~

Bn501

lim

r f,,(z) I2plll(Z) dv ~ C,

II->OOJB

where ~1 is an arbitrary sub domain of the domain result,

(fo'/o) Thus,

=

fo(z)

~, ~1 c~.

As a

fB IfoCzWp"'Cz)dv "" C. = limf,,(z) E A(r, 111). /1-> 00

We will now show that A(r, m) c

t Recall

mer, m).

that a cylindrical domain associated with a given fibering is a subset with the following properties: (1) the projection of P onto the base !!)' of the fibering is strictly contained in !!)' (i.e., its closure belongs to !!)'), (2) the intersection of P with any fiber is contained in the bounded holomorphic hull of some orbit of the group of parallel translations.

pc!!)

175

AUTOMORPHIC FORMS

Let T be some sufficiently small cylindrical domain with base Q in some r-rational component fF. We will assume that T is chosen so that if two points Zl and Zz are equivalent with respect to r (i.e., Zz = '}'Zl' where ,},Er), then ,},Er1(fF). When Tis selected this way, we have the inequality (3) Map the domain f» onto the corresponding canonical Siegel domain S. After substitution of variables, integral (3) takes the form (4) where A(W) is a solution of equation (1) in the domain S, E>y(w) = J(¢(W))jlll(W) , and dv is an invariant volume. The function E> yew) is invariant under the transformations in the group r 3(fF) so it may be expanded in a Fourier-Jacobi series E>y(W) = t/J /u, t) eZ1ti (p,z), (5)

I

p

It is easy to verify that A(W) = A1(y-ReLi(u, U))Az(t) and that an invariant volume has the form

Let A(yo, uo, to, e) be the set of points W = (z, U, t) E S such that

u-uol < e,

t-tol < e,

IY-Ayol < e,

A ~ 1,

xEL,

where L is the fundamental domain of the group r 4(fF). Finiteness of the integral

f

A(yo, 110, to; B) i

It/Jp(U, t)eZ1ti(p,Z)lzAIIl-l(W)dXdYdUlduzdtl dt z

(6)

P

follows from (4) when Yo, uo, to, and e are suitably chosen, and, therefore (after integrating with respect to x), the sum

ff

A'''-'(w)i'''p(u, tW e- 4 ,,(p,y) dy du, dU 2 dt, dt2

(7)

176

THE GEOMETRY OF CLASSICAL DOMAINS

is also finite. It is easy to verify that the individual terms in this sum may be finite only when P E V'. It follows from this and Lemma 2 of Chapter 3, Section 5 that the function 03""(z) is finite in any cylindrical domain with base in fF. Thus, we have proved that (j, f) < CfJ implies that condition b is satisfied. It is not difficult to interpret the space mer, m) as the set of all crosssections of some analytic fibering into complex lines over M, where M is the space introduced into Chapter 3. As a result, if M is compact, the dimension ofm(r, m) is finite. The dimension of A(r, m) is afOl·tiori finite. The technique ofFourier-l acobi series (see Section 3 ofthe present chapter) can be used to reduce computation of the dimension of mer, m) to computation of the dimension of A(r, 111). A. Selberg [1] gave a method making it possible to do this for the case in which f» is a symmetric domain. We should note that, as a rule, dimA(r,m) = CfJ for nonsymmetric domains. We will now briefly summarize his method. Let km(z, u) denote a function of z E f» and u E f» that has the following two properties:

(1) the integral operator

f..

km(z, u)f(u) dv

(8)

commutes with the operators Tgf(z) = f(gZ)j;'(Z); (2) the function k(z, u) is analytic with respect to z. It is easy to show that property (1) is equivalent to the following functional equation: knlgz, gu)j;l(z)j";m(u) = k lll (z, u)

for all g E G (G is the full group of analytic automorphisms of the domain f»). Apparently, this proposition can be proved for all arithmetic groups. As far as the author knows, however, the argument has never been completed. We should note that, for arithmetic groups, asymptotic formula (14) is a corollary of general theorems on the dimension of the zero-dimensional cohomology group of a coherent sheaf. R. P. Langlands [1] gave an exact formula for the dimension of a space of automorphic forms of weight m that is applicable to the case in which the fundamental domain of the group r is compact.

AUTOMORPHIC FORMS

177

Section 3. The Theorem. on Algebraic Relations It is well known Siegel [7] that the theorem on algebraic relations is a simple corollary of the following estimate of the dimension N m of the space mer, m) (1)

where Nis the complex dimension of the domain f». In the present section we will prove estimate (1) for a certain class of discrete groups (quasinormal discrete groups) in symmetric domains f». It can be shown that all arithmetic groups are quasinormal discrete groups. We will not present the proof of this proposition, for it is very similar to the arguments that we used to prove the Andreotti-Grauert conditions for arbitrary arithmetic groups. Definition of quasinormal discrete groups: Let f» be a symmetric domain, and let r be a discrete group of analytic automorphisms of the domain f». The group r is said to be quasinormal if there exists a finite number of homogeneous r-rational fiberings with bases g; 1, ... , g; p that satisfy the following conditions: (l) there exist, corresponding to the given fibe_ring, cylindrical sets = f»jr is entirely contained in

T 1 , ... , Tp with the property that the fundamental domain B

(2) (2) the fundamental domains g;dr~ are compact, where r~ is the group induced on g;k (see Chapter 3, Section 2); (3) there exist, corresponding to the given fibering, cylindrical sets T~, K = 1, ... ,j}, with the properties that Tk C T~ and for any'}' Er and any k and I (3)

where C1 is constant and Ct:lz) is the Jacobian of the mapping of the domain f» onto the corresponding Siegel domain. Let jEm(r, m), and set 0 k(z) = f(z)Ct:"m(z), 1 ~ k ~ p. It follows . from the definition of mer, m) that 0 k (z) is bounded in any cylindrical domain corresponding to the given fibering with base g;k' We now turn to the proof of estimate (1).

178

THE GEOMETRY OF CLASSICAL DOMAINS

Theorem 1. Let r be a quasinormal discrete group of analytic automorphisms of some classical domain f». The dimension of the space EJ! of aut0l11OJ1Jhic forms of weight p, is finite and does not exceed (4) where C2 is a constant depending onlv on rand N is the complex dimension of the domain f». Proof Let M' be the maximum modulus lE>k(Z)1 in the domains Tk and let Mbe the maximum of Iek(z) I in the domains Tk. It follows from (3) that for any functionf(z) E E that is not identically zero we have the inequality (5)

where C 3 = Inle!l. The function E>k(Z) may be expanded in a Fourier-Jacobi series. It follows from Lemma 3 of Section 1 that if all the coefficients t/J /u, t) of the functions E>iz) vanish when Ipi ~ 'r, then sup \E>k(Z)\ < zeT k

C4

e- CST sup \E>k(z)l,

(6)

zeT'k

where C4 and c 5 are constants. It is clear from (5) and (6) that there exists a C6 such that if all the Fourier coefficients t/J p(u, t) of the functions ek(z) are equal to zero when Ipi < C 6 p, for all k(1 ~ k ~ p), thenf(z) == O. It remains for us to note that in view of Lemma 5 of Section 1, it is sufficient to impose no more than C 7 p,"+lII+k = C 7 p,N linear homogeneous conditions on fez) for these coefficients to become zero. The theorem follows from this.

CHAPTER 5

Abelian modular functions Section 1. Statement of Fundamental Results

One of the most interesting and important classes of arithmetic groups is the class of groups to which the theory of abelian functions can be applied.t This chapter is devoted to such groups. First of all, we will recall the classical connection between elliptic functions and ordinary modular functions; in essence, this connection led Gauss to discover the latter. As we know, elliptic functions are doubly period meromorphic functions of one complex variable. The set of elliptic functions with a given period lattice forms a field. A pair of periods COl' CO2 is said to be fundamental if any vector in the period lattice is an integral combination of these periods. Modular functions and modular groups appear naturally in the study of the manifold F of all nonisomorphic fields of elliptic functions. Modular functions are functions on the manifold F, or, in other words, functions on a pair offundamental periods with identical values on pairs of periods corresponding to isomorphic fields of elliptic functions. As we know, two fields of elliptic functions are isomorphic if and only if the period lattices corresponding to them can be obtained from each other by a linear transformation (z -+ o::z) of the complex plane. As a result, we can construct the manifold F in the following manner. Consider the set Q of pairs of complex numbers (COl' CO 2 ) such that the ratio C02/C01 is not real. Fields with fundamental periods (COl' CO2) and (co~, CO2) are isomorphic if and only if there exists a complex 0:: and an integer matrix A with determinant ± 1 such that (1)

t Recall that an abelian function is a merom orphic function in CP that has 2p periods that are linearly independent over the field of real numbers. 179

180

THE GEOMETRY OF CLASSICAL DOMAINS

The manifold F can clearly be obtained by identification of all pairs (COl' CO 2 ) that correspond to isomorphic fields of elliptic functions. This identification can be carried out in two ways. First we identify the pairs (COl' CO2) and (aco l , aco 2 ), where a is an arbitrary nonzero complex number. This leads to a set consisting of the complex plane with the real axis removed. Let K denote the connected component of the set we have obtained. For example, we can assume that K is the upper halfplane: Im7: >0. It is easy to verify that two points 7:1 and 7:2 correspond to isomorphic fields of elliptic functions if and only if there exists an integer matrix A with determinant = ± 1 such that a7: l +b

7:2=--C7: l +d'

Thus, F can be treated as a factor space Kjr, where K is the upper halfplane and r is a discrete group of analytic automorphisms of the domain K. The group r obtained with this method is called a modular group. Unfortunately, the technique we have given above does not generalize immediately to the case of p > 1 variables. The difficulty lies in the fact that the set F of nonisomorphic fields of abelian functions of a given number p of variables is not a manifold when p ~ 2. The reason for this is that not any 2p vectors that are linearly independent over the reals in a p-dimensional complex space can be periods of a nondegeneratet abelian function. It is well known that systems of periods of nondegenerate abelian functions form an everywhere dense set in the manifold of all systems of 2p vectors independent over the reals in a p-dimensional complex space. We must introduce the following change into our method. In the set of systems of periods we select analytic manifolds n and apply the technique described above for constructing an ordinary modular group to each such manifold. In this case, we do not only identify systems of periods that correspond to isomorphic fields of abelian functions, but systems of periods in which the isomorphism of fields of abelian functions extends to some neighborhood. (The neighborhood is selected in the given manifold n.)

t An abelian function of p variables is said to be nondegenerate if there are p analytically independent shifts in its set of shifts, i.e., functions of the form/(z + r), where rECp.

ABELIAN MODULAR FUNCTIONS

181

The manifold F obtained with this method from Q can be treated, as we will prove in this chapter, as a factor space Kjr, where K is some classical domain and r is a discrete group of analytic automorphisms of the domain K. This group r is called the modular group corresponding to the manifold Q. Thus, there is an infinite number of different modular groups associated with the abelian functions of a given number of variables. Some of these modular groups, in contrast to the ordinary modular groups, have compact fundamental domains. Sections 2 and 3 of this chapter contain a classification of all modular groups associated with abelian varieties. We will now present a technique for constructing modular groups associated with abelian functions more concretely. Our description of the manifold Q uses certain notions drawn from the theory of "complex multiplication", which notions we will now restate. Let OJ denote a matrix whose columns are the fundamental periods of some abelia,n function of p variables. As we know, the Riemann-Frobenius conditions for the periods of a non degenerate abelian function of p variables consists in the following: There exists a rational 2p x 2p skew symmetric matrix R such that OJROJ'

= 0,

iwROJ'

>

o.

(2)

The matrix R is called the principal matrix for matrix OJ. Generally speaking, there is no single principal matrix for a matrix OJ. The set of all principal matrices for a given matrix OJ forms a cone [l}l. Let A be an integer square matrix for which there exists a p x p complex matrix a such that OJA

=

aOJ.

(3)

The set D of all such matrices A forms a ring. Such matrices A are sometimes called "multipliers". This is connected with the following circumstances. Let P denote the set of all meromorphic functionsJ(z) in CP whose periods are the columns in a matrix OJ. The mapping (z)f -+ J(az), where a is defined by (3), is clearly an endomorphism of the field P. Let Wdenote the set of all matrices of the form

IrkAk k where the rk are rational numbers.

182

THE GEOMETRY OF CLASSICAL DOMAINS

Henceforth we will call the algebra mthe algebra of endomorphisms of a field of abelian functions. It is well known that the algebra has a positivet involution A ---7 A a = RA'R - 1

m

where R is an arbitrary principal matrix. These properties are characteristic, i.e., we have the following theorem. Theorem. 1. Let mbe an algebra of rationalm.atrices of order 2p, and let R be a skew symmetric matrix. If the lnapping (4) is a positive involution of the algebra m, then there exists a matrix w for which R is a principal matrix and mis the algebra of all endomorphisms (A. A. Albert [1]). Let Q(m, R) denote the set of all matrices w for which (1)

wRw'

m

0,

iwRw' > 0,

Em

is the algebra of endomorphisms for w, i.e., for any A (2) there exists a p x p complex matrix a such that wA = aw. It is clear that if WE Q(m, R), then {3w E Q(m, R), where {3 is any nondegenerate complex matrix. We agree to say that the matrices w and {3w are equivalent. We denote the set of classes of equivalent matrices wEQ(m,R) by K(m,R). We have the following theorem. (Section 2 of this chapter.) Theorem 2. K(9.(, R) is the product of classical domains of the first three types (see Chapter 2): The dOl1iain K(m, R) depends only on the real span m of the algebra m. It is clear that the fields of abelian functions corresponding to the matrices wand {3w, where {3 is an arbitrary nondegenerate matrix, are isomorphic. However, this is not the only case in which they are isomorphic. It is well known that two fields of abelian functions with period matrices w 1 and W 2 are isomorphic if and only if there exist matrices U and {3 such that (5) where {3 is a complex nondegenerate matrix and U is a unimodular integer matrix.

t An involution A ---7 Acr of the algebra Sll is said to be positive if S(AAcr) > 0, where S(B) is the trace of the matrix B.

ABELIAN MODULAR FUNCTIONS

183

It follows from (5) that vm 2 V- 1 =

m1 , Vf!lt2 V' =

f!lt 1 ,

(6)

where m1 and m2 are the corresponding algebras of endomorphisms, while f!lt 1 and f!lt 2 are the corresponding sets of principal matrices. It can be shown that the algebras of endomorphisms for all co EQ(m, R), except for the union of a countable number of submanifolds of smaller dimension, coincide with m. If one principal matrix for a matrix co is Ro and the algebra of all endomorphisms is m, then, as we can show with little difficulty, any principal matrix is of the form AR o, where A Em, AD' = A, and all characteristic roots of A are positive. The cone of principal matrices is therefore uniquely defined by one principal matrix Ro and the algebra mof endomorphisms. As a result, for all co E (m, R) with the same position (i.e., for which the algebra of endomorphisms coincides with m), the cone f!lt of principal matrices is the same. Let L(m, R) denote the set of all unimodular integer matrices V such that V2i.v- 1

=

m,

Vf!ltV'

= f!lt.

(7)

With each matrix V E L(m, R) we naturally associate the following transformation in Q EL(m, R): co

---7

coV.

(8)

Some analytic automorphism in K(m, R) clearly corresponds to each transformation of the form (8). We should note that some of the transformations of the form (8) induce identity transformations in K(m, R). We denote the group of transformations obtained in K(m, R) by rCA, R). We have the following assertion. Theorem 3. The group R) is an arithmetic group (Section 2). Note that, as a rule, R) does not coincide with the set of all integer matrices of some linear algebraic group; instead, it is some extension of this set. In what follows we will call the group R) a modular group, and we will call the meromorphic functions that are invariant under R) modular functions. The following theorem is a consequence of Theorem 3.

rem, crm,

rem,

rem,

184

THE GEOMETRY OF CLASSICAL DOMAINS

Theorem. 4. Afield of modular functions is afinite algebraic extension of a field of rational functions of n variables, where n is the complex dimension of the domain K(m, R). So-called modular abelian functions are interesting in certain cases. We will now define them. Consider the space C(m, R) of pairs (co, z), where co EQ(m, R) and z is a p-dimensional complex vector. A function f(co, z) that is meromorphic on C(m, R) is said to be a modular abelian function if it is invariant with respect to the following transformations:

(co, z) -7 (J3co, J3z)

(1)

(9)

where J3 is any nondegenerate matrix of order p, (10)

(2) where

CO k

is any column vector in the matrix co,

(co, z) -7 (coU, z)

(3)

(11)

where UEL(m, R). We have the following proposition. Theorem 5. A field of modular abelian functions is a finite algebraic extension of a field of rational functions of n + p variables, where n is the complex dimension of K(m, R). Theorem 5 can be used with an ordinary method to prove that each modular abelian function can be represented in the form of the ratio of modular abelian forms. These are functions that are holomorphic in C(m, R) and multiplied, under transformations (9), (10), and (11), by some factor. An example of such functions is the theta-function. The general theory of modular abelian forms is very interesting and, as far as we are concerned, may have important applications to the theory of numbers.

Section 2. The Domains K(9J, R) In this section we will present an explicit description of the domains K(m, R) and, in particular, we will prove Theorem 2 of Section 1. Let 1 be an algebra over Q of rational matrices of order 2p, and let Rl be some skew symmetric matrix defining a positive involution in 1 ; assume, moreover, that 2 is some other algebra and that R2 is a skew symmetric matrix defining a positive involution in m2 •

m

m

m

ABELIAN MODULAR FUNCTIONS

185

We agree to say that two pairs (m!, R!) and (m 2 , R 2 ) are equivalent if there exists a unimodular integer matrix U such that (1)

where R k , k = 1,2, denotes the set of all skew symmetric matrices R defining positive involutions in mb k = 1,2. If, however, there exists a rational and not necessarily unimodular matrix U for which (1) is valid, then we agree to say that the pairs (m!, R!) and (m 2 , R 2 ) are isomorphic. Let Si( denote the real span of the algebra m, i.e., the set of all matrices of the form

where the rk are real numbers. R is similarly defined. We agree to say that two pairs (m!, R!) and (m 2 , R 2 ) are equivalent over the reals if there exists a real matrix U such that (2) It is not difficult to prove that:

(1) If two pairs (m!, R 1 ) and (m 2 , R 2 ) are equivalent over the reals, the domains K(m 1 , Rd and K(m 2 , R 2 ) are analytically equivalent. (2) If two pairs (m 1 , R 1 ) and (m 2 , R 2 ) are isogenous, the modular groups r(~!, R 1 ) and r(m 2 , R 2 ) are commensurable. The algebra 21 is always semisimple and, therefore, is the sum of simple algebras m1 , .•. , As a result, there exists a basis in which all of the matrices in mhave the form

mm.

(3)

It is easy to verify that, in this basis, the matrix R is always of the form (3). We will show that

(4)

186

THE GEOMETRY OF CLASSICAL DOMAINS

In order to do so, it is sufficient to show that each class of equivalent matrices OJ contains a matrix of the form

OJ

=(:'

~).

0

o

(5)

OJ m

Let OJ be some matrix in Q(m, R), and consider the algebra of all complex matrices (X such that (XOJ = OJA, where A E m. This algebra is isomorphic to the algebra mand, consequently, is also the sum of simple algebras isomorphic to the algebras m1 , ... , mm. In this proper basis, all matrices (X of this algebra are of quasidiagonal form

(6)

A change of basis is equivalent to substitution of f30J for OJ, where f3 is some p x p complex matrix. We have thus showed that every class of equivalent matrices OJ in Q(m, R) contains a class in which all matrices (X are of quasidiagonal form (6). It immediately follows from the relationship (XOJ = OJA that such an OJ must be of the form (5). We have thus proved (4). It follows from our result that when we are trying to find K(m, R), it is sufficient to restrict the discussion to the case in which mis a simple algebra. We will now show how to reduce all cases to the case in which ~( is a division algebra. As we know, every simple algebra is a matrix algebra over a division algebra. Let mbe a simple algebra consisting of order 2p matrices. By 1 we denote its maximal division subalgebra. Then there exists an integer m such that all of the elements A E mare of the form

m

_(a~l

1m

a

)

A-. a m1

, •••

amm i

(7)

ABELIAN MODULAR FUNCTIONS

Consider the following involution in

A

-+

AO' = (b KS )'

187

m: bKS = a~K'

(8)

mi'

where a -+ aO' is a positive involution in lt is known that any positive involution is of the form

A-+A' = H-iAO'H where HO' = Hand His positive definite (this means that any representation maps H into a matrix with positive characteristic roots). We select a basis in the space of representations so that the matrices A E are of the form

m

(9) where au -+ Au is a fixed (the same for all i, j) representation of the algebra lt is easy to verify that there exists a matrix R Ef!ll that, in this basis, is quasidiagonal:

mi'

R=

(~1 o

~) ..

(10)

Rl

We will now show that each equivalence class of matrices contains a matrix of the form

WE Q(m, R)

(11)

Let W be an arbitrary matrix in Q(m, R). The algebra of all matrices ex such that exw = wA, where A E m, is isomorphic to the algebra and, consequently, is a matrix algebra over some algebra isomorphic to the algebra

a

mi'

188

THE GEOMETRY OF CLASSICAL DOMAINS

In the proper basis, all matrices (X of this algebra are of the form

(X

(

(x~i

•••

(Xim)

=. (XIII 1

, ..•

(12)

(XIII1/!

mi'

where the (Xij generate an algebra isomorphic to the algebra As we have already noted above, a change of basis is equivalent to substitution of f30J for OJ, where f3 is some p x p complex matrix. It remains to note that if (XOJ = OJA, where A is of the form (9) and (X is of the form (12), then OJ must be of the form (11). We have proved that K(m, R) = K(m i , R1)' Consequently, to find K(m, R), we can limit discussion to the case in which is a division algebra. Let be some division algebra over the field of rational numbers Q, and assume that it has a positive involution a -+ aO'. Note that all remaining positive involutions a -+ at are of the form at = baO'b -1, where bO' = band S(b k ) > 0 for all k, where S(b) denotes the trace of the element b. Our problem consists in describing all representations a -+ A(a) of the algebra mfor which there is a skew symmetric matrix R such that

m

m

(13)

where a -+ aO' is the given involution in m. Henceforth we will call the matrix R a principal matrix. It is clear that if there exists a matrix R such that (13) holds for one positive involution, then there is also a matrix R with analogous properties for any other positive involution. Let R denote the set of all skew symmetric matrices R defining positive involutions in the given space of representations of the algebra m. We agree to say that two representations a -+ A1(a) and a -+ A2(a) are isogenous if there exists a rational (not necessarily unimodular) matrix U such that (14)

VR i V' = R 2 , where R1 and R2 are cones of principal matrices.

(15)

189

ABELIAN MODULAR FUNCTIONS

Since, for any two representations of the algebra mby matrices of the same order, there exists a matrix U for which (14) holds, classification of nonisogenous representations of the algebra reduces to the following problem. Given some representation of the algebra m describe all possible cones R for this representation, where the cones Rl and R2 are clearly not assumed to be different if there exists a rational matrix U such that

m

UR 1 U ' = R 2 ,

UA = AU

for all

A Em.

(16)

Note, moreover, that each cone R has a skew symmetric matrix Ro such that for all A E m (17) where a ---7 aD' is some fixed (to the end of the proof of the theorem) positive involution in the algebra m. This matrix is naturally defined nonuniquely. The general form of such matrices is (18) where aD' = a and a belong to the center of the algebra m. In the space in which a representation of the algebra is defined, consider the scalar product (x, y) = x'Ro 1 y. It clearly has the following properties:

m

(x, y)

= -(y, x),

(Ax, y) = (x, AD'y)

for all

(19) A Em.

(20)

The converse is also clear, i.e., it is clear that a scalar product with properties (19) and (20) can easily be used to restore the matrix Ro. As we know, any representation of the algebra mcan be described in the following manner. Consider the vectors

(21)

where XiEm. The set of vectors of the form (21) forms a linear space X of dimension kl' over the field of rational numbers, where l' is the rank of the algebra mover the field Q.

190

THE GEOMETRY OF CLASSICAL DOMAINS

With each element a E

mwe associate the following transformation:

x= (Xl) xa = (X~ a). -+

Xk

(22)

Xk a

As we know, the algebra !l' of transformations that commute with the transformations of mconsists of matrices of the form

(b~lXl+"'+blkXk)

X.l') X

=

.

-+

Bx =

( . XIe

.

(23)

,

.

bk1 Xl + ... + bkk Xk

where B = (bij) is a square matrix with entries from m. We introduce an involution into the algebra !l' by means of the following formula: (24) Our problem consists in describing all scalar products (x,y) with the following properties: (x, y)

- (y, x),

(25)

(xa, y) = (x, yaG).

(26)

Let e i denote a vector whose elements are all equal to zero, except for the element in the ith place, which is equal to e (e is the identity of the algebra 520. We have, by (26), (27)

m;

where x, yE (e i , ejY) is a linear function of the algebra exists an hij E msuch that (eb ej y)

S(h ij y),

where Sea) denotes the trace of the element a in the algebra It follows from (27) and (28) that (e x, e y) = S(hij yx G) = S(xGhij y), i

j

m, so there (28)

m. (29)

ABELIAN MODULAR FUNCTIONS

191

whence it follows that (X,y)

= L(eiXi,ejy) = LS(xf hijY) = S(x<1Hy) ij

(30)

where H is an H x H matrix with entries h ij E ~ and x<1 is a row vector with elements xL ... ,x". We will now explain the limitations that result from (25). We have (x, y)

= S(x<1Hy) =

S(y<1H<1x),

(y, x)

whence

= S(y<1Hx), (31)

i.e., the matrix H is skew symmetric. It is clear that scalar products (30) with matrices Hi and H2 lead to isogenous representations if and only if (32) where p belongs to the center of the algebra ~ and all characteristic roots of the matrix p are positive. In concluding the present section, we will describe the domains K(~, R). As we have already noted, K(~, R) is entirely determined by the real span 2Y of the algebra~. As a result, it is sufficient to limit discussion to the case in which 2Y is a division algebra over the field of real numbers. By the classical Frobenius theorem, there exist three nonisomorphic division algebras over the field of real numbers: (1) the field of real numbers, (2) the field of complex numbers, (3) the field of quaternions. (1) ill is the field of real numbers. In this case there exist representations by matrices of any even order 2p. A representation of the algebra 2Y by matrices of order 2p is given in the following manner:

A -+ AE 2P ' The algebra f£ of matrices that commute with ~ consists of all real matrices of order 2p. Transposition provides an involution in this algebra. The skew symmetric elements of this algebra are skew symmetric matrices. It is well known that they are all equivalent to each other. Thus, in this case, all representations are equivalent. Let an involution be given by the matrix I,

1=( -E0 E). 0

192

THE GEOMETRY OF CLASSICAL DOMAINS

o(m,1) consists of all complex matrices

0,

OJIOJ' =

We set OJ = imply that

(OJ! OJ 2 ),

where

OJ!

OJ

iwIOJ'

and

OJ2

such that

> O.

(33)

are P x P matrices. Relations (33) (34)

It is easy to show that the matrices OJ! and OJ2 are nondegenerate. As a result, each class of equivalent matrices OJ contains only one matrix of the form (35) where ep denotes thep xp unit matrix. Relations (34) imply that the following relations hold true for such an OJ: i(w! - OJD > O. OJ! = OJ~, Set OJ! = z =x+iy, where x and yare real p xp matrices. We then find, finally, that (36) z' = z, Y> O. Thus, the domain K(m, J) is, in this case, analytically equivalent to a classical domain of the third type. is the field of complex numbers. In this case there exist repre(2) sentations by matrices of any even order 2p. The algebra !l' is isomorphic to the algebra of all complex matrices of order p. An involution in this algebra is provided by transposition plus conjugation. The skew symmetric elements are skew Hermitian matrices. Nondegenerate skew Hermitian matrices are clearly equivalent if and only if their signatures coincide. As a result, representations are given by one number s (0 ~ s ~ p), which we use to denote the number of positive characteristic roots of the matrix i Y. A representation of the field of complex numbers by matrices of order 2p is of the form

m

. ( x+ Yl-+

xEp - yEp

YEp) . xEp

An involution is given by the matrix

R=( -H0 H), 0

Es

H= ( o

ABELIAN MODULAR FUNCTIONS

193

where Ek is the unit matrix of order k. It is desirable for us to replace this representation by an equivalent one in which the involution is given by the matrix 1. In order to do this, we transform the involution by the matrix

We obtain the following representation of the field of complex numbers:

x + yi ---7 (

xEp - yH

YH). xEp

The involution is given by the matrix I. Note that K(m,1) can be treated as the part of Siegel's upper halfplane of degree p that consists of the stationary points of the transformation given by the symplectic matrix

H)

T= ( 0 -H 0

It is not difficult to show that the set of stationary points of this transformation is a classical domain of the first type, i.e., the set of complex matrices Z with s rows and p-s columns such that ZZ* < Es.

(37)

(3) mis the algebra of quaternions. As we know, each element of m can be written in the form x = XO+Xl i+x 2j+x 3 k, i 2 =j2 = k 2 = -1, ij = -ji = k. An involution is defined in the following manner:

iG' = - i,

r=

- j,

kG' = - k.

m

The dimension 2p of the space in which representations of the algebra are defined is divisible by 4. Thus, we can set p = 2s. The algebra .P is isomorphic to the algebra of quaternion matrices of order s. We will show that all skew symmetric elements in .P are equivalent. Indeed, every skew symmetric quaternion matrix is equivalent to a diagonal skew symmetric matrix. Moreover, using, for example, the geometric interpretation of quaternions in the form of rotations in three-dimensional space, we can show with little difficulty that all skew G

194

THE GEOMETRY OF CLASSICAL DOMAINS

symmetric elements in 2r are equivalent. As we did in Sections 1 and 2, we can show that K(2r, R) is a symmetric domain of the second type, i.e., the set of all s x s matrices Z such that Z*Z < E s '

Z' =

-z.

The considerations of Sections 1, 2, and 3 also lead us to the following theorem. Let 2r be some algebra of rational matrices of order 2p, and let R be a skew symmetric matrix defining a positive involution in 2r. Let @(2r, R) denote the set of all real unimodular matrices U of order 2p that commute with the matrices A E2r and are such that URU' = R. With each matrix U E @(2r, R) we associate the following transformation in Q(2r, R) : w~wu.

(38)

It is clear that each transformation (38) induces an analytical automorphism of the domain K(2r, R). We have the following theorem. Theorem 6.. The group @(2r, R) is transitive in K(2r, R) and contains the connected component of the identity of the group of all analytic automorphisms of the domain K(2r, R). In order to prove this theorem, it is clearly sufficient to consider only the cases in which 2r is the field of real numbers, 2r is the field of complex numbers, and 2r is the field of quaternions. Section 3. The Modular Groups <M(2r, R)

In this section we will use the same notation as we did in the preceding section. We denote the subgroup of the group @(2r, R) consisting of integer matrices by @z(2r, R). The group @(2r, R) is a semisimple algebraic group defined over the field Q. As a result (A. Borel, Harish-Chandra [1]), the factor space @(2r, R)/@z(2r, R) has finite volume. The group @z(2r, R), as we can easily see, is a subgroup of the group r(2r, R). It follows from Theorem 6 and the fact that the factor space @(2r, R)/@z(2r, R) has finite volume that the group @z(~{, R) is a subgroup of finite index in r(2r, R). Using only the definition of r(2r, R), we can show with no difficulty that @z(2r, R) is a normal divisor in r(2r, R). Note that the factor group r(2r, R)/@z(2r, R) is, as a rule, nontrivial.

ABELIAN MODULAR FUNCTIONS

195

U sing the same method as we used in Section 2, we can show with no difficulty that: (1) if the algebra 2r is isogenous to the sum of simple algebras 2r 1 , ... ,2rIll , the modular group r(2r, R) is commensurable with the product of the groups r(2rk , R k ), k = 1, ... , m, and (2) if the algebra 2r is a simple algebra and 91 1 is a maximal division sub algebra, then the modular groups r(2r, R) and r(2rl' R 1 ) are commensurable. Thus, classification of all modular groups up to commensurability reduces to classification of the modular groups r(2r, R), where 2r is a division algebra over the field of rational numbers. Let a --* aO" be a positive involution in a division algebra 2r. Consider the algebra of m x m matrices 2 over the algebra 2r. We define an involution in the algebra 2 as follows: (1)

The matrices HE 2 such that HO" = - H are said to be skew symmetric. Let a be some order in the algebra 2r. Consider the set r(O, H) of all matrices B = (b km ) E 2 such that: (1) bkm E a and (2) BO" HB = H. Moreover, let 2 denote the real span of the algebra 2 and let CM(H) denote the set of all BE 2 such that BO" HB H. As we showed in Section 2, the group CM(H) is defined in some classical domain K. The group r(O, H) is a discrete subgroup of the group CM(H). In Section 2 we actually proved that all modular groups r(2r, R) are commensurable with groups of the type r(O, H). If two matrices Hl and Hz are equivalent, i.e., there exists a matrix BE 2 such that Hl = pBO" Hz B, where p is an absolutely real element in the center of the algebra 2r, then, as we can easily show, the groups r(O, H 1) and r(O, Hz) are commensurable. With few exceptions, the converse is also true, i.e., commensurability of the groups r(Ol' H 1 ) and r(Oz, Hz) implies that the matrices Hl and Hz are equivalent. Very little is known at present about modular functions. It is surmised that a field of Siegel's modular functions is isomorphic to a field of rational functions. As far as the author knows, this has been proved only for small dimensions (A. 1. Lapin [1]). At the present time, however, the general problem of the structure of fields of modular functions is not clear. For example, as far as the author knows, there is still no answer to the following problem. By analogy to the theory of quadratic forms, we say that two matrix algebras 2rl and 2r z are contiguous if the pairs (9(1' R 1 ) and (2r z, R z) are equivalent

196

THE GEOMETRY OF CLASSICAL DOMAINS

over the field of real numbers and the ring of integral p-adic numbers for any p. (Rl and R2 are cones of principal matrices.) How are these fields related to modular functions? Somewhat more is known about the structure of the field of definition of a field of modular functions; namely, that is possible to show that it is always the field Q(E), where E is some root of unity. We will now give a brief outline of the proof of the theorem that the field pf definition of modular functions is contained in the field k obtained by adjoining all roots of unity to the field of rational numbers Q. Let I denote a 2p x 2p skew symmetric matrix with determinant at + 1. In addition, let Q2p denote the set of matrices of the form AE2p , where AE Q. As we know, the group r(Q2p,1) is Siegel's modular group. Consider the set P of all of Siegel's modular functions that are representable in the form of a ratio of modular forms with rational Fourier coefficients. As we know, the field K contains N + 1 functions 10, 11, ···,IN' where N = tp(p+ 1), with the following properties: (1) a polynomial relation between the functions 10, ... ,IN has rational coefficients and (2) any of Siegel's modular functions is representable in the form of a rational function of1o, ... ,/11" With each real matrix U such that UIU' = AI, A > 0, we can associate an analytic automorphism w -+ wU of the manifold Q(Q2P,1). This transformation induces some analytic automorphism of the domain K(Q2P, 1), which automorphism we will denote by u(z). It is easy to show that the set r 2p U r 2p is the union of a finite number of sets of the form r 2pUv, v = 1, ... , r. Let I(z) EP. Then any elementary symmetric function s of I(Ul(z)), ... ,/(Ur(z)) is one of Siegel's modular functions. We will show that the function s is representable in the form of a ratio of modular forms whose Fourier coefficients belong to the field k. In order to do this, it is sufficient to show that all of the functions l(uvCz)) are representable in the form of ratios of forms that are modular with respect to certain congruences of a subgroup of the group r 2p and have Fourier coefficients in the field k. As we know, every rational matrix U satisfying the relationship UIU' = AI, A > 0, can be reduced by means of multiplication by matrices in r 2p to the form (2) As a result, we can assume that the matrices uv , v = 1, ... , r are of the

ABELIAN MODULAR FUNCTIONS

197

form (2). We wiil use the well-known interpretation of K(Q2p, J) as the upper halfplane:

Z = X + iY,

Y > O.

(3)

An analytic automorphism of the domain K(Q2p, I) of the form Z --* AZA' + T,

T = T'

B£i2-l,

(4)

corresponds to each U of the form (2), where A and T are rational matrices. It is clear that if ¢(Z) has rational Fourier coefficients, then the Fourier coefficients of ¢(U(Z)), where U is of the form (4), belong to the field k. What we have said implies that an elementary symmetric function s of /(U1(Z)), ... ,/(uz(Z)) is representable in the form of a rational function of /0'/1' ""/N with coefficients in the field K. We will now turn to proving that the field of definition of modular functions is the field k. We will first consider the case in which the center of the algebra 2r is either the field of rational numbers or its imaginary quadratic extension. The natural imbedding of O(2r, J) in O(Q2p, J) induces an analytic imbedding of K(2r, J) in K(Q2p, J). Consider the subgroup r' consisting of the transformations in the group r 2p that map K(2r, I) into itself. As we showed at the beginning of this section, r' is a subgroup of finite index in r(2r, I). As a result, it is sufficient to show that a field of functions automorphic with respect to the group r' is defined over k. The functions /EP separate all points in K(Q, J) that are not equivalent under the group r 2p' This implies that the restriction of Siegel's modular functions to K(2r, J) generates the field of all r'-automorphic functions of K(2r, J). Consider the field p' of all functions representable by the restriction of functions / EP to K(2r, J). It is sufficient for us to show that the additional (in comparison to those already in P) relations between the functionsfEP' are defined over k. Let F denote the subgroup of the group of analytic automorphisms of the domain K(Q2P' J) that consists of the transformations that leave every pointZ E K(2r, J) fixed. The rational automorphisms are dense in F. Indeed, with each A E 2r such that AA = AE, where AE Q, we can associate a rational automorphism of the domain K(Q2P' J) and, as we can see with no difficulty, such A form an everywhere dense subset in F. Let/o'/1' "',/N be a system of functions in P such that all of Siegel's (f

198

THE GEOMETRY OF CLASSICAL DOMAINS

modular functions are rational functions of them. Let U be a rational automorphism of the domain K(Q2p, J) that is contained in F. Then

1,/Z) = 1,/U(Z)), 0

~ 11 ~

N,

ZEK(91,J).

(5)

We will show that relationships (5) correspond to certain algebraic relations between the restrictions of the functions/o, ""/N to K(91,I). As we noted above, the set r 2pur 2p consists of a finite number of cosets r 2p Uv , v = 1, ... , l', and the coefficients of the polynomial r

wet,!, U) =

IT (t-f(Uv(z))),

f(Z)EP,

v= 1

can be rationally expressed in terms of /0, ... , /" with coefficients in k. It follows from (5) that (6) Relations (5) and, therefore, (6) form a basis for additional relations in P'. Ordinary methods can be used to show that all of relations (6) follow from a finite number of such relations. A somewhat more complicated form of this argument can be used to show that a field of functions automorphic with respect to a congruence of a subgroup of the group r' is also defined over k. This implies that a field of functions automorphic with respect to r(91, R), where R is any skew symmetric matrix, is defined over k. The same proof can be generalized in the case of an arbitrary algebra 91. In this argument an imbedding of K(91, R) in K(K, R), where K denotes a maximal absolutely real subfield of the center of the algebra 91, is used instead of the imbedding of K(91, R) in K(Q, R).

CHAPTER 6

Classification of bounded homogeneous domains Section 1. Introduction

As we noted above, and as we will show in the appendix, there is a one-to-one correspondence between normal j-algebras and bounded homogeneous domains. Thus, classification of bounded homogeneous domains reduces to classification of all normal j-algebras. In this chapter we will construct algebraic apparatus convenient for construction of examples and classification of normalj-algebras. We will use this same algebraic apparatus to describe homogeneous imbeddings of bounded homogeneous domains ~ in the Siegel disk Kw In particular, we will show that there is always a finite number of different homogeneous imbeddings of the Siegel disk Kn in the Siegel disk Km. We will also note that there is a continuum of homogeneous imbeddings of the n-dimensional ball Iz 112 + ... + IZnl2 < 1 in the Siegel disk K,'Z' and, because of this, there is a continuum of different homogeneous bounded domains. Let us now turn to a section-to-section survey of the contents of this chapter. Section 2 contains certain auxiliary propositions from linear algebra. In Section 3 we will introduce the notion of a complex, show that a complex corresponds to each normal j-algebra, and that each such complex is uniquely determined. Section 4 presents the construction ofthej-algebra corresponding to a given complex and contains a discussion of several examples. Section 5 presents a description of all homogeneous imbeddings of a given bounded homogeneous domain in the Siegel disk Kw Section 6 considers the problem of the characteristics that distinguish j-algebras that are transitive in a given domain~. We will show that the number of essentially differentj-algebras of this type is finite. 199

200

THE GEOMETRY OF CLASSICAL DOMAINS

Section 2. Isometric Mappings

Let X and Y be Euclidean spaces. A nonnegative scalar product defined on their tensor product X x Y is called an isometric scalar product or isometry if (x x y, x x y) = (x, x)(y, y).

(1)

We should note that isometric scalar products may be degenerate. The simplest example of an isometry is the following scalar product: (2) where X k and Ys form orthonormal bases in X and Yand (jkm is the Kronecker delta. Another example of an isometric scalar product can be constructed in the following manner. Let X be an (not necessarily associative) algebra over the field of real numbers in which there exists a scalar product with the following property: (3) All such algebras are well known, i.e., such an algebra is either the algebra of complex numbers, or the algebra of quaternions, or the algebra of Cayley numbers. Then the following scalar product can be defined on the tensor product Xx X: (4) We should note that the isometric scalar product obtained in this manner is always degenerate. Note that with each isometric scalar product defined on the tensor product of two spaces X and Y we can associate a bilinear mapping of X and Y into some Euclidean space Z. This mapping is defined in the following manner. Let Zo denote the subspace of the space X x Y consisting of all Z E X X Y such that (x x y, z) = 0

for any

x, y.

(5)

CLASSIFICATION OF BOUNDED HOMOGENEOUS DOMAINS

201

Let Z denote the factor space X x YjZo. The scalar product inherited by Z from Xx Y is no longer degenerate. Note that the bilinear mapping (x, y) --* (x y) E Z, which bilinear mapping is the composition of the tensor product X x Y and a homomorphism of the space X x Y onto Z, has the following property: 0

(x 0 y, x 0 y) = (x, x)(y, y).

(6)

We will call such mappings isometric mappings. It is also clear that an isometric scalar product in the tensor product of X and Y corresponds to each of their isometric mappings into some Euclidean space Z. So-called continuable transformations will playa very important role in what follows. Let X, Y, and Z be Euclidean spaces, and let (x,y) --* (x o y) EZ be an isometric mapping. Definition 1. A linear transformation x --* ax of the space X is said to be continuable if there exists a transformation [3 of the space Z such that a(x) 0 y = [3(x 0 y), a*(x) 0 y

(7)

[3*(x 0 y).

Here a* and [3* denote the adjoints of a and [3. We can obtain a complete description ofa continuable transformation in the following manner. Let ny,y" y, y' E Y denote the linear transformation of the space X that is induced by an isometric mapping, i.e., such that (ny,y'(x), x') = (x 0 y, x' 0 y')

for all x, x' E X. We have the following lemma. Lemma 1. A transformation x --* ax is continuable

(8)

if and only if (9)

for all y, y' E Y. Proof Let a be a continuable transformation; then

(ny,y,(a(x)), x') = (a(x) 0 y, x' 0 y') = ([3(x 0 y), x' 0 y')

= (x y, [3*(x' y')) = (x y, a*(x') = (nV,y'(x), a*(x')) = (any,y'(x), x' 0

0

0

0

y')

(10)

202

THE GEOMETRY OF CLASSICAL DOMAINS

It remains to show that if (9) holds, then the transformation a is continuable. Let fi denote the transformation defined on the space Xx Y in the following manner: fi(x x y) = a(x) x y. (11) We will now show that

jj maps

the space Zo into itself. Note that

if and only if (12)

for any y E Y. Indeed, if

then (z, x x y) = 0 for any x E X, Y E Y. As a result,

0= (z,xoy) = I(XkoYk,Xoy) k

whence follows (12). _ It is clear from (12) that if a satisfies (9), then f3 maps Zo into itself. and, consequently, induces some linear transformation f3 onto Z = Xx YjZo. Lemma 1 is proved. We also have the following elementary proposition. Lemma 2. The bilinear function ny,y' defines an isometric scalar product if and only if (13) ny,y = (y, y)E, for any y, y' E Yand (14)

where Yb ... 'YIII are arbitrary vectors in Y and Xl, ... 'X IIl are arbitrary vectors in X. We leave the almost obvious proof of this proposition to the reader. Let Xl' X 2 , ••• , Xp be Euclidean spaces. We assume that an isometric

CLASSIFICATION OF BOUNDED HOMOGENEOUS DOMAINS

scalar product is defined on X k x X k + 1 (1 these scalar products are compatible if k-l k+l n a , a' n b, b'

=

~

203

k ~ p) and agree to say that

k+l k-l n b, b' n a , a'

(15)

where n~,~,l and n~,t,l are induced linear transformations in X k, i.e., such that for all x, x' E X k (n~~:(x), x')

= (a x x, a ' x x'),

(n~,t,l(x), x')

a, a ' E X k- 1,

(16)

(x x b, x' x b'), b, b ' E X k+ l'

We will show that if the scalar products defined on X k x X k + 1 are compatible, then we can uniquely define scalar products on all tensor products of the form X k XX k+ 1 X

... X k+ s'

1 ~ k < k+s ~ p,

(17)

in such a manner that (Xk

X ... x k+ s, x~ X •.. x~+s)

= (n~k.X'JXk+ 1) x ... x k+ s, x~+ 1 X ... x~+s),

(18)

(19) where nk and nk + s respectively denote the linear transformations induced on X k + 1 and X" +s -1 by the isometric scalar product in X k x X k + 1 and X k +s 1 X X k +S' First of all, we should note that relationships (18) uniquely define the scalar products. We need only verify that these scalar products are nonnegative and that relationships (19) follow from (18). We will carry out the proof by induction on p. For p = 2 our assertions are direct consequences of the definitions. Assume that p > 2 and that our assertions have been proved for p 1. It is clear that it is sufficient for us to prove that the scalar product defined by (18) on Xl x X 2 X .•• X Xp is nonnegative, and that (19) is valid for this scalar product. Successively applying (19), we can easily obtain the equation (20)

where band b ' are vectors in X 3 •

204

THE GEOMETRY OF CLASSICAL DOMAINS

Let z = Ix~ k

x... x;x ... x;; then

Here we have used (20) and (15). (n~, I) composed of the matrices n~, 1 is nonnegative definite. The matrix pi = (nt, I) has the analogous property. It follows from (15) that

It follows from (20) that the matrix p3 =

(21) Using (21), we can easily show that the matrix P = (nt, 1 n~,I) is nonnegative definite. We have thus proved that the scalar product defined on Xl x ... x Xp is positive definite. Successively applying (18) to both parts of (19), we can easily obtain a relationship that follows from (15). We leave the details to the reader. Note that in certain cases there is a complex structure in one of the spaces X or Y. It clearly carries over to their tensor product, and the definition of the isometric scalar product requires introduction of the additional requirement of invariance with respect to this complex structure. Section 3. Complexes

By a complex of rank p we mean a set consisting of: (1) Euclidean spaces A km , 1 ~ k < m ~ p, (2) Hermitian spaces Ck , 1 ~ k ~ p, and (3) isometric scalar products defined on the tensor products of the spaces (1) where the scalar products defined on Atk X A km

and

A km X Ams

(2)

are compatible. We should note that the dimension of each of the spaces A km and C k may be equal to zero. In this section we will describe the construction of the complex K( G) corresponding to a given normalj-algebra G, and we will show that the algebra G is uniquely determined by its complex.

205

CLASSIFICATION OF BOUNDED HOMOGENEOUS DOMAINS

We should also note that the homogeneous domain corresponding to the complex K is a Siegel domain of genus 1 if and only if dim em = 0, 1 ~ 111 ~p. Let us briefly recall certain results obtained in Section 3 of Chapter 2. Let G be a normalj-algebra, K its commutator, and H the orthogonal complement of K. The algebra H is commutative and its representation onto K is completely reducible, so K can be represented in the form of the sum of root spaces Ka. Each of the spaces Ka consists of all x E K such that [h,x]

CI.(h)x

foraIl

hER.

(3)

The linear forms CI.(H) are called roots. Let CI.!, ... , Cl. p denote all roots such thatjKalll c H. When the are appropriately labeled, every root has the form t(Cl.k±Cl.IIl) ,

1 ~ k < 111 ~ p,

Cl. 1 , ... , Cl. p

tCl.m , Cl. lm 1 ~ m ~ p.

(4)

Notation:

Moreover, we set (6) (7) In addition, let

xgn

denote the orthogonal complement of I,X ktm in t

Xkl1Z' and, similarly, let ZI~ denote the orthogonal complement of I,Zms in Z11l' Moreover,

ZO = IZ~. k

XO = I x2s, k<s

(8)

In addition, We set Xl = [XO, x°], x 2 = [XO, Xl], etc. Similarly, let Z! = [XO,ZO], Z2 = [XO,Zl], etc. We will now show that the union X of the spaces xk" k = 0, 1, ... , coincides with I, X klll , and that the unionZ of the spaces zk, k 0,1, ... , coincides with I,Zk' It is not difficult to k

use induction to show that X is a Lie algebra. Note that X kk + 1 = x2k+ 1 and, therefore, belongs to X. We will now show that X ks C X by induction on s-/c Thus, assume that we have proved that X ks C X under the assumption that s- k < d; we now show that this is true when s-k = d. We have X ks = x2s+ IX kts . t

206

THE GEOMETRY OF CLASSICAL DOMAINS

By definition, X~s c X and, moreover, X kts = [XkP Xts] c X, because X is a Lie algebra and, by the induction hypothesis, so is X kt and Xts C S. We will now show that Z = I,Zk' First we will show that [X,Z] c Z. By the definition of Z, it follows that [Xo,Z] c Z. We use induction on n to show that [X", Z] c Z. Assume that we have proved that [X"-l,Z] c Zfor n < no. We now show that [x,z] c Zfor XEX", ZEZ. It is clear that it is sufficient to verify this for [XO,X Il - i ], wherexoEXo, x"- 1 E X II - 1 ; we have

[[XO, X

ll

-

i

],

z] = [XO, [x -l, z]] + [x -l, [XO, z]]. ll

ll

By the induction hypothesis, [xIJ - l ,Z]EZand, therefore, [XO,[xIJ - l , Zl]EZ; similarly, [xo, z] EZ and, therefore, [xll -l, [xo, zl] EZ. We will now show that Zk c Z for all k. We prove this by induction on k, beginning with k = n. It is clear that ZII c Z, because ZII = Z~. Assume that we have already proved that Zk C Z for all k > s. We will show that Zs c Z. We have (9)

By definition Z~ c Z. Moreover, Zst = [Xst' Zt] c Z, because Zt c Z, by the induction hypothesis. In addition [Xk1m rill] = Yk1ll • The following lemma summarizes what we have proved. Lemma 1. A normal i-algebra G is generated by its subspaces iRk, R k, X~m' andZ~. As we noted in Section 2 of Chapter 2, the form w on G is not uniquely defined. In what follows, it will be convenient for us to fix it so that w(rk) = .h k = 1, ... ,p, where rk denotes an element of Rk such that [irk' tk ] = rk· We have the following relationships (see Section 3 of Chapter 2): [x,jz]

j[x,z]

([x,z], [x,z]) = (x,x)(z,z)

where

X

EX k1ll ,

ZEZm +

I

(10) (11)

(Xms+ Y IlIS )'

m<s~p

Consider the set of Euclidean spaces X~1II and Hermitian spaces Z~.

It follows from (11) that the commutation operator in G induces isometric scalar products in the spaces XE1II x ZI~ and X~m x X~s' We will

207

CLASSIFICATION OF BOUNDED HOMOGENEOUS DOMAINS

show that these spaces form a complex, and that the isometric scalar product of this complex can be used for unique restoration of any algebra G. Notation: p

Um

Zm+

I (X mt + Y mt ), t=m+l

Gm = jRm+ Um+R,w

(12)

As we know, Gm is an elementary i-algebra and

[Gm,U k] c Uk

for

k < m.

(13)

Consider the i-algebra G = Gt+Gk+G,w We set

L = Rt+Rk+ Ytk , U = U;+ Uk'

(14)

where U; is the orthogonal complement of X tk + Ytk in Ut. Then

Gt+Gk = L+jL+U.

(15)

As we showed in Section 6 of Chapter 2, the representation ofiL onto U is complex linear and

[jL, Gill] = 0,

[Gill' U]

C

U.

(16)

The operators adg on U are symplectic operators, so their commutability with if implies that they commute with (jf)*. Let x, x' E X tk ; it is then clear that the operators nx,;x' = (adx')*ad x carries Uk into itself and commutes on Uk with any operator adg, g EGm • The operator n x , Xl is clearly uniquely defined by the relation

(nx,x{u), U') = ([x, u], [x', U/]),

x, x' EX tk , U, U' E Uk.

(17)

The following lemma states the proposition we have obtained. Lemma 2. The operators adg, g E Gm and nx,x' commute. It immediately follows from this lemma that the spaces X km , Yklll , and Zk are invariant under nx,x'. It also follows from Lemma 2 that

nx,x'([x kr , X rm ]) = [nx,x,(Xkr), x rm ]'

x, X E X tk ,

(18)

where Xkr E X kn Xrlll E X nll , and, therefore, the spaces X krm are invariant under the operators nx,x' . The space X~/1I is the orthogonal complement of L X krm and, therefore, is also invariant under the operators nx Xl. I"

'

We will now show that the set of spaces X~m and Z~ forms a complex.

208

THE GEOMETRY OF CLASSICAL DOMAINS

In order to do this, it is sufficient to show that the isometric scalar products in the spaces X?k x x2m

and

x2m x X~IS'

(19)

X?k

and

x2m x Z,~

(20)

X

x2m

are compatible. Expression (19) immediately follows from Lemma 1 of Section 2 and (18). Expression (20) can be proved analogously. We will now show that an algebra G is uniquely determined by its complex. We set where kl < k2 < ... < k r • It is clear from (18) that the spaces X~l ... kr are invariant under the operators nx,x' and are mutually orthogonal. Using (18), we can easily use induction to show that the scalar products in X21 ... k r are uniquely determined. This also shows that the commutation operation in X is uniquely defined. Similarly, it can be shown that the commutation operation is uniquely defined if x E X km , ZEZm. Moreover, using the fact that the representation of Gm onto Uk is symplectic, we can easily show that the commutation operation in G is uniquely defined by the commutators [x km , xms] and [Xkm, ZIIJ, where XkmE X km , zmEZIII. We leave the details to the reader. Section 4. Construction ofj-algebras

In this section we will present a method for constructingj-algebras by means of a complex, and we will also consider several examples. First of all, note that a complex of rank 1 consists of one Hermitian space C 1 . An elementary j-algebra G = jR+ C1 +R corresponds to such a complex. We will now introduce the notion of an ideal of a complex. As we will show below, the ideals of the complex of the algebra G are complexes of the j-ideals of this algebra. Henceforth, we agree to denote the complex span of a space A by [A]; note that if A is a Euclidean space, then [A] is a Hermitian space. An ideal of a complex of rank p is any complex of rank 'p, 1 ~ P < p, that is obtained from the initial complex in the following manner: Akm=Akm , l~k<m~p, Ck=C k+ I [AkIllJ, (1) p
where A and

E denote the spaces constituting the new complex.

CLASSIFICATION OF BOUNDED HOMOGENEOUS DOMAINS

209

Although construction of the j-algebra corresponding to a complex is not difficult, it unfortunately requires multi-index notation. It is therefore necessary to be patient and carefully follow the argument. Let k1 < k2 < ... < k r. We define a scalar product by the method given in Section 2 in the tensor product Aklk2 x Ak2k3 X ••. Akr-lkr' Let AZ1... kr be a maximal subspace of the space Aklk2 x ... Akr-lkr in which the scalar product is equal to zero. We set (2) Similarly, we define the space Ct .. kr the spaces Aklk2 x ... A kr _ 1 k2 X Ckr ' and the factor space Ck1 "' kr ' The tensor product for the spaces Ak! ... k r and A kr ... ks induces a natural bilinear mapping of them into Ak! ... k s ' A similar proposition holds for the spaces A k1 ... kr and Ckr ... ks ' We now define root spaces for what will be the algebra G. (3)

Summation is over all sequences k1 < k2 < ... < kr such that k < k1' kr < m in the first sum and k < k1 in the second. The bilinear mappings defined above induce the bilinear mappings (4) We set X = L X km , Z = L = Zk' We now define the structure of a Lie algebra in X +Z in such a manner that: (1) the mapping Xknp Xms --* [Xk1m xms] coincides with the naturally available mapping of X km and Xms into X ks ; (2) [Xkm , Xms] = 0 if m i= nand k i= s; (3) the mapping (Xkm, zm) --* [Xkm, zm] EZ k coincides with the naturally available mapping; (4) [Xkm,ZIl] = 0 if m i= n; and (5) [z, z] = O. It is not difficult to show that these properties uniquely define the structure of a Lie algebra in X + Z. In addition, let Ykm denote the space of linear functionals on X km • Set Y = LYkm . The space X is Euclidean, so Y is also Euclidean and there exists a natural isomorphism (l between the spaces X and Y. We now consider the space X + Y +Z and define an endomorphism j of the complex structure in this space so that (1) jy = (l(Y) for y E Y and (2) j induces the same complex structure in Z as the one already there. This defines the structure of a Hermitian space in X + Y +Z. Moreover, let G denote some p-dimensional Hermitian space. Set G = C+X + Y +Z. G is clearly a Hermitian space. We now introduce the structure of a normal j-algebra into G. First we select an ortho-

210

THE GEOMETRY OF CLASSICAL DOMAINS

normal basis 1'1, .,., rp in C and fix it. We now denote the one-dimensional space generated by the vector l'k by Ric and let R = f R k • We define commutation in G so that (1) R + Y is a commutative ideal, (2) [z, z] c R + Y, and (3) X + Y + Z + R is a nilpotent subalgebra and the factor algebra X + Y +Z + R/ R + Y is isomorphic to the Lie algebra X +Z defined above. A commutation operation satisfying these conditions in G is defined by the following relations:

[Zk' zn [x kllP YkmJ

= 21m (Zk' z~)rk'

(5)

= 2(a(x km ), Ykm)r k ,

Here the indices denote membership in the corresponding root spaces. Commutation is introduced into the remaining root spaces so that (1) if these root spaces are different and are contained in X +Z, the commutation operation coincides with the commutation already in X +Z, and (2) the operator (6)

is symmetric. Here (7) 111

We leave it to the reader to verify that (5), (6) and (7) define the structure of a normalj-algebra in G. We will now consider several examples. 1. Let X be an associative division algebra over the field of real numbers. As we know, there is a scalar product in X such that for any x, YEX (xy, xy)

= (x, x)(y, y).

(8)

This scalar product clearly induces an isometric scalar product in the tensor product X x X. Consider a complex ofrankp in which A kk + 1 = X, dimA klll = 0 if m-k~2 and the isometric scalar product in Akk+1XAk+1, k+2, 1 ~ k ~ p - 2 coincides with the isometric scalar product defined in

sr

CLASSIFICATION OF BOUNDED HOMOGENEOUS DOMAINS

211

Xx X above. Using the fact that the algebra X is associative, we can easily show that these isometric scalar products are compatible. It can be shown that the construction leads to classical domains of the first, second, and third types that admit mappings onto Siegel domains of genus l. If the algebra X is not associative but (8) holds, i.e., X is the algebra of Cayley numbers, it is possible to define a complex st' of rank p = 3 in which A12 = A 23 = X, A 13 = 0. 2. Let X be an n-dimensional Euclidean space, and let U be an m-dimensional Hermitian space; we assume that a linear transformation Px of the space U is associated with each x E X, and for any x EX

PxP; = (x, x)E.

(9)

Consider the complex st' of rank 2 in which A12 = X, C 1 = 0, C2 = U, and the scalar product in X x U is defined in the following manner: (x x u, x'

X

u')

(pxCu), Px,(u '».

(10)

It follows from (9) that this scalar product is isometric. As we know, m must be a multiple of 2 V where v = [tn-I]. It is not difficult to explicitly describe the form of the Siegel domain corresponding to this complex. Section 5. Homogeneous Imbeddings of Bounded Domains in the Siegel Disk Kn Let ~ 1 and ~ 2 be bounded homogeneous domains in C'l and Cn2 (nl < n2)· By a homogeneous imbedding of ~1 and ~2 we mean a one-to-one analytic mapping of ~ 1 into ~ 2 with the following property: the set @1 of all analytic automorphisms of the domain ~ 1 that are continuable to analytic automorphisms of the domain ~2 is transitive in ~ 1. In this section we will describe all homogeneous imbeddings of' a bounded homogeneous domain ~ in the Siegel disk K". Recall that 1(" consists of all 12 x 11 complex symmetric matrices Z such that

ZZ<E.

(1)

Note that, as we showed in Chapter 2, the group of automorphisms of Kn coincides with the real symplectic group of order ll. We will now show that the description of all homogeneous imbeddings of the domain ~ in the Siegel disk 1(" reduces to description of all

212

THE GEOMETRY OF CLASSICAL DOMAINS

normal symplectic representations of the normal i-algebra G corresponding to the domain ~. We will first recall the definition of symplectic representations. Let U be an n-dimensional complex Hermitian space with Hermitian norm h(Ul' U2) and an endomorphism i of the complex structure. Recall that a real linear transformation P of the space U is symplectic if (2 )

where p(u b U2) = 1m h(u 1 , u2 ). Let G be some i-algebra. We will call the mapping g -+ P g of the algebra G into the set Sn of symplectic transformations of the space U a symplectic representation if P[91,92] = PgIPg2-Pg2Pgl'

(3)

Pjgj = O.

(4)

Pg+jPgj+jP jg

Condition (4) means that the set of operators P g is ai-subalgebra of the algebra SII' Let G be a normal i-algebra; its symplectic representation g -+ Pg is said to be normal if the characteristic roots of the operator P g are always real. All of the symplectic representations discussed in this section are normal, so, in this section, by "symplectic representation" we will mean normal symplectic representation. It is clear that an analytic imbedding of the domain ~ corresponding to a normal i-algebra G in the Siegel disk corresponds to each faithful symplectic representation of G. We will show that the converse of this statement is also true. Consider some homogeneous imbedding of the domain ~ in the Siegel disk K/I' Let @ denote the set of automorphisms of the domain KII that map ~ into itself, and let be the Lie algebra of the group @. We will show that is an algebraic subalgebra of the is not an algebraic Lie algebra. algebra S/I' Indeed, assume that Let denote a minimal algebraic subalgebra of the algebra S/I that contains G. As we know, Gu is defined in the same domain and, therefore, must coincide with G. Let P denote a subalgebra of G that leaves every point of ~ fixed. Then P is a compact ideal of the algebra and, therefore, is a direct summand of a. We have

a

a a

au

a

a = P+G

1,

where G1 is some algebraic algebra that is transitive in the domain ~. It is clear that and G1 arei-subalgebras of the algebra S/I' A maximal

a

CLASSIFICA nON OF BOUNDED HOMOGENEOUS DOMAINS

213

solvable and split (over R) subalgebra of the algebra G1 must clearly be isomorphic to the normalj-algebra G. Thus, we have proved that a faithful symplectic representation of the algebra G always corresponds to a homogeneous imbedding of the domain ~ in K,P and therefore, description of homogeneous imbeddings of the domain ~ in the Siegel disk 1(" reduces to description of all symplectic representations of the normalj-algebra G. Let G be a normal j-algebra of rank p. In this section R k, jRk' X km , Ykm , Zb X~'P and Z~ have the same meaning as in Sections 3 and 4. Consider symplectic representation g -* g(u) of the algebra G in the space U. The restriction of this representation to the subalgebra fjRk+R" is also a symplectic representation. As a result, by Lemma 2 of Section 3, Chapter 2, the space U can be presented in the form of the sum of orthogonal Euclidean spaces X" and Yk = jX" and a Hermitian space Uo, where

jrk =

r-1E

on

X k,

0

on

U 0+

1E

on

Y k,

j

on

X k,

1

rk= 0

on

I

s*k

(X s + Y s),

Y k+ U o + I(Xs+ Ys)·

(5)

(6)

We will now show that if x E Xb g E X km , then g(x) E Xm' Assume that g E X km ; then [jrs, g] = !(b"s-bms)g, where bks is the Kronecker delta. As a result, (7)

Assume that x E X k ; it then follows from (7) that (8) jrsCg(x») = -1(b ms ) g(x) and, therefore, g(x) E X m. Let X! denote the subspace of Xm that is generated by vectors of the form g(x), where g E X km , X E X k, and let X~ = Am denote the orthogonal complement of f X! in Xm' We can use the same technique as we used in Section 3 to show that the following propositions are true. 1. The bilinear mapping (x, g) -* g(x), where XE X~, g E X~s, is an isometric mapping and, therefore, induces an isometric scalar product

214

THE GEOMETRY OF CLASSICAL DOMAINS

in the tensor product X~ x X,~S (with the corresponding normalization of scalar products in G and U); similarly, the bilinear mapping (x, g) ~ g(x), where x E X,~' g EZ,~, is isometric. 2. The isometric scalar products in X~ x X,~S and X~ x Z~ uniquely define a symplectic representation of the algebra G. 3. Let Cia a', where a, a' E X~, denote the isometric scalar product in X~ x x~s that is induced by a system of linear transformations of the spaces X~s' and let f3~. x', where x, x' E Xs~, S < t ~ P denote the isometric scalar product induced inX~s x X~ by a system oflinear transformations of the space X,~s' Then Cia, a' f3~, x' = f3~, x' Cia, a',

S


(9)

where a, a' E X,~, x, x' E X~. In addition, we can use methods from Section 4 to show that given any system of Euclidean spaces Am' 1 ~ m ~ Po, and an isometric scalar product in Am X X~s and Am X Z,~ for which (9) is true, we can construct a symplectic representation of the algebra G. We will now present criteria for irreducibility of symplectic representations. A symplectic representation of an algebra G of rank p is irreducible if and only if the following two conditions are satisfied: (1) The family of spaces x2, 1 ~ k ~ p, has only one O-dimensional space, say X~. (2) Let Ci,:, x', where x, x' E X~m' r < m, denote the system of transformations of the space X~ that is generated by an isometric scalar product in X~ x X~lI and let f3z z', where z, z' EZ~, denote the system of transformations of the space X~ that is generated by an isometric scalar product in X~ x X~1I' Let mr be the associative algebra of linear transformations of X~ that is generated by the transformations Ci':,x' and f3~z" Then X~ has no nontrivial subspaces generated by transformations in r • In order to prove this proposition, it is sufficient to examine the construction of a symplectic representation by means of the spaces x2. It should also be noted that symplectic representations of the algebra G are, generally speaking, not faithful. We will now give a simple method for finding the kernel of any normal symplectic representation. Recall that any normalj-algebra G ofrankp is representable in the form of the semidirect sum of p elementary j-algebras Gk , 1 ~ k ~ p. The kernel of a symplectic representation is clearly a j-ideal. As we showed in Section 4 of Chapter 2, every j-ideal Go has the form 2: Gk (, where Gkt

m

CLASSIFICATION OF BOUNDED HOMOGENEOUS DOMAINS

215

is an elementary j-algebra contained in the semidirect decomposition of the algebra G. We have the following proposition. An elementary algebra Gk is contained in the kernel of a symplectic representation if and only if for any s, 1 ~ s ~ k, either dimX~=O or dim X sk = O. Note the following curious consequence of this proposition. There exist j-algebras that do not have faithful irreducible symplectic representations. For example, this is the case for the j-algebra G of rank 3 whose complex consists of the two one-dimensional spaces X 13 and X 23 • We will now give several examples. (1) Let G =jR+Z+R be an elementary j-algebra. All symplectic representations of the algebra G can be described .by giving a Euclidean space A and an isometric scalar product in A x Z. It is clear that if dim A ~ 2, and dimZ ~ 2, then there is a continuum of such isometric scalar products. (2) Here is another example. Let G be a normalj-algebra acting transitively in the Siegel disk Kw As we can show with no difficulty, in this case we have dim Xfk+ 1 = 1, dim X~1I = 0, m - k ~ 2, and Zk = O. If the normal j-algebra G is such that dim xfm ~ 1, dimZ2m ;;;;; 1, then, as we can easily see, it has only a finite number of symplectic representations. As a result, there is only a finite number of homogeneous imbeddings of KII in Km for all 11 and m. In Section 6 of Chapter 2 we showed that description of all normal j-algebras with a given j-ideal G 1 and factor algebra G' reduces to description of all symplectic representations of the algebra G' in some space U, where it was additionally required that the representation operators commute with a previously given set of complex linear transformations of the space U. This raises the problem of describing, for a given symplectic representation in the space U, all complex linear transformations that commute with it. It is clearly sufficient, however, to restrict discussion to the case in which U has no invariant spaces and dim xg = 0 for all k except for one, which we will denote by r. We have the following proposition for this case. Every complex linear transformation 'Y of the space U that commutes with all of its representation operators carries the space X~ into itself. If the restriction of'Y to X~ is the identity transformation, then 'Y is the identity transformation. The transformation ¢ of the space X~ induced by some complex linear transformation 'Y of the space U commutes with the representation operators if and only if the transformation ¢ can be extended to a transformation of any of the spaces of the form X~ x X~,

216

THE GEOMETRY OF CLASSICAL DOMAINS

x~ X z~, r < t, carrying the kernel of the scalar product into itself. In other words, the transformation ¢ must commute with all transformations of the form

a':, x',

where

X,X'EX~m'

/3z,z"

where

Z,Z'EZ~.

Section 6. Algebraic i-algebras Let f!) be a bounded homogeneous domain in C". As we noted in Section 4 of Chapter 2, even in the simplest case, in which f!) is an n-dimensional ball, there exists a continuum of nonisomorphic ialgebras that are not essentially different. This is connected with the fact that the usual notion of isomorphism for i-algebras is unnecessarily fine. It is more natural to use the notion of i-algebra equivalence introduced in the present section. We will show that for any domain f!) there exists only a finite number of i-algebras that are transitive in this domain and are not equivalent, and we will present an algorithm that will make it possible to find them. As we noted above, any algebra i can be imbedded in an algebraic i-algebra by extending it by a certain number of its differentiations. All algebraic i-algebras G that are transitive in a given domain f!) can be described in the following manner. Let B denote the normal i-algebra corresponding to the domain f!). Let K be the commutator of the algebra B, H the orthogonal complement of K, and let (1)

be the decomposition of the algebra B into the sum of root spaces. In this section we will show that any algebraic i-algebra G can be represented in the form G=H+IKcz+ IK-cz+C,

(2)

czeA

where A is some subset of the set of roots of the algebra B, Cis a compact subalgebra commuting with H, and the algebra G'

= H+ I Kcz+ I K-cz+C czeA

(3)

czeA

is reductive. It is also easy to show that the algebra G is uniquely determined by the set A and the compact subalgebra C.

CLASSIFICATION OF BOUNDED HOMOGENEOUS DOMAINS

217

We should also note that any j-algebra can be obtained from some algebraic j-algebra G for which, consequently, (2) is valid in the following manner. Let P be a subalgebra (of the algebra H + C) whose image under a homomorphism of H + C onto H fills all of H. Consider the followingj-algebra sub algebra Gp of the j-algebra G: Gp = P+ IKcz+ I K_cz+C.

(4)

czeA

It is not difficult to see that Gp is transitive in the domain f!). It can be shown that every j-algebra is of the form (4). The most important difference between two j-algebras defined in a single domain is undoubtedly related to differences in the set A. In connection with this, it is reasonable to say thatj-algebras for which the sets A coincide are equivalent. We will now show that (2) holds for any algebraic j-algebra G. As we know, any algebraic Lie algebra G can be represented in the form G = G'+N,

where N is a maximal nilpotent ideal, G' is a reductive algebra, and a representation of G' onto N is completely reducible. Let H be a maximal commutative subalgebra of the algebra G' such that the operators ad h, hE H, are semisimple and have only real characteristic roots. As we know, G' can be represented in the form G' = H+ IG;.+ IC-;.+C, where the A are positive roots, G;. and G _;. are the corresponding root spaces of the algebra G', and C is a compact subalgebra in the centralizer of H. A representation of H onto N is completely reducible, so there exist linear forms a1(h), ... , ak(h) on H such that N = "LN i, where Ni consists of all x EN such that [h, x] = ai(h)x for all hE H. In order to prove (2), it is clearly sufficient to show that none of the equalities ai(h) A(h) and a/h) - A(h) , where A > 0 is possible. Assume, for example, that aio(h) = Ao(h), where .1..0 > O. Consider the following sub algebra : k

H+IG-;.+

I N i=1

i•

This subalgebra is clearly solvable and factorable over the field of real numbers. In addition, in the set of all algebras with this property it is maximal. As a result it is isomorphic to the normal algebra B. On the

218

THE GEOMETRY OF CLASSICAL DOMAINS

other hand, its set of roots contains two, Aio and - Ao, whose sum is equal to zero, which is impossible. Thus, we have proved that the equation a io = Ao is impossible, and a similar approach can be used to prove that the equations ai = - A, where A > 0, are also impossible. We have thus proved decomposition (2). It is also clear from our argument that the algebra

H+IG).+IG-).+C is reductive. In conclusion, we will present, without proof, certain simple propositions about the structure of A. As usual, we agree to let a!, ... , ap denote roots such that jKa. c H. We will also assume that the roots ab 1 ~ k ~ p, are labeled in such a manner that if dim Kt(a.s-a. m) =I- 0, then s < m. We also agree to write s < 111, if s = 111 or s < 111 and dim Kt(a.s-a. m) =I- 0. (1) If -!(ak- am) E A and k < s < t < 111, then -!Cat - at) E A (E. B. Vinberg [6] and 1. 1. Pyatetskii-Shapiro [16]). (2) If one of the inclusions ak EA,

-!ak EA,

-!( ak+ czm) EA,

occurs, then for any sand t such that k < s < t we have the inclusions as EA,

-!as EA,

(1. 1. Pyatetskii-Shapiro [16]).

-!( as + at) EA.

Appendix Introduction 1. Siegel Domains of Genus 1 and 2 2. Decomposition of a i-algebra Associated with a Commutative Ideal 3. Algebraic i-algebras 4. Decomposition of a i-algebra Associated with a Commutative Ideal (continuation) 5. Representation of a Homogeneous Domain in the Form of a Siegel Domain of Genus 1 References

219 225 231 238 242 253 254

Introduction

1. A bounded domain f!) in n-dimensional complex space C" is called homogeneous if for each pair of its points there exists an analytic automorphism of domain f!) which transfers one of these points into another. From Riemann's theorem on conformal mappings it follows that all one-dimensional, singly connected, bounded domains are analytically equivalent to one another and homogeneous. It is also known that there are no other homogeneous bounded domains in C 1 • Poincare noted [19J that in C 2 there are at least two homogeneous bounded domains; a hypersphere and a bicylinder, which cannot be analytically mapped onto one another. Cartan showed [12J that every homogeneous bounded domain in C 2 is analytically equivalent to one of the two aforesaid domains. He in fact classified (accurate to analytic equivalence) the homogeneous bounded domains in C 3 • All these domains proved to be symmetric. Cart an classified all the symmetric domains in C" space and posed the problem: is every homogeneous' bounded domain symmetric? Borel [lJ and Koszul [16J showed that if a bounded complex 219

220

THE GEOMETRY OF CLASSICAL DOMAINS

domain admits a transitive semisimple group of automorphisms, it is a symmetric domain. Borel's and Koszul's results were strengthened by Hano [33J, who replaced the requirement of semi simplicity by a requirement of unimodularity. Pyatetskii-Shapiro [22J constructed the first example of a nonsymmetric homogeneous bounded domain (in C 4 ). It later proved to be the case that symmetric domains are in a certain sense the exception [23J, [25J and [26J. Thus, when n ~ 7 a continuum of analytically nonequivalent homogeneous bounded domains exists in C' space, whereas there is only a finite number of analytically nonequivalent symmetric domains in each dimension. The construction of what are known as Siegel domains, introduced in connection with the problems of the theory of automorphic functions [20J and [25J, proved to be extremely important to the theory of homogeneous bounded domains. Siegel domains are generalizations of the ordinary upper half-plane 1mz> O. It became apparent that a realization in the form of a Siegel domain of genus 1 or 2 is most natural for a homogeneous nonsymmetric bounded domain [26J. Such a realization is homogeneous with respect to affine transformations. The principal result of the present paper is that every homogeneous bounded domain ~ can be realized in the form of a Siegel domain of genus 1 OJ' 2. For each domain f!) this realization is unique, accurate to the affine transformations. With certain additiona,l assumptions concerning domain f!) this statement was proved previously by PyateskiiSha piro [26J. Using the results of the present paper, it is also possible to give a description of the full group qj(f!)) of the analytic automorphisms of domain f!). Namely, qj(f!)) is a group of all "quasilinear" transformations of a certain canonical model of domain f!) into a Siegel domain of genus 3, the base of which is a symmetric domain. (For a definition of Siegel domains of genus 2 and their quasilinear transformations see [25J). We shall note that by means of realizing bounded homogeneous domains in the form of Siegel domains of genus 1 and 2 it was possible to find an explicit form of integral formulae for arbitrary bounded homogeneous domains [8J, [9J. Constructions of Siegel domains are closely associated with homogeneous convex cones in n-dimensional real space R". These cones have been studied by Kocher [13J-[15J, Koszul [17J, [18J, Vinberg [2J,

APPENDIX

221

[4J-[6J and certain other authors [21 J, [30J, [31 J, [34]. Vinberg [6J has constructed a generalized matrix calculus which enables each homogeneous convex cone to be regarded as a cone of positively defined Hermitian generalized matrices. This apparatus was used in particular to obtain a description of the full group of automorphisms of a homogeneous convex cone. Some results of the theory of homogeneous convex cones are used in the present article. 2. As stated already, the main result of the present paper is that every homogeneous bounded domain is analytically equivalent to a certain Siegel domain of genus 1 or 2. 'We shall give here a precise definition of Siegel domains of genus 1 and 2. Let V be a homogeneous convex cone in n-dimensional real space R". We shall assume that cone V does not contain an entire line. Domain §)(V)= {x+iy: yE V}

(1)

in n-dimensional complex space e" is called a Siegel domain of genus 1 connected to cone V. Domain §)(V) is analytically equivalent to a bounded domain in ell and is homogeneous with respect to a group yt of complex affine transformations generated by transformations of the form x+iy~ (x+a)+iy (aER") (2) and x+iy~Ax+iAy, (3) where A is the automorphism of cone V (see [25J). To define a Siegel domain of genus 2 we shall introduce the concept of a V-Hermitian function. Let V c RII be a homogeneous convex cone. The function F defined on vector pairs u, v of complex space em and taking values in C ' is called V- Hermitian if the following conditions are fulfilled: (1) FO'l Ul + A2u2, v) = A1F(u l , v) + A2F(u 2, v) (Ai' A2 are arbitrary complex numbers); (2) F(u,v)=F(v,u); (3) F(u, U)E V (Vis the closure of cone V); (4) F(u, u) 0 only when u = o. Domain §)(V,F)={(x+iy,u): y-F(U,U)EV} (4) in (m+n)-dimensional complex space ell x Gil is called a Siegel domain, of genus 2 associated with the convex cone Vand the V-Hermitian function F: This domain is analytically equivalent to a bounded domain.

222

THE GEOMETRY OF CLASSICAL DOMAINS

The automorphisms A of cone V for which there exists a linear transformation B of space C m such that

AF(u,v) = F(Bu, Bv), (5) form a group. If this group is transitive in cone V, then domain ~(V,F) is homogeneous with respect to the complex affine transformation group generated by transformations of the following three types: (x+iy,u)~((x+a)+iy,u)

(aERll),

(x + iy,u) ~ (Ax+ iAy, Bu) (A, B satisfy condition (5)), (x+ iy, u) ~ (x+i(y+2F(u, c) + F(c, c)),u+c) (CECm)

(6) (7)

(8)

(see [25J). 3. Our proof is based on the fact that with each homogeneous bounded domain ~ it is possible to associate some algebraic object known as a j-algebra. Let cg be some transitive group of automorphisms of domain ~ and % be a stationary subgroup of some point Zo E f!). We shall denote by G and Kthe Lie algebras of groups-cg and % respectively. Making use of the fact that the tangential space to domain f!) at point Z1 is naturally identified with the factor space G/K, an operator j and a linear form w, which possess the properties enumerated below (see [16J and [17J), can be defined on space G: (1) jK c K,j2 == -1 (modK); (2) [k,jxJ ==j[k,xJ(modK)for any kEK, xEG; (3) [jx,jyJ==j[jx,yJ+j[x,jyJ+[xyJ(modK)for any x, YEG (con~ dition of integratability); (4) w([k,xJ) = Of01" any kEK, xEG; (5) w([jx,jyJ) = w([x,yJ)for any x, YEG, (6) w([jx,xJ) > ofor any XEG, xf/=K. The Lie algebra G with the noted subalgebra K, operator j and form 0) for which conditions (1)-(6) are fulfilled is called aj-algebra. Henceforth a j-algebra will be denoted by {G, K,j, w}, {G, K,j} or simply by

G. Two j-algebras {G V K 1,jl,wd and {G 2,K2,j2'W 2} are called isomorphic if there exists an isomorphism ¢ of Lie algebra G1 onto Lie algebra G2 such that ¢ (K1 ) = K2 and ¢j1 ==j2¢ (modK2). It is easy to verify that this definition of a j-algebra is equivalent to the definition of a general j-algebra given in [27J.

APPENDIX

223

We shall further introduce the concept of a j-subalgebra of a j-algebra Let G1 be some sub algebra of a Lie algebra G which satisfies the condition jG 1 c G1 + K. In space Glone can then define operator jl such that jx =jlx(modK) at all XEG 1 • We shall define the linear form W 1 on G1 as a restriction of form wand assume Kl K G1 • It is easy to see that {G 1 , K 1 ,jl' wd is a j-algebra. It is called a j-subalgebra of j-algebra {G, K,j, w}. It can be shown that a homogeneous bounded domain is uniquely restored over its j-algebra. However, not every j-algebra {G, K,j} corresponds to a homogeneous bounded domain. For this, in any case, the following condition must be fulfilled: (P) Every compact semisimple j-subalg ebra G1 of alg ebra G is contained in K. In actual fact, we shall assume that {G, K,j} is aj-algebra of a bounded domain f!) with a transitive automorphism group eg. We shall denote by egl a connected compact subgroup of group eg corresponding to G 1 , and we shall examine the orbit eglZ0 of point Zo Ef!) under transformations from eg l' As can easily be seen, this orbit is a compact complex submanifold of domain~. Every analytic function is constant on eg 1 zo, and orbit eg 1 Zo must therefore consist of one point Zo0 This means that egl c % and G1 c K. We shall call a j-algebra proper if it satisfies condition (P). It follows from Borel's results [1] that every semisimple proper j-algebra is the j-algebra of a bounded symmetric domain. It can easily be deduced from the results of the present article that every proper j-algebra is the j-algebra of some homogeneous bounded domain. 4. In this paragraph we shall layout the plan proving the main theorem. Let f!) be some bounded homogeneous domain and {G, K,j} be any j-algebra corresponding to it. In Section I we find the conditions under which a j-algebra {G, K,j} corresponds to some Siegel domain of genus 1 or 2. We shall formulate these conditions in a slightly different form, making in addition the insignificant assumption that Lie algebra G is algebraic (see Section 3). Algebra {G, K,j} corresponds to a Siegel domain of genus 1 or 2 when and only when an element go possessing the following properties exists in G: (1) transformation x ~ [g 0' x] is semisimple; (2) its eigenvalues are 1, 0 or 1-; (3) if [go, x] = AX then [go,jx] = (1-A)X. {G,K,j,w}.

n

224

THE GEOMETRY OF CLASSICAL DOMAINS

It is easy to show the equivalence of these conditions to the condition of Theorem 2 (the part of element go is played by jr). We shall note that if transformation x ~ [gl, xJ has no eigenvalues 1, {G, K,j} corresponds to a Siegel domain of genus 1. Thus, in order to show that domain f!) is analytically equivalent to a Siegel domain of genus 2, it is sufficient to establish the existence in G of an element with the above-mentioned properties. Element go also admits a curious geometric interpretation. We shall quote it now although we shall have no need of it in the future. We shall examine a one-parameter transformation group e tgo in the affine transformation group C§ of a Siegel domain of genus 2. In this case etgo

(x+iy, u) = (l (x+iy), eft u)

(A)

(see notations in Paragraph 2). We shall return to setting out the plan of the proof. If algebra G is semisimple, our statement follows easily from Borel's results [1 J (see also [25J and [26J). One can therefore confine oneself to the case when algebra G is not semisimple. Then there is a commutative ideal R in G. It is easy to show that G1 = K+R+jR is aj-subalgebra of algebra G. In Section 2 it is proved that an elementjr, which satisfies the relation co (Ur, aJ) = co (a)

for all a ER,

possesses all the properties of element g 1 in j-algebra Gl' Hence it follows that G1 corresponds to some Siegel domain of genus I or 2. Further proof is based on a study of the eigenvalues of transformation x ~ [jr, x J on factor space G/ Gl' The main result here is that they are all equal to 0 -or 1. In Section 3 it is established that every j-algebra can be extended to an algebraic j-algebra acting in the same domain (Theorem 3). This result is not only of direct interest, but it also enables considerable technical simplifications to be introduced. In particular, if G is an algebraic j-algebra one can consider, without loss of generality, that operator x ~ Ur, x J is semi-simple and all its eigenvalues are real. In Section 4 it is proved that every proper j-algebra decomposes into the semi-direct sum of a j-ideal corresponding to a Siegel domain of genus I or 2 and a semisimple j-subalgebra corresponding to a symmetric bounded domain (Theorem 5). This decomposition corresponds to an analytic fibering of a homogeneous bounded domain into Siegel domains of genus I or 2, the base of which is a symmetric domain.

225

APPENDIX

On the basis of the results of Section 4, in Section 5 we prove the main theorem, that every homogeneous bounded domain is analytically equivalent to some Siegel domain of genus 1 or 2 (Theorem 6). 5. We shall agree about certain notations which will be used in this article without special reservations. Lie algebras will be denoted by letters G, H, ... , and their corresponding Lie groups by the written letters C§, :Yt,... Adjoint operator x ~ [g, x] (x E G) will be denoted by ad g for g E G. If subspace ReG is invariant with respect to ad x, adR x will denote the restriction of operator ad x on R. The symbol exp will denote the exponential mapping of a Lie algebra onto the appropriate group. From the definition of aj-algebra it follows that the setting of operator j in j-algebra {G, K,j} is real only in modulus K. Bearing this in mind we shall sometimes understand j as a nonunique operator. In other cases we shall use such expressions as "we shall define operator j such that ... ". If x is an element of vector space then (x) will denote the subspace spanned onto x. The Kronecker delta will be denoted by (japo Section 1.

Siegel Domains of Genus 1 and 2

1. The concept of a Siegel domain of genus 1 (see introduction) . admits the following natural generalization. Let U be an affine homogeneous convex domain in n-dimensional real space R" not containing a whole line. We shall examine the cylindrical domain ~(U)={x+iy:

yE U}

(1)

in n-dimensional complex space C'. In the particular case when U is a cone, ~ (U) is a Siegel domain of genus 1. In the same way as for Siegel domains of genus 1, it is shown that domain ~ (U) is analytically equivalent to a bounded domain in e" and is homogeneous with respect to an affine transformation group C§ generated by transformations of the form x+iy ~ (x+a)+iy

(2)

x+iy ~ Ax+iAy,

(3)

and where a runs over space R", and A runs over any transitive group of automorphisms of domain U. H

yt

226

THE GEOMETRY OF CLASSICAL DOMAINS

2. We shall give a description of the j-algebra G of a cylindrical domain f!fi (U). The elements of the algebra G can be understood as infinitesimal affine transformations of the space CII. Transformations (2) correspond to infinitesimal transformations

Da(x+iy)= a,

(4)

where a runs over space R , and transformations (3) correspond to infinitesimal transformations Il

Dc(x+iy)= C(x)+iC(y),

(5)

where C runs over Lie algebra H of the differentiations of domain U corresponding to group yt of its automorphisms. If D 1 , D2 are two infinitesimal affine transformations their commutator, as can easily be seen, is given by formula

(6) where D i (i = 1,2) denotes the linear part of transformation D i • From formula (6) the following commutation relations are obtained in algebra G:

[DaDa] = 0,

[Dc, Da] = DCa,

[D cl , Dc2 ] = D[cl,c2]'

(7)

where C is the linear part of transformation c. From here it follows that Lie algebra G decomposes into a semidirect sum of commutative ideal R, formed by the transformations DO' and subalgebra Hformed by transformations Dc:

G=R+H.

(8)

Let Yo E U; then Zo= iyo E~ (U), and every transformation of ytwhich preserves point Zo has the form of (3) with such an A that Ayo = Yo. Consequently, subalgebra KeG corresponding to a stationary subgroup of point Zo is contained in H and is isomorphic to the stationary subalgebra of point Yo of domain U. Finally we shall compute operator j in space G which is induced by the complex structure of domain f!fi (U). Element j Dais defined from condition

(jDa)(ZO) = iDa (Zo) = ia.

(B)

From here it is evident that jDaE H. More accurately,

jDa= Do

where

C(Yo)= a

(9)

APPENDIX

227

(element jDa is defined with an accuracy to the addend of K). From (9) it follows that

H=K+jR.

(10)

3. Theorem 1. Let j-algebra G admit decomposition into the direct sum of subspaces. (11)

G=K+jR+R,

and let R be the commutative ideal. Then it is isomorphic to the j-algebra of some cylindrical domain f!) (U), where U is an affinely homogeneous convex real domain which does not contain a line. Domain U is a cone, and domain f!) (U) is a Siegel domain of genus 1 when and only when an element r E R exists such that [ja, rJ = a for all aER

(12)

Proof We shall take an arbitrary point Yo E R and for every element h = k +ja (k E K, a E R) we shall define an infinitesimal affine transforma-

tion Ch of space R from formula (13)

We shall compute the commutator of transformations Chi and Ch2 , where h i = ki+ja; (i 1,2). According to (6) [Chi' Ch2J (y)= [hi' [h 2,y-YoJ

+a2J

[h 2, [h 1 ,y-YoJ +alJ

= [[h 1 ,h 2J,Y-YoJ+[h 1 ,a2J-[h2,alJ.

(C)

On the other hand, from axioms (2) and (3) (see page 222) for a subalgebra we find: [h 1 ,h 2J =:j[h 1 ,a2J-j[h2,alJ

(modK).

(D)

From here it follows that [h 1 ,h 2JEK+jR and (14)

Thus, transformations Clz , hE K +jR, form an affine Lie algebra. We shall denote it by H. Let Yr be an affine group corresponding to H, and U the orbit of point Yo under transformations from Yr. From (13) it can be seen that the mapping (15)

228

THE GEOMETRY OF CLASSICAL DOMAINS

of group Y't into space R has a rank at point Yo equal to the dimension of space R. Consequently, orbit U is a domain in R. We shall prove that domain U is convex and does not contain an entire line. According to Koszul [17J for this it is sufficient to construct a linear differential form a= LaidYi on U, closed invariant with respect

to.Yt and such that quadratic form Da =

L(daJdYj) dYi dYj is positively

defined. If the form exists, it is uniquely restored in its value ao at point Yo. Form ao must be invariant with respect to the stationary subgroup of point Yo. It is easy to see that the condition is reduced to the fact that

ao([k,aJ)= 0 for all

kEK, aER.

(16)

In order to represent the condition of closure of form a in a form which is convenient for us, we shall examine the pre-image f3 of form a under mapping (15). Form f3 is a left-invariant differential form on group Yr. If X and Yare left-invariant vector fields on Yr then according to the well-known Maurer-Cartan equations:

df3(X, Y)= -!f3([X, YJ).

(E)

We shall denote by f30 the value of form f3 in an identity of group Yr. Form f3 is closed when and only when

f30 ([ ChI' Ch2 J) = 0

(F)

for any hi' h2 E K +jR. On the other hand the closedness of form f3 is equivalent to the closedness of form a. Since f30 (CII ) = ao (CII (Yo)), the closedness condition of form a can be represented in the following manner: (G)

for any h i ,h 2 EK+jR. In view of (16) this is equivalent to the fact that

aO([ja i ,a2J)= ao ([ja 2, aiJ) for all ai' a2 ER. (17) Finally, for the positive definition of quadratic form Da it is necessary and sufficient that for all h = k +ja E K +jR, where a =I- 0, the following condition be fulfilled (H) ao(C\C,,(yo))> 0 (see [17J formula (2.3)) or what is the same thing,

ao([ja,aJ»

0 for aER, a =I- 0

(18)

(see formula (16)). As a i we shall take form ill set on G. Conditions (16)-(18) are then fulfilled in view of axioms (4)-(6) (see page 222) for a

APPENDIX

229

j-algebra. In this way it is proved that U is a homogeneous convex domain not containing a line. It is easy to verify that j-algebra G is isomorphic to the j-algebra of a cylindrical domain ~ (U). Domain U is defined with an accuracy to the shift (due to the arbitrary nature of the selection of point Yo). It is a cone when and only when transformations Ciz are linear, i.e., [ja,yo] = a for all aER. This brings us to condition (12). 4. Let f!) (V, F) be a homogeneous Siegel domain of genus 2 associated with a convex cone V and a V-Hermitian function F (see introduction), and C§ be a transitive group of affine automorphisms of domain f!)'(V, F) generated by transformations (6), (7) and (8) of the introduction. In the same way as in Paragraph 2 for cylindrical domains it is easy to verify that the j-algebra of domain f!) (V, F) is constructed in the following manner: (19) G K+jR+R+W, where R is the commutative ideal, K +jR a subalgebra, Wan invariant subspace and the following relations are fulfilled: [K +jR , W] c W, [R, W] = 0, [W, W] cR. (20) In the notations given in the introduction R, K +jR and W correspond to transformations (6), (7) and (8) respectively. Cone V can be considered to lie in R, and function F is set on WX Wand takes values in R +JR. The commutator of elements u, v E Wis then defined by formula [u,v] =2j(F(v,u)-F(u,V))ER. (21) Finally the j-subalgebra K +jR + R of algebra G corresponds to cylindrical domain f!) (V), i.e., a Siegel domain of genus 1 associated with cone V. 5. Theorem 2. Let j-algebra {G, K,j} admit decomposition (19), in which (1) R is the commutative ideal,' (2) there exists an element r E R satisfying condition (12),' (3) W is a j-invariant subspace; (4) relations (20) hold. Then j-algebra {G, K,j} is isomorphic to the j-algebra of some Siegel domain of genus 2 with a transitive affine automorphism group. Proof First of all, from axioms (2) and (3) (see p. 222) for aj-algebra and condition (1) of the theorem it follows that H = K +jR is a subalgebra. In actual fact, when k E K, a E R [k,ja] =j[k,a] (modK) (1)

230

THE GEOMETRY OF CLASSICAL DOMAINS

and when a, bER [ja,jb] =j[ja,b]+j[a,jb]

(modK).

(J)

In view of condition (2) of the theorem, j-algebra K +jR + R is isomorphic to the j-algebra of Siegel domain f!) (V) associated with homogeneous convex cone V (see Theorem 1). Cone V can be considered imbedded in R; then it is the orbit of affine group H generated by infinitesimal transformations (13) which at Yo = r take the particularly simple form:

C/i(y) = [h,y]

(hEH).

(22)

We shall assume now that for any u, v E W F(u, v)

= -!-(ju, v] +j[u, v])ER+jR

(23)

and prove that F is a V-Hermitian function (see introduction). The integrability condition written for elements u, v E W, gives in projection onto R [ju,jv]

= [u, vJ.

(24)

By means of this identity for function F properties (1) and (2) of a V-Hermitian function are verified. Further, F(u, u) = -!-[ju, u],

°

(K)

so that if u =I- 0, F(u, u) =I- (see axiom 6 for a j-algebra). Thus, it remains to verify property (3). We shall examine the affine representation C of Lie algebra G/ R + G in space R + W defined by formula C-(a+u)

where aER, UE W, and element condition

= [g,a+u]+w, WE

(25)

W depends on g and is defined by

g =jw(mod R+H).

(26)

The affine group X generated by infinitesimal transformations Cacts transitively in some domain U of space R + W containing point Yo' In the same way as in paragraph 3 it is established that domain U is convex and does not contain a line. Inasmuch as R is the ideal in algebra G, the linear parts of transformations in X preserve subspace R. From here it follows that if

APPENDIX

231

A EX, Ayo E R then ARc R. The transformations A E X for which AR c R form a subgroup xx; its Lie algebra, as can be seen from (25), coincides with H. The previous remark shows that group xx acts transitively in the convex domain U II R of space R. The connected component of the identity of group xx coincides with group yt which acts transitively in cone V (compare formulae (22) and (25) at g = h). From here it follows that U II R = V. Now let WE Wand A be a real number. Then

_ (L) (see, for example, [6J, Chapter 1, formula (18)). From the convexity of domain U it follows that

iCy;. + Y -;.)

= Yo +!A2[jW, WJE U II R = V.

(M)

As this is fulfilled for any A, then [jw,

wJ= 4F(w, W)E V.

(N)

In this way it is proved that F is a V-Hermitian function. From axioms (2) and (3) (see page 222) for a j-algebra it is easy to obtain the relation [h,juJ =j[h, uJ, (27) which is valid for any hE H, u E W. From this relation it follows that [h, F(u, u)J = F([h, uJ, u) +F(u, [h, uJ).

(0)

We shall assume A = exp adRh, B = exp adwh. Then AF(u, u) = F(Bu, Bu).

(P)

Thus, function F satisfies the condition for homogeneity (formula (5) of the introduction). We shall examine a homogeneous Siegel domain 2) (V, F). It is directly established that its j-algebra, described in Paragraph 4, is isomorphic to G. In particular, formula (21) is verified with the aid of relation (24). Section 2. Decomposition of a j-algebra Associated with a Commutative Ideal

1. We shall call j-algebra {G, K,j} effective if K contains no non-nil ideals of algebra G. This corresponds to the assumption that group C§ acts in a homogeneous space C§ yt effectively.

232

THE GEOMETRY OF CLASSICAL DOMAINS

We shall establish some properties of an effective j-algebra {G, K,j}. We shall examine the bilinear form (x,y)

co ([jx,y])

(1)

in space G. It follows from the properties of a j-algebra that thi$ form is symmetric, vanishes at x E K and induces a positively defined scalar product on GjK. It is also easy to see that for any k E K, x, Y E G ([k, x],y) + (x, [k,y]) =

o.

(2)

This shows that an invariant Riemann metric exists on homogeneous space C§ :It. It is well known that in this case (and on condition of effectiveness) a positively defined scalar product exists in space G which is invariant with respect to ad K. The following statement follows from here and it is one which we shall use several times. Lemma 1. The adjoint representation of Lie algebra K and of any subalgebra of it is reducible in space G. We shall also need. Lemma 2. The adjoint representation of Lie algebra G is precise. Proof Let g E G be an element such that adg = O. Then [jg,g] = 0 and it follows from axiom 6 for a j-algebra that g E K. The subspace spanned onto g is the ideal of algebra G. In view of our assumption concerning effectiveness it follows from here that g = O. Lemma 3. If R is a commutative ideal of algebra G, R n K o. Proof Let a ERn K. Then (ada)2 = 0, and, on the other hand, in view of lemma 1 operator ad a is semisimple. Consequently, ad a = O. Lemma 2 now shows that a O. 2. Let {G, K,j} be an effective j-algebra and R any commutative ideal of Lie algebra G. From properties 2 and 3 of a j-algebra (see introduction) it is easy to deduce that subspace H = K +jR is a sub algebra in G. We shall examine j-subalgebra

GC = R+H = K+jR+R

(3)

of j-algebra G. By Theorem 1 it corresponds to some cylindrical domain f!fi (U). We shall call commutative ideal R an ideal of the first kind if U is a cone (accurate to the shift) and an ideal of the 2nd kind otherwise. In view of Lemma 3, scalar product (1) defines the structure of a

233

APPENDIX

Euclidean space on R. element l' E R for which

There exists, and moreover it is unique, an

(1', a)

= w (a) for all a E R.

(4)

Element r will play the main part in the constructions of this section. We shall prove some properties of element t. First, from relation (2) it follows that

[K,r] =0.

(5)

Axiom 2 for a j-algebra then shows that

[jr,K] c K. Lemma 4.

(6)

With the appropriate definition of operator j, [jr,K] = O.

(7)

Proof We shall examine subspace L K+(jr) of algebra G. From relation (6) it follows that [K,L] c K. In view of the complete reducibility of the adjoint representation of subalgebra K in algebra G (Lemma 1) there exists an element 1 = jr+ko (k o EK) such that subspace (1) is invariant with respect to ad K. Obviously [K, 1] = O. Element 1 can be taken for jr with the appropriate definition of operator j. Lemma 5. [jr, r] = r. Proof For any aER ([jr, r], a) =w([ja,[jr, r ]]) = w([[ja,jr], r]) + w([jr,[ja, r ]]) = w([j[a,jr] +jUa, 1'], r]) + w([ja, r]) = w([a,jr] + [ja, 1']) + (a,

(Q)

1') = (r,a).

3. We shall now require one result obtained in [28]. At this point we shall give an accurate formulation of this result. Let Z be an even-dimensional real space in which are set operator j and skew symmetric bilinear form p(x,y) connected by the following relations

j2

= -1, p(jx,jy) = p(x,y), p(jx, x) )

o.

(x #- 0).

(8)

We shall call the linear transformation p of space Z symplectic if it is a differentiation of form p(x,y), i.e.,

p(px,y) + p(x,py) for any x, YEZ.

= 0,

(9)

234

THE GEOMETRY OF CLASSICAL DOMAINS

The result which we require consists of the following: if p and q are symplectic transformations of space Z satisfying conditions

[p,qJ = q,

(10)

[j,p-t[j,qJJ = 0,

(11)

space Z can be broken down into a direct sum of subspaces:

Z =z+ +Z- +Zo

(12)

in such a manner that (1) subspaces Z+, Z- and ZO are invariant with respect to p; (2) The real parts of the eigenvalues of operator p on subspaces Z+, Zand ZO are t, - t and 0 respectively; (3) /q- = Z+, jZO = ZO; j on Z-, (4) q= { OonZ++Zo. The proof of this statement is given on pp. 460-462 of [28J. 4. In this paragraph we shall study the structure of j-algebra G' =K+jR+R. In fact it will be shown that

G' = G,o+G,t+G'l,

(13)

where G'l is the biggest subspace on which the real parts of operator adjr are equal to A. In this case (14)

and (with suitable definition of operator j)

G'o = K+jR 1 ,

G't = jRt +Rt,

Gil = R

1.

(15)

In addition to which, (16) We shall assume that Z = G' /K and define the skew symmetric form p(x,y) on Z by the relation

p(x,y) = w([x,yJ).

(17)

Obviously conditions (8) of the previous paragraph are in this case fulfilled. Let r E R be the element constructed in Paragraph 2. Operators

p = adjr-t,

q = adr

(18)

235

APPENDIX

preserve G' and K and induce linear transformations of space Z, which we shall also denote by p, q. Lemma 5 shows that [p,q] = q. It is easily deduced from the axioms for a j-algebra that operators p and q are symplectic in space Z and satisfy condition (11). Thus decomposition (12) holds. Property (2) of decomposition (12) shows that the real parts of the eigenvalues of operator adjr on factor space G'IK can take only values 0, -t, 1. Together with lemma 4 this gives decomposition (13). It follows from property (3) of decomposition (12) that with appropriate definition of operator j (19) Equality [r, K + R] 0, when compared with property (4) of decomposition (12), leads to inclusion (R) whem;e follows (14). Relations (15) are deduced from (14) and (19). Finally, formula (16) is equivalent to the first part of property (4) of decomposition (12). 5. In this paragraph it will be shown that the real parts of the eigenvalues of operator adjr on factor space GIG' are or t. We shall denote by G)' the biggest subspace in G on which the real parts of the eigenvalues of operator adjr are equal to A. Then

°

G

= IG A

(20)

(a direct sum) and (21)

If NeG is a subspace invariant with respect to adjr then N = INA, where N A = N n GA. Lemma 6. For any aER operator adja is commutative withj on factor space GIG' Proof From the condition of integrability we obtain: [ja,jx]-j[ja,x] :=j[a,jx]+[a,x]

(modK).

(S)

Obviously, the right side of this equality lies in G'. We shall examine j-invariant subspace

Q ={qEG: [q,r],[jq,r]ERt}

(22)

of algebra G. It is easy to see that

Q n G'

= K+jRt+tR.

(23)

236

THE GEOMETRY OF CLASSICAL DOMAINS

On the other hand

(24)

G=G'+Q. In actual fact, let XE G, [x, r] = a l Then x-a2-jal ERt. We shall show that

+ PI' [jx, r] = a2 + P2(ai ER

[jr, Q]

c

Q.

I

,

Po ER~).

(25)

Let qE Q. From Lemma 5 it follows that

[[Jr, q], r] = [r, q] + [jr, [q, r]] E Rt.

(T)

SincejqEQ it is similarly proved that [[jr,jq], r] ERt. Now, making use of the condition of integrability and formula (16), we obtain:

[j[jr, q], r] = [[jr,jq], r] - [j[r,jq], r] E Rt.

(D)

This shows that [jr,q] E Q. We have: Q = IQ\ where QA = Q n GA. From Lemma 6 and formula (23) it follows that

jQA

c

QA+K+jR-5:+Rt.

(26)

A

Lemma 7. If A =1= 0, 1, then w(G ) = O. Proof By axiom 5 of a j-algebra, for any x E G

w([jr, x]) = w([jx, r]).

(V)

If x EjRt+Rt, then [jx,r]EG't 0; inasmuch as operator adjr is nondegenerate on jRt + Rt = G't, it follows from here that

w(jRt + Rt) = O. Now let XE QA, A =1= O. Then in view of (26) [ix, r] E [QA, r] c R A+\

(27)

(W)

inasmuch as rER1. From (27) it follows that w(R A+ 1) =0, and so w([jr, x]) = O. Since operator adjr is nondegenerate on QA, W(QA) = O. This completes the proof of the lemma. Now we can prove that QA can be non-zero only when A = 0 or 1Let qE QA, q =1= O. From (26) we have

jq = q' +p' -jp+k,

(X)

237

APPENDIX

where q' E Q-\ p, p' E Rt, k E K. follows

This equality can be rewritten as

j(q+p) = q' +p' +k.

(Y)

Therefore

[j(q+p),q+p] = ['q,q] + [q',p] + [p',q] + [k,q+p] EG 2A +RA+t +[K, G] (Z) From here it follows that either q+ p EK, or W(G 2A ) =1= 0, or W(RA+t) =1=0 (see axioms 4 and 6 for aj-algebra). In view of Lemma 7 this is possible only when A = or -!-. 6. From the results of Paragraph 5 it follows in particular that Ri is an ideal in Lie algebra G. In actual fact, for any A

°

[Ri, GA]

c

RA+i.

°

(AA)

--!- and RA + 1 = or Ri. Therefore, [Ri, G] C Ri. Commutative ideal Ri is an ideal of the 1st kind (see Paragraph 2), since element r E Ri constructed in Paragragh 2 satisfies condition (12) of Theorem 1 (see formula (16)). It is also obvious that if R is an ideal of the 1st kind, R = Ri. Finally, if R =1= 0, then Rl =1= either, inasmuch as rER'. We shall assume that R is a commutative ideal of the 1st kind. In this case algebra G is broken down into the direct sum of subspaces: If G =1= 0, then in any case A =1=

°

G=S+jR+R+W,

(28)

in 'vvhich (1) S+jR = GO, (2) (3)

(4) (5)

R G 1 , W Gt; S = {SEGo: [s,l') = o}; S is a j-subalgebra containing K; jW= W; j-subalgebra K +jR + R + W satisfies all the conditions of Theorem 2 and corresponds to some Siegel domain of genus 2.

In actual fact, in the case under examination Rt = 0, and the definition of subspace Q (formula (22)) takes the form

Q = {qE G: [q, r], [jq, r] = O}.

(29)

We shall assume that s = QO, W = Qt = Gt. Then decomposition (28) holds. Property (2) of this decomposition follows from formulas

238

THE GEOMETRY OF CLASSICAL DOMAINS

(16) and (29). Since GO is a subalgebra, it follows from property (2) that S is also a subalgebra. Formula (26) shows that with the appropriate definition of operator j jSc S, jW= W.

(BB)

Verification of the remaining properties of decomposition (28) is trivial. A deeper study of decomposition (28) will be made in Section 4, and on the basis of this the main theorem of the present article will be proved in Section 5. Section 3. Algebraic j-algebras 1. Let {G, K,j} be an arbitrary j-algebra, GC = G + iG a complex extension of Lie algebra G and Z ~ Z an antilinear automorphism of algebra GC leaving all the elements of G in their place. We shall examine a complex subspace G+ of algebra GC , consisting of elements of form x+ijx, where XEG. (The j operator is here understood to be nonunique). Axioms 2 and 3 for aj-algebra are equivalent to the fact that G+ is a subalgebra of Lie algebra GC • Mapping Z ~ Z induces an antilinear isomorphism of subalgebra G+ on subalgebra G-, consisting of elements ofform x- ijx. The following relations hold: (1) where K = K+iK: We shall continue form w over linearity onto space GC • From axioms (4)-(6) (see p. 222) for aj-algebra it is easy to deduce that C

w([G+, G+]) = w([G-, G-]) = 0 and that when Z E G+ , Z ¢= K

(2)

C

iw([z, z]) ) O.

(3)

2. In this paragraph we shall obtain an auxiliary result. Let G be a j-algebra. For every g E G we assume p(g)x = [jg,x]-j[g.x]

(XEG).

(4)

From the axioms for a j-algebra it follows that operator p(g) preserves K and is commutative withj in factor space G/K. Lemma 1. In every j-algebra G there exists an element go such that operator p(go) is nondegenerate in factor space G/K.

239

APPENDIX

Proof' First of all let Lie algebra G be semisimple. We shall determine element go from condition: B(go,x) = w(x) for all XEG

(5)

where B denotes the Cartan scalar product. From equality w([jgo,go])

= B(go, [jgo, go]) = 0

(CC)

it follows that go EK. One can therefore consider thatigo = o. Then p(go) = - iadg o, and it is sufficient to show that operator adg o is nondegenerate on G/K. Let XEG be an element such that [go,x]EK. We shall select ix in such a way that B(K,ix) = O. Then w([jx,x])

= B(go, [jx,x])

-B([go,x],ix)

= 0,

(DD)

and consequently x E K. If algebra G is not semisimple, a non-nil commutative ideal of the 1st kind, say R (see Paragraph 6 Section 2), exists in it and decomposition (28) of Section 2 holds. Let r be the element constructed in Paragraph 2 Section 2. Operator per) preserves i-subalgebra S and, as can easily be seen, is nondegenerate on G/S. We can assume that for a i-algebra of smaller dimensions the lemma is proved. Then there exists an element So E S such that operator peso) is nondegenerate on S/K. With a suitable A, element go = r+Aso then satisfies the requirement of the lemma. 3. We shall agree henceforth to consider i-algebra G effective (see Paragraph 1 Section 2). Then the adjoint representation of algebra G is accurate (Lemma 2 of Section 2). We shall call i-algebra {G, K,i} algebraic if linear Lie algebra ad G is algebraic. Theorem 3. Any eflective i-algebra {G, K,i} can be included in the form of a i-subalgebra into an algebraic i-algebra {Ga' Ka,i} in such a way that (6)

If the i-algebra G is proper,

then i-algebra Ga is also proper (see introduction). Proof We shall use the notations of Paragraph 1 and shall identify the elements of algebra G C with their images in the case of adjoint representation of algebra GC • Then algebra GC and all its sub algebras realized in the form of linear Lie algebras are in space GC • Mapping

240

THE GEOMETRY OF CLASSICAL DOMAINS

z -+ Z is a

restriction on GC of the involuntary transformation A -+ A of the set of all endomorphisms of complex space GC • (Here Au = Au for uEG C ) . Let G~ and G! be algebraic hulls of Lie algebras GC and G± respectively.t Then G: G+ a' We shall prove that (7)

It is known that G ::;) [G~, G~J (see [35J, Theorem 13, Chapter 2). From the obvious inclusions C

(EE) it follows that G! + G: is a Lie algebra. Since G! and G: are algebraic Lie algebras, algebra G! + G: is also algebraic (see [35J, Theorem 14, Chapter 2) and therefore coincides with G~. We shall assume (8)

(We note that K~ is not, generally speaking, an algebraic hull of algebra KC). Let go E G be the element which figures in lemma 1. We shall take an element Zo = go + ijgo E G+ and prove that operator adz o is nondegenerate on GC/G+. Let z = X+iYEG(' (x, y,EG). Then [zo,zJ = [go,zJ+i(p(go)z+j[go,zJ) == ip(go)z (modG+). (FF) (Here operators peg 0) and j are considered to be extended in linearity onto GC). If [zo, z J E G+, then p(go)z=p(go)x+ip(go)YEG,

(GG)

whence p(go)y-jp(go)x ==p(go)(y-jx) == 0 (modK). (HH) By the property of element go this implies that y - jx EK, and means thatzEG+. From the nondegeneracy of operator adz o on GC/G+ it follows (see [35J supposition 9 and consequence of supposition 15, Section 4, Chapter 6) that algebra G+ contains some Cart an subalgebra H C of algebra GC • Let H~ be the algebraic hull of algebra H C • Then (see [35J supposition 21, Section 4, Chapter 6) G~ = GC + H~ and furthermore (9)

t The algebraic hull of a linear Lie algebra H is the name given to the smallest algebraic Lie algebra containing H.

APPENDIX

241

whence (II)

and finally (10)

We shall now prove that (11)

Let zEG! n G-; then [ z, zJ E [ G: , G+J = [G +, G+J,

(JJ)

and reI ations (2) and (3) show that (12)

Let Ga be the algebraic hull of algebra G and Ka = Ka n Ga. Then Ka+iKa. From (10) and (12) we obtain:

G~ = Ga+iG a and K~

(13) In addition to which (14) We shall extend form w arbitrarily from algebra G to Ga and prove that {G a, Ka,j, w} is a .i-algebra. In view of (14) only the verification of axiom 4 is trivial. We have [Ka'

GaJ c [G: n G;, G: +G;J c [G:, G:J+[G;, G;] = [G +, G+J + [G - , G-J

(KK)

and w([Ka , GaJ) = 0 (see formula (2)). Finally, if j-algebra G is proper, then the j-algebra Ga which we constructed is also proper. In actual fact let L be a compact semisimple j-subalgebra of algebra Ga. Then L = [L,LJ c [G a, GaJ c G and L cKcKa • 4. Theorem 4. Let {G, K,j} be an algebraic j-algebra. If j-subalgebra G' of algebra G contains K, it is algebraic in the adjoint representation G. Proof

011

As in the proof of theorem 3 we shall regard all the Lie algebras under examination as the Lie algebras oflinear transformations of space G The Lie algebras associated with G' will be given the same C

•

242

THE GEOMETRY OF CLASSICAL DOMAINS

notation as the corresponding Lie algebras associated with G except that they will be given a prime. In the case under examination, algebras G and GC are algebraic; it follows from (11) that algebra G+ is also algebraic. We must show that algebra G' is algebraic as well. We have (LL) (analogue of equality (9)). Further, G~ + G~c c G+ + G'- . More accurately G~c =

C

G; = G+, and therefore

(G+ n G~C)+ G'-

= G+ n(G'+ +G~-)+G'-

c

G,c+(G+ n G-)

= G'c.

(MM)

Section 4. Decomposition of a j-algebra Associated with a Commutative Ideal (continuation)

1. In Paragraphs 1-6 of this section we shall restrict ourselves to the study of algebraic j-algebras. First let {G, K,j} be an effective algebraic j-algebra corresponding to a Siegel domain of genus 1. The following decomposition holds (see Section 1). G = K=/=jR+R,

in which R is a commutative ideal and linear lie algebra. H = adR G ~ K +jR generates a group which acts transitively in a convex cone VcR. Algebra H is algebraic and therefore contains a triangular subalgebra T which generates a simply transitive group of automorphisms of cone V (see [6J, Section 7, Chapter 1). With the appropriate definition of operator j'algebra T can be identified with jR, and then jR + R is a j-subalgebra of algebra G. There exist such elements fa E R(a = 1, ... , m) and such a decomposition of space R into the direct sum of subspaces : (1)

that Raa

= (r a) and for

evelY x E R [jra,xJ = 1((\p+bay )X,

(2)

[jra,jxJ = 1(bap -bay )jx.

(3)

In this case

(4)

APPENDIX

243

These facts can be obtained without difficulty by means of the matrix calulcus constructed in [6J or by deduction from the results of [28]. 2. Now let {G, K,j} be an arbitrary effective algebraic i-algebra and R a commutative ideal of the first kind of algebra G. According to Theorem 4 j-subalgebra G' = K +jR + R is algebraic in the adjoint representation on G. Algebra G' can be noneffective. The kernel of noneffectiveness for it is ideal

Ko = {kEK: [k,RJ = O}.

(5)

We shall denote by Z(Ko) the centralizer of ideal Ko in algebra G'. Since the adjoint representation of Ko in G is completely reducible (Lemma 1, Section 2), then (6)

It is also to see that algebra Z(Ko) is algebraic (in the adjoint representation on G).

We shall define operator j such that jR

c

Z(Ko). Algebra

Z(Ko)j(Ko n Z(Ko)) satisfies the conditions of Paragraph 1. . Consequently, elements raER, jraEjR exist such that relations (2) and (3) are satisfied over modulus Ko n Z(Ko). Further, there exist replicas c~ + c; of element jra in algebra Z(Ko) (see [25J Section 14 Chapter 2) such thatjra = c~+c~, operator ad c~ is semisimple and has real eigenvalues, whilst operator ad c; has purely imaginary eigenvalues. When element jra is replaced by c~ relations (2) and (3) continue to remain valid over modulus Ko nZ(Ko)' From (2) it follows that [c~, rJ = ra; therefore c~ :=jraCmod Ko nZ(Ko)), and, correcting operator j if necessary, it may be considered that jra = c~. U sing this we shall henceforth assume without special reservations that operators adjra are semi-simple and have real eigenvalues. It follows from (2) that operators adjra are commutative on R. This shows that

[jra,jrpJ EKo nZ(Ko)

and

[jra, [jra,jrpJJ = O.

(NN)

From the semisimplicity of operator adjra it then follows that (7)

244

THE GEOMETRY OF CLASSICAL DOMAINS

Algebra G decomposes into a direct sum of weighting subspaces with respect to operators ad jra: (8)

where A = {Aa} are different collections of real numbers and when xEG A

[jra'X] = Aax .

(9)

A

Those A for which G i= 0 we shall call weights. Obviously

[G A , G M ]

c

GA + M •

We shall denote by jj.P a collection of numbers for which i\.~ =b ap . It follows from formula (2) that collections ±(i\.a+ i\.P) are weights on R. Further, formula (3) shows that the weights on subspace H = (Ko nZ(Ko))+jR have the form ±(i\.a_i\.p), where ex ?:f3; in this case H O = (Ko n Z(Ko)) + l}jra)

(00)

(direct sum). We shall define operator j so that jR

=

L(jr )+ L a

HA.

(10)

A*O

Then jR will be a subalgebra. 3. We shall ascertain what the weights A on factor space GIR can be. First of all operators adjra are commutative withj onto GIG' (Lemma 6, Section 2), and therefore

jG A

c

GA + G' .

(11 )

Lemma 1. IfgEG\ thenjg = g' +a+jb+k, where g,EG\ kEK and (a)

a, bERA+ IRA+L\

(PP)

a

Proof We shall write the condition of integrability for elements ra' g and project it onto R. Then we shall obtain:

[Jra' a] = Aaa + era, g].

(QQ)

Since [ra,g] ERA+L\", for any ex (P)

(mod L RA + L\ ). jJ

(RR)

245

APPENDIX

The confirmation of the lemma for element a follows from here. For element b the proof is similar inasmuch as

jg' = -g+b-ja+k'

(k' EK).

(SS) A Lemma 2. 1fvveight A is other than !.lex or !.lP-!.lY,w(G ) = 0. Proof First of all it is easy to show that W(RA) =1= only for A = !.lex, i.e. for RA = Rexex. Further, using the notations of Lemma 1, we have:

°

w([jrex,g]) = w([jg, rex]) = w([g',rex])+w([jrex,b]).

(TT)

Since [}rex,b] = Aexb+[rex,g'], for any rx

Aiw(g)-w(b)) = 0. If A

=1=

(UU)

0, it follows from here that

w(g)

web),

whence with the aid of Lemma lour statement is obtained.

Lemma 3. For every weight A on factor space G/R at least one of the following conditions is fulfilled: (1) W(G 2A ) =1= 0, (2) W(R 2A +L\"') for some rx. Proof Let g E G\ g E R. In the notations of Lemma 1

°

j(g-b)

g' +a+k,

whence

[j(g-b),g-b] = [g',g]+[a,g]

[g',b]+[k,g-b]

(VV)

and

w([g',g])+w([a,g])+w([b,g']) 2A

It remains to note that [g', g] E G

0.

=1=

(WW)

and

[a,g],[b,g']ER 2A +LR 2A +L\ ex

(ex) •

(XX)

Lemma 4. Every weight 071 factor space GjR has the form 11.\2 or !(!.lex -!.lP). The proof is obtained by comparing Lemmas 2 and 3. 4. In Paragraph 6 of Section 2 we obtained the decomposition G=S+jR+R+W

(12)

246

THE GEOMETRY OF CLASSICAL DOMAINS

Where S is a j-subalgebra containing K, and W is a j-invariant subspace. In our notations S+jR =

I LA"

G\

I

R

=0

rA"

G\

W=

=1

I LA"

(13)

GA. =t

We shall assume thatjR is a subalgebra defined by condition (10). Lemma 5. If A = ±([\a- [\P), a ~ /3, then

(S+jRt = (S n GA)+(JRt.

(YY)

Proof Let g E (s +jRt. If A 0, then [g, raJ = cara (c a are real numbers) and g - Icajra E S. Since jra E (JR) 0 at A = 0 everything is proved by this. Now let A = ±(L\.a_L\.p), a
(S+jRt

c

S+(jR)-A.

(ZZ)

Proof For every g E (S +jRt we have: [g, ryJ = 0 when y =ft /3, and [g, rpJ E Rpa. There exists an element a E Rpa such that Cia, rpJ = [g, rp]. In this case ja E (jR) - A and g - ja E S. 5. We shall examine the ideal So = {SES: [s,RJ = O}

(14)

of algebra S. In this paragraph we shall show that factor algebra Sf So is compact (i.e., it is a Lie algebra of a compact Lie group). Linear Lie algebra adRjR = T generates a group which acts transitively in convex cone VcR (Paragraph 1). We shall prove that algebra ad R G = ad R (S +jR) containing T consists of the differentiations of cone V. From one result due to [6J (Proposition 17 Chapter 1) it follows that for this it is sufficient to show that for any s E S:

SpadRs = O.

(15)

We shall introduce the following notation:

a(g) = SpadRg.

(16)

APPENDIX

247

Obviously

a([G, GJ) = 0

(17) A

and, inasmuch as when A =1= 0 operators in adR G are nilpotent,

a(G A ) = 0 when

A

=1=

O.

(18)

Lemmas 5, 6 and formula (18) show that it is sufficient to prove equality (15) for the case SES, aER

j[s, aJ = [s,jaJiR' Lemma 7. aER

Let A

= 1(L\_ClL\P), ex ;:;;'{3.

(19) Then for any

a([s,jaJs) = O.

SES n

G\ (20)

Proof If a = r y ' then either [s,ja] = 0 or A =1= 0 and [s,ja J E S n GA. In the latter case formula (20) follows from (I8). Now let a ERyo, 'Y(c5. Then [s,jaJE(S+jR)M, where M has the form 1(LV'-L\'I) when 8(1]. Relation (20) follows in this case from Lemma 5 and formula (18). We shall examine the bilinear form (a, b) = a(j[ja, bJ)

(21)

on space R. It is symmetric and positively defined (see [6J Section 1, Chapter 2). Lemma 8. When s E S n GO operator ad R s is skew symmetric in metric (21). Proof By means of formula (19) and (17) we find:

([s, aJ, b)+(a, [s, bJ) = a(j[j[s, aJ, bJ) +a(j[ja, [s, bJJ) = a(j[s, [ja, bJJ) - a(j[[s,jaJs, bJ) = - a([s,j[ja, bJsJ) + a([[s,ja ]s,jb ]s)'

(AAA)

From Lemma 7 it follows that a([s,j[ja, b]sJ) = O. It remains to prove that

a([[s,ja Js,jb ]s)

= O.

(22)

Let aER/. Then [s,ja] E(S+jR)/)., where A = 1(L\.Cl_L\.P), and formula (22) follows from Lemmas 5 and 7. Since the trace of a skew symmetric operator is zero, it follows from Lemma 8 that a(S n GO) = O. As already noted above, the validity of equality (15) for all S E S follows from this. Consequently, transforma-

248

THE GEOMETRY OF CLASSICAL DOMAINS

tions of adR S are differentiations of cone V. In this case [S, rJ = 0, so that the automorphism group of cone V generated by adR S is contained in a stationary subgroup of point rEV and is therefore compact (see for example [6J, Section 7, Chapter 1). From the obvious isomorphism adRS:::::.SfSo it now follows that Lie algebra Sf So is compact, which is what we set out to prove. 6. We shall assume that R is a maximal commutative ideal of the 1st kind and prove that in this case Lie algebra S decomposes into a direct sum of a semisimple ideal Sl' having no compact components and lying in So, and a compact ideal K 1 : (23) It will then follow from Borel's results [lJ that Sl and K1 arej-invariant, and therefore if j-algebra G is proper (see introduction), K1 c K.

°

Lemma 9. Let W = and R be a commutative ideal of the first kind contained in Sl. If r is an element of ideal Rfor which [ja, rJ a for all iiER,

[jr,RJ =

°

(24)

with suitable selection of element jr. Proof Let

G=S+jR+R+

W

be a decomposition of type (12) associated with ideal R. Element jr can be selected in such a way that operator adjr is semisimple and has real eigenvalues (Paragraph 2). Then

o on ~+jR, adjr =

1 on R,

{±on Jill:

Ideal R, being invariant with respect to adjr, decomposes into a direct sum of its intersections with subspaces S+jR, R and Jill: Since R c So, R n R = 0. It is now sufficient to show that R n W = 0. Let x ERn W. Then [jx, x JE R, and on the other hand, [jx, x JE [w, WJ c R. From here it follows that [jx, x J = and so x = 0. If it is not assumed that W = 0, one can examine j-subalgebra

°

G" = S+jR+R.

(25)

APPENDIX

249

It is not obliged to be effective, and therefore Lemma 9 is not directly applicable to it. Let Koo be the noneffectiveness kernel of j-algebra Gil, i.e. the biggest ideal of Lie algebra Gil contained in K. We shall denote by Z(Koo) the centralizer of ideal Koo in algebra Gil' Since the adjoint representation of Lie algebra Koo is fully reducible in G, it is also fully reducible in Gil, and so

(26) (compare formula (6) Paragraph 2). For j-algebra Z(Koo) its centre Z = Koo n Z(Koo) serves as the kernel of effectiveness. We shall set Soo = So n Z(Koo) and prove that factor algebra Soo/Z is semisimple. It is easy to see that So is an ideal in Gil. From here it follows that Soo/Z is an ideal in Z(Koo)/Z. If Lie algebra Soo/Z were not semisimple, there would exist an ideal R of algebra Z(Koo) such that Z

eRe Soo, R =1= Z

(BBB)

and R/Z is a commutative ideal of algebra Z(Koo)/Z. In view of the remark made in Paragraph 6 of Section 2, R/Z can be considered as an ideal of the first kind in j-algebra Z(Koo)/Z. There exists an element rE R defined correct to the addend in Z, such that

[jG,r]==G(modZ) for all GER

(27)

Lemma 9 enables one to state that element jr can be defined in such a way that

[jr, R] == 0 (modZ).

(CCC)

Since R n Z = 0, it follows from the last relation that

[jr,R]

= 0

(28)

In the same way as in Paragraph 2, it is shown that with the appropriate definition of element jr operator adjr is semisimple and has real eigenvalues on G. On this condition we shall set G~

= {XEG": [jr,x] = AX}.

(29)

Then (30)

moreover G'{ eRe So

(31)

250 and element

THE GEOMETRY OF CLASSICAL DOMAINS

r can be corrected in such a way that rEG~_

(32)

We shall set p

= adwjr,

q

= adwr.

(33)

Operators p and q are symplectic (Paragraph 3 Section 2) with respect to skew symmetric form

p(u, v) = w([u, vJ)

(34)

(u, V E W).

In actual fact it follows from (28) that p(pu, v)+p(u,pv) = w([jr, [u, vJJ)

0,

(DDD)

as [u, vJ E R. The symplecticity of operator q is proved in the same way. We shall apply the result of [28J, given in Paragraph 3, section 2, to operators p and q. It is easy to verify that conditions (10) and (11) of Section 2 are satisfied for operators p and q. Consequently, space W is decomposed into a direct sum W= W++W-+W o (35) in such a manner that (1)

(2) From an examination of the eigenvalues of operators adjr and adjr it is easy to obtain that subspace R' = + R + W+ is a commutative ideal of algebra G. Operator ad j(r + r) is identical on R', and therefore R, is an ideal of the first kind. This contradicts our assumption that R is a maximal commutative ideal of the first kind. Consequently, Lie algebra Soo/Z is semisimple. Since [Z, SooJ = 0, algebra Soo is broken down into a direct sum of its centre Z and semisimple ideal Sz. Ideal Sz is in turn broken down into a direct sum of semisimple ideal Sl' having no compact components, and compact semisimple ideal KzObviously, Koo c So, therefore

Gr

(EEE)

APPENDIX

251

Algebra Koo + K2 is a compact ideal in So. Since factor algebra S /So is compact, factor algebra S/S1 is also compact. Decomposition (23) follows from here. 7. Theorem 5. Let {G, K,j} be an effective proper j-algebra. Lie algebra G can be decomposed into a semisimple sum (36) in such a way that (1) S1 is a semisimple j-subalgebra having no compact components;

(2) Gl is a j-ideal corresponding to a homogeneous Siegel domain of genus 1 or 2 with an affine automorphism group. (3) K= KnS 1 +Kn Gl • Algebra G1 is decomposed into a direct sum of subspaces Gl = Kl +jR+R+ W,

(37)

in which Kl = K n G1 , R is a maximal commutative ideal of the first kind of j-algebra Gt and all the conditions of Theorem 2 are satisfied. In this case (38)

Proof First of all let G be an algebraic j-algebra, comparing (12) and (23) we obtain the decomposition (39)

Since Sl C So, [Sl,R] = 0, [Sl' W] C W, and to prove the theorem it is sufficient to verify that the subspace G1

= Kl +jR+R+ W

is a subalgebra and that (40) First of all we shall prove relation (40). Since [jR, So] c So and S 1 is a semisimple ideal in So, [jR,Sl]

C

Sl'

We shall show that the weights of operators adjra on Sl are zero.

t It is easy to show that in fact a maximal commutative ideal of the first kind is unique.

252

THE GEOMETRY OF CLASSICAL DOMAINS

From the j-invariance of subspace S 1 and formula (11) it follows that jS~ c S1+(K n Sl)'

(FFF)

By correcting operator j one can achieve (GGG)

Then [jS~, Sn c Sill. If A =1= 0, 2A is not a weight on GjR (see Lemma 4). Consequently, [jSt, stJ = 0 and c Sl n K when A =1= O. We shall prove that

st

CSt, SlJ

c

Sl n K when A =1= O.

cst,

(41)

As Sl = ISr it is sufficient to verify that SrJ c Sl n K for any M. If M =1= 0 this follows from the fact that Sr c Sl n K. If M = 0, then SrJ c Sl nK. Set N of elements nE S 1 n K, for which en, s 1J c s 1 n K, is an ideal of algebra Sl' Since Sl has no compact ideals, N = O. Formula (41) means that c N when A =1= O. Consequently =1= 0 only when A = O. Thus, all the weights on S 1 are zero, i.e. [jra, S 1J = 0 for all a. We have

cst,

cst

st

st,

st

I

jR = (jR)O +

(jR),\

(HHH)

A*O

moreover (jR)O =

I

(jra)

(Ill)

(Paragraph 1, 2). If A =1= 0,

[(jRY\ SlJ

cst = O.

(JJJ)

In this way relation (40) is proved. We know that the subspace Sl +K1 +jR = S+jR = GO

(KKK)

is a subalgebra. Since [SbK1J = 0 and as proved [Sl,jRJ = 0, subspace K1 +jR is the centralizer of subalgebra Sl in GO and is consequently a subalgebra. From here it is easy to deduce that G1 = K1 +jR+R+ W

is also a subalgebra. Now let {G, K,j} be an arbitrary effective proper j-algebra and {Ga' Kmj} be an algebraic j-algebra containing G as a j-subalgebra and

253

APPENDIX

satisfying the conditions of Theorem 3, i.e.

Ga = G+Km [G a, GaJ

c

G.

(42)

By Theorem 3 j-algebra Ga is proper. It can also be regarded as effective. In actual fact, let N be an ideal of algebra Ga contained in Ka. Since j-algebra G is effective, N n G = 0, and we can examine GaiN instead of j-algebra Ga. Algebra Ga , as proved, decomposes into a direct sum

Ga = Sl +Ga1 , moreover S 1 is a semisimple j-subalgebra and

Gal = Kal +jR+R+ W,

(LLL)

where R is a maximal commutative ideal of the first kind of algebra Ga. Relations

Sl = [Sl, SlJ,

R = [jr, RJ,

W = [jr, WJ

(MMM)

show that Sl +R+ We G. Consequently G = Sl +G n (Kal +jR)+R+ W.

(NNN)

Correcting operator j, we obtain decomposition

G = Sl +Kl +jR+R+ W,

(000)

where Kl = G n K a1 . Obviously it satisfies all the requirements of the theorem. Section 5. Representation of a Homogeneous Domain in the Form of a Siegel Domain of Genus 2

In this section we shall obtain the fundamental result of the present article. It is easily deduced from Theorem 5. Theorem 6. Every homogeneous bounded domain !!fl in n-dimensional space e" is analytically equivalent to some homogeneous Siegel domain of genus 1 or 2. Proof Among all the connected algebraict transitive automorphism groups of domain !!fl we shall select the group cg with the smallest dimension. Let:/{ be a stationary subgroup of some point of domain !!fl and {G, K,j} be a j-algebra corresponding to cg and:/{o From the algebraicity ofj-algebra G it is easy to deduce that algebra K is algebraic tHere and henceforth by an algebraic Lie group we understand a Lie group whose adjoint group is the component of the identity of some algebraic linear group.

254

THE GEOMETRY OF CLASSICAL DOMAINS

in the adjoint representation on G. From here it follows that group % is compact. We shall decompose algebra G in accordance with Theorem 5: (1)

and prove that Sl = o. Decomposition (1) corresponds to a decomposition of group C§ into a semi direct product of semisimple subgroup gland normal divisor C§ l' Asj-algebra Gis proper,j-algebra Sl is also proper, and it follows from Borel's results [IJ that group g 1 acts transitively in a symmetric domain. We shall set Ks = K n S l' We know that group g 1 can be decomposed into the product of subgroup :Y('s (corresponding to Lie algebra Ks) and an algebraic solvable group :!I which intersects :Y('s only in the identity. Obviously group :!IC§ 1 is transitive in domain f!}. In view of our assumption concerning the minimum nature of group C§, it follows from here that:/{ s = {e} and S 1 = o. Thus, G= G 1 =K1 +jR+R+W.

(2)

By Theorem 2 there exists a homogeneous Siegel domain of genus 2 whose j-algebra is isomorphic to {G, K,j}. Let C§' be a connected automorphism group of domain f!}', whose Lie algebra is isomorphic to G, and :/(' be a stationary subgroup of C§'. It is easy to show that group :Y(" is connected and is the maximal compact subgroup in C§' • Group C§ is the covering group for C§'. Let re be a natural homomorphism of C§ on C§'. Then re(:/{) c;j['. Since group % is compact, it follows from here that re(:/{) =:/{'. The mapping of re consequently induces a covering mapping of domain f!} onto domain f!}'. This covering, however, is trivial, for domain f!}' is homomorphic to a Euclidean space. In this way we have proved that f!} ~ f!}'. f!}'

REFERENCES 1. BOREL, A. Kahlerian coset spaces of semisimple Lie groups. Proc. Nat. Acad Sci USA, 40, 12 (1954) 1147-1151. 2. VINBERG, E. B. Homogeneous cones, DAN 133, No.1 (1960). 3. VINBERG, E. B. Morozov-Borel theorem for real Lie groups, DAN 141, No.2 (1961). 4. VINBERG, E. B. Convex homogeneous domains. DAN 141. No.3 (1961).

REFERENCES

255

5. VINBERG, E. B. Automorphisms of homogeneous convex cones, DAN 143, No. 2 (1962). 6. VINBERG, E. B. Theory of homogeneous convex cones. Trudy Mosk, matem. o-va 12 (1963). 7. VINBERG, E. B., and GINDIKIN, S. G. On some nonassociative algebras occurring in the theory of homogeneous domains. UMN 17, No.6 (1962) 229-230. 8. GINDIKIN, S. G. Integral formulae for Siegel domains of genus 2, DAN 141, No.3 (1961). 9. GINDIKIN, S. G. Integral formulae for complex homogeneous bounded domains. UMN 17, No.3 (1962),209-211. 10. GrNDIKIN, S. G., and KARPELEVICH, F. 1. On some special functions of several variables associated with Lie groups. Theses Reports at the Second AlIUnion Conference on the Constructive Theory of Functions, 1962. 11. SIEGEL, K. L. Automorphic functions of several complex variables, Moscow, Gostekhizdat, 1954. 12. CARTAN, E. Sur les domaines borll(~s homogenes de l'espace de n variables complexes, AM. Math. Sem. Ham. Univ. 11 (1935), 116-162. 13. KOCHER, M. Positivitatsbereiche im Rn, Amer. Journ. Math. 79, 3 (1957), 5-75 596. 14. KOCHER, M. Analysis in reellen Jordan Algebren, Nachr. Wiss. Gottingen 5 (1958), 67-74. 15. KOCHER, M. Die Geodatischen von Positivitatsbereichent, Math. Ann. 135, 3 (1958), 192-202. 16. KOSZUL, J. L. Sur la forme hermitienne canonique des espaces homogenes complexes, Canad. Joum. Math. 7, 4 (1955), 562-576. 17. KOSZUL, J. L. Domaines bornes homogenes et orbites de groupes de transformations affines, Bull. Soc. Math. France 89, 4 (1961), 515-533. 18. KOSZUL, J. L. Ouverto convexes des espaces affines, Math. Zeitschr. 79 (1962), 254--259. 19. POINCARE, A. Les fonctions analytiques de deux variables et la representation conforme, Rend. die bircolo mat. di Palermo 23 (1907), 185-220. 20. PYATETSKll-SHAPIRO, 1. 1. On evaluating the dimension of a space of automorphic forms for certain types of discrete groups, DAN 113, No.5 (1957). 21. PYATETSKil-SHAPIRO, 1. 1. Some questions of harmonic analysis in homogeneous cones. DAN 116, No.2 (1957). 22. PYATETSKll-SHAPIRO,1. 1. On a problem of E. Cartan. DAN 124, No.2 (1959). 23. PYATETSKil-SHAPIRO, 1. 1. The geometry of homogeneous domains and the theory of automorphic functions. The solutions of a problem of E. Cartan, UMN 14, No.3 (1959), 190-192. 24. PYATETSKII-SHAPIRO, 1. 1. The theory of modular functions and contiguous questions of the theory of discrete groups. UMN 15, No.1 (1960),99-136. 25. PYATETSKII-SHAPIRO, 1. 1. The geometry of classic domains and the theory of automorphic functions. Moscow, Fizmatgiz, 1961. 26. PYATETSKII-SHAPIRO, 1. 1. On the classification of bounded domains in ndimensional complex space, DAN 141, No.2 (1961). 27. PYATETSKil-SHAPIRO,1. 1. On bounded homogenous domains in n-dimensional complex space, lzv. AN, ser. matem, 26, 1 (1962), 107-124. 28. PYATETSKII-SHAPIRO, 1. 1. Structure of j-algebras, lzv. AM, ser, matem. 26, 3 (1962), 453--484. 29. PYATETSKll-SHAPIRO, 1. 1. Generalized upper halfplanes in the theory of many complex variables, Transactions of the Stockholm Mathematical Congress, 1963.

256

REFERENCES

30. ROTHAUS, O. S. Domains of positivity, Bull. Amer. Math. Soc. 64, 2 (1958), 85-86. 31. ROTHAUS, O. S. Domains of positivity, AM. Math., Sem. Univ. Hamburg 24 (1960), 189-235. 32. FucHs, B. A. Theory of analytic functions of many complex variables, 2nd ed., vols. 1,2, Moscow, Fizmatgiz, 1962, 1963. 33. HANO, J. On Kahlerian homogeneous spaces of unimodular Lie groups Amer. Journ. Math. 79, 4 (1957), 885-900. ' 34. HERTNECK, CH. Positivitatsbereiche und Jordan Strukturen, Math. Ann. 146, 6 (1962), 433-455. 35. CHEVALIER, K. Theory of Lie Groups, vols. 2, 3, Moscow Foreign Lit. Publishing House, 1958.

Index Page numbers in bold type refer to chapter headings, those in italics to section headings and those asterisked to footnotes. Abelian funct}on(s) 179* isomorphic fields of 13 modular-see Abelian modular functions non-degenerate 12*, 180* non-isomorphic fields of 12, 180 Abelian modular function(s) 1, 179 field of 184 fundamental results 179 Adjoint representation of Lie algebra 232 Affine automorphisms 229 Affine complex space 15 Affine representation(s) 67 of Lie algebra 230 Affine transformation(s) 17, 29, 67, 101, 161 group 225 infinitesmal 67* unimodular 161 Algebra(s), bigraded 11 division 210 factor 246, 249 involution in 195 isogenous representations of 191 j- -see j-algebra(s) Jordan 21 left-symmetric 21 Lie-:-see Lie algebra(s) matrix 10 non-associative 21

non-isogenous representations of 189 of Cayley numbers 200, 211 of complex numbers 200 of quaternions 200 simple 186 T- 10, 21 uniquely determined by a complex 204, 208 Algebraic j-algebra(s) 216, 238 decomposition of 218 effective 242 Algebraic Lie algebra 217-see also Lie algebra(s) Algebraic relations, theorem on 2, 3, 4 163, 177, 223 Analytic automorphisms 2, 34 discrete groups of 131 of bounded domain(s) 46, 131 of component 105, 106 Analytic curve 84 * Analytic functions, linear space of 171 dimension of 171, 176 Analytic homogeneous Siegel domain 26, 28-see also Siegel domain(s) Analytic normal spaces 136 Analytic set 84 Analytic subset 136 Andreotti-Granert method 4, 159, 160 Antilinear transformations 77 Arithmetic group(s) 131, 162 257

258

INDEX

Andreotti-Granert conditions 160 arbitrary 4, 160 fundamental domains of 4, 153 in symmetric domains 153, 163 Associate vectors 169 Automorphic forms 163,173 analytic at infinity 147, 149 construction of 141 for discrete groups 163 functional equation of 174 F-automorphic 140, 167, 169 separating points 152 space of 174, 178 Automorphic functions 1, 2, 131, 159, 160 Automorphisms, affine 229 analytic-see Analytic automorphisms group of 220 Bergman manifold 8, 48, 49 Bergman metric 8 Borel-Satake construction 5 Boundary component 84, 85*, 93, 99 -see also Component Boundary points, pseudo-concave 159 Bounded domain(s), analytic automorphisms of 34, 131 -see also Bounded homogeneous domain(s) Bounded holomorphic hull(s) 6,41,43 Bounded homogeneous domain(s) 8, 30, 219, 253 canonical models of 76 classification of 199 holomorphism of 69 homogeneous imbeddings of 199, 211 integral formulae for 220 non-symmetric 90 realizations as Siegel domains 76, 220, 225, 253 symmetric 219 Canonical models, of bounded homogeneous domains 76

of symmetric domains 80 Canonical realization(s) 96, 151 of classical domains 117, 125 of unit circle type 83 Cayley numbers 21 algebra of 200, 211 Chebotarev's lemma 17 Classical domain(s), analytically equivalent component 115 analytic automorphisms of 117, 121 canonical realizations of 117, 125 geodesics 122 geometry of 83 of the first type 91 of the second type 114, 120 of the third type 114, 123 Riemann distance 122 Classification of symmetric domainssee Symmetric domains Commutative ideal 55, 157, 231, 232, 237, 242 maximal 158, 248, 251* of the first kind 232, 237, 248, 251 * Complex(es) 9, 10, 199, 204 Complex space, affine 15 Component 86, 87, 90, 91 analytical equivalence of 115 analytic automorphisms of 105, 106 boundary-see Boundary component "distance" 87 infinitely distant 100, 103, 111 mapping 94 membership of points in 95 zero-dimensional 90, 126 Cone(s) 16* affine homogeneous 21 affine transformations of 17, 29 dual 148 homogeneous 18, 21, 220 irreducible 18 non-self-adjoint 21 polynomial inequalities defining 73 rank of 70* self-adjoint 21 Continuable space-see Space(s)

INDEX

Continuable transformation-see Transformation(s) Cylindrical domain(s) 69, 110, 146 j-algebra of 226, 227 Degree of transcendence of a field of automorphic functions 2 131 D~recting subgroup-see Subgro~p(s) DIscrete group(s), of analytic automorphisms 163, 131 quasinormal 177, 178 Distance, between points-see Component Riemann-see Riemann distance Division algebra-see Algebra(s) Domain(s), bounded-see Bounded domain(s) bounded homogeneous-see Bounded homogeneous domain(s) classical-see Classical domain(s) cyli?drical-see Cylindrical domain(s) eqUIvalent 29, 185 fundamental 141 geometry of-see Geometry of domains homogeneous-see Homogeneous domains K(\ll, R)

184

of arithmetic groups 153 of holomorphy 69 pseudo-concave 159 Siegel-see Siegel domain(s) symmetric-see Symmetric domain(s) Dual cone 148 Dual manifold 97 Elementary j-algebra(s) 53, 56 commutator of 53, 56 j-homomorphisms of 65 Elliptic functions 11, 179 Equivalent, domains j-algebras matrices pairs -see subheadings

259

Euclidean spaces 59, 204 Extension of the factor space 9fr 3, 133 Factor algebra 246 compact 246 semisimple 249 Factor space 9fr 2,3, 131, 133 Fibering, automorphisms preserving 81 * cylindrical domain associated with 174* r-rational 133, 134 homogeneous-see Homogeneous fibering Form(s), automorphic-see Automorphic forms T-automorphic 140, 167 non-degenerate 31 semi-Hermitian 30, 31 vector 31 Fourier-Jacobi series 4, 7, 163, 164 coefficients 164 Frobenius theorem 191 Function(s), abelian-see Abelian function(s) and Abelian modular function(s) analytic at infinity 139 bilinear 202 elliptic 11, 179 T-equivalent 138* T-invariant 138*, 167, 168 Jacobian 165 modular-see Modular function(s) and Abelian modular function(s) separating points 137* theta 4, 184 vector-see Vector functions V-Hermitian-see V-Hermitian function(s) Fundamental, domain parallelipiped -see subheadings T-automorphic forms-see Automorphic forms

260

INDEX

T-equivalent function 138* T-invariant function 138*, 167, 168 T-rational homogeneous fibering-see Homogeneous fibering Geodesics 111, 122 connecting points 89, 112, 113 Geometry of domains, classical 83 fundamental results 45 homogeneous 1,45 symmetric 1 Granert-see Andreotti-Granert Group(s), action of 78 affine transformation 225 arithmetic-see Arithmetic group(s) automorphisms 131, 220 commutant of 24* discrete-see Discrete group(s) "guilty" 44 Lie-see Lie group(s) modular-see Modular group(s) nilpotent of class 2 24* pseudo-concave 132, 160, 162 semisimple 220 transitive 29, 38, 77, 220 unimodular 220 "Guilty" group 44 Hartog's theorem 33 Hecke operators, generalized 8 Hermitian scalar product-see Scalar product Hermitian spaces 204 Holomorphic hull(s)-see Bounded holomorphic huIles) Homogeneous bounded domain(s)-see Bounded homogeneous domain(s) Homogeneous cone(s) 18 affine 21 convex 21, 220 irreducible 18 self-adjoint 21 Homogeneous domain(s) 1*, 8 bounded-see Bounded homogeneous domain(s)

corresponding to a complex 205 geometry of 1, 45 representation in the form of a Siegel domain 253 -see also Bounded homogeneous domain(s) Homogeneous fibering, analytic 6, 76 base of 81 fiber of 81 T-rational 133, 173 natural 45, 76 Homogeneous imbeddings-see Imbeddings Homogeneous Siegel domain(s) 26, 30, 44, 76, 229, 253 analytic 26, 38 bounded 26, 30 of genus one 253 of genus two 66, 253 symmetric 26 -see also Bounded homogeneous domain(s), Homogeneous domain(s) and Siegel domain(s) Hull(s)-see Bounded holomorphic hull(s) Ideal(s) 46 commutative-see Commutative ideal compact 248 in Lie algebra 237 j- 47,64 j-invariant 248 maximal 159 nilpotent 159 of the first kind 232, 237 one-dimensional 55 semisimple 248 Imbeddings, homogeneous 199, 211 Infinitely distant component 100, 103, 111 Isogenous pairs 185 Isometric mapping(s) 200, 213 Isometric scalar product(s) 9,200,202, 204,213 bilinear function defining 202

INDEX

defining symplectic representation 214 invariance 204 Isometry-see Isometric scalar product Isomorphism for j-algebras 216 Jacobian functions 165 j-algebra(s) 8, 46, 157, 158, 222 algebraic-see Algebraic j-algebra(s) associated with a cominutative ideal 231,242 complete 66 construction of 208 construction of differentiation 159 corresponding to a given complex 199 decomposition of 55, 67, 227, 231, 242 difference between two 217 effective 232, 239, 242, 249, 251 equivalent 216 ideals of 46 isomorphic 47,216,229 j-homomorphism 65 non-effective 243, 249 non-isomorphic 51 normal-see Normalj-algebra(s) of bounded symmetric domain 223 of cylindrical domain 226, 227 proper 224, 239, 248, 251 structure of 234 subalgebra 217 universal 73, 75 -see also Normalj-algebra(s) j-homomorphism 65 j-ideals 47, 64 Jordan algebras 21 j-subalgebra(s) 46, 47, 223, 239 algebraic 241 Kernel, of algebra 243,249 Kocher effect 167 Kocher's theorem 167 Left-symmetric algebras

21

261

Lie algebra(s) 8, 67, 209, 222 algebraic 217 algebraic hull of 240* decomposition of 248, 251 isomorphic 48 normalizer of 83 reductive 83 representations of 67, 230, 232 semisimple 239, 250 Lie group(s), algebraic 253* orbits 41 representations semisimple 43 Linear transformations 34, 106, 108, 121, 144, 161, 233 Manifold, Bergman-see Bergman manifold dual-see Dual manifold Mappings, conformal-see Conformal mappings isometric 200, 213 Matrix (matrices), algebras 10 class of 97, 116, 124 equivalent 97, 116, 124 in algebra \ll 185 integral 4 mapped onto a point of a boundary component 99 normalization 103* period 165 principal 188 Maurer-Cartan equations 228 Metric, Bergman-see Bergman metric Riemann-see Riemann distance Modular function(s) 179 abelian-see Abelian modular function(s) connection with elliptic functions 11, 179 field(s) of 184, 195, 196 Modular group(s) 2, 179 classification of 195

262 (}j

INDEX

commensurable 185 ('Jr, R) 194 Siegel's 2, 132

Non-associative algebra(s) 21 Non-isomorphic division algebras 191 Non-self-adjoint cone 21 Normalization of matrix 103* Normalj-algebra(s) 9,46,51,205 axioms for 51 canonical decomposition of 67 classification of 63, 199 construction 9 dimension 53, 59, 63 direct sum of 60, 63 elementary 53 irreducible 64 j-ideals of 64 non-isomorphic 63 representations of 9, 56, 59, 60 subalgebra(s) of 52 -see also j-algebra(s) Normal spaces, analytic-see Analytic normal spaces Normal subgroup-see Subgroup(s) Normal symplectic representations-see Symplectic representations One-parameter subgroups 41 Operators, Hecke-see Hecke operators symplectic-see Symplectic operators Orbits, Lie group-see Lie group(s) Pairs, equivalent 185 isogenous 185 Parallelipiped, fundamental 148* Parallel translations 39, 77, 80, 108, 114, 161 Period matrix 165 Poincare series 2, 140 construction of automorphic forms 141 separation of r-non-equivalent points 142 Principal matrix 188

Proper subset 145 Pseudo-concave, boundary points 159 domain 159, 160 group 132, 160, 162 Quasilinear transformation(s) 34, 76, 105, 121, 167, 220 matrix correspondence 108 normal divisor of the group of all 37 properties of 34 Quasinormal discrete groups-see Discrete groups Quaternions, algebra of 200 Realizations, canonical-see Canonical realizations Relations, algebraic-see Algebraic relations Representation(s), affine adjoint of homogeneous domain symplectic -see subheadings Riemann distance (metric) 111, 122 invariant 232 Riemann's theorem on conformal mappings 219 Ringed space 136 Root spaces 61, 209 dimension of 61 Satake extension 3, 132 Satake subalgebra 156 Satake subgroup 5, 153, 156 Scalar product(s) 52, 209 isometric-see Isometric scalar product(s) Self-adjoint cones 21 Semi-Hermitian forms 30,31 Semisimple Lie group 43 Separation of points 137, 142, 150 152 by a function 137*

INDEX

Series, Fourier-Jacobi Poincare -see subheadings Set, analytic-see Analytic set Siegel circle 50, 64 Siegel disk 49 Siegel domain(s) 15, 46, 66 affine transformations of 101 analytic 26, 38 analytic automorphism of 27 automorphic forms in 164 bounded 26, 30 canonical 86, 96, 151 construction of 7, 220 directing subgroup of 41 homogeneous-see Homogeneous Siegel domain(s) inhomogeneous 44 models in the form of 45 quasilinear transformations of 76, 167 -see also Siegel domains of genus 1, 2, 3 Siegel domains of genus 1 15, 16, 126, 221, 225 analytic automorphism of 16 definition of 16 homogeneous 253 linear transformations of 34 skeleton of 16 -see also Siegel domain(s) Siegel domains of genus 2 7, 15, 21, 123, 129, 221, 225 definition of 22 homogeneous 66, 253 involution of 27, 29 linear transformation of 25, 34 nilpotent group 24 parallel translations for 24 symmetric 29 -see also Siegel domain(s) Siegel domains of genus 3 15, 30, 120 base an analytic boundary component 45 construction of 39

263

definition of 32 fibering of 76 homogeneous 30 nilpotent group of 33 parallel translations for 33 principal 40 quasilinear transformations of 34 -see also Siegel domain(s) Siegel's generalized upper half-plane 20, 127 Siegel's modular group-see Modular group(s) Space(s), affine complex 15 continuable 201 Euclidean 59, 204, 206 factor-see Factor space Hermitian 204,206 normal 136 ringed 136 root-see Root spaces Subalgebra(s), commutative 9, 61 j- -see j-subalgebra(s) of j-algebras 217 nilpotent 9 Satake 156 Subgroup(s), commensurable 4 directing 80 maximal unipotent normal 156 one-parameter 41 Satake 5, 153, 156 solvable and split 5* Subset, proper-see Proper subset Symmetric domain(s) 1*, 18, 29, 80 analytic automorphisms of 163 arithmetic groups 153, 163 Cartan's classification of 18 canonical models of 80 geometry of 1 homogeneous fibering of 81 Symplectic operators 56 Symplectic representation(s) 46, 50, 56, 214 construction of 214

264 kernal of 214, 215 normal 57, 59 irreducible 215 Symplectic transformation(s) 233, 234

INDEX

50, 57,

T-algebras 10, 121 Theorem on algebraic relations-see Algebraic relations Theta functions 4, 184 Transcendence of field, of automorphic functions 131, 132, 159, 160 of meromorphic functions 160 Transformation(s), affine-see Affine transformation(s) antilinear 77 commutator of 227 continuable 201 linear-see Linear transformations principal part of 151 quasi linear-see quasilinear transformations

symplectic 50, 57, 233, 234 transitive 76 unimodular 161 unitary 144 Translations, parallel-see Parallel translations Transitive group 29, 38, 77, 220 Unimodular affine transformations 161 Unimodular matrices, integral 4 Unitary transformations 144 "Unit circle" canonical realization 83 Universalj-algebras 73, 75 Vectors, associate 169 Vector forms 31 semi-Hermitian 31 Vector functions, consistent 32, 40 V-Hermitian 22, 221, 229, 230 V-Hermitian functions-see Vector functions, V-Hermitian Zero-dimensional component

90, 126