Algebraic Structures of Symmetric Domains [Course Book ed.] 9781400856800

Citation preview

ALGEBRAIC STRUCTURES OF SYMMETRIC DOMAINS

PUBLICATIONS OF THE MATHEMATICAL SOCIETY OFJAPAN 1. The Construction and Study of Certain Important Algebras. By Claude Chevalley. 2. Lie Groups and Differential Geometry. By Katsumi Nomizu. 3. Lectures on Ergodic Theory. By Paul R. Halmos. 4. Introduction to the Problem of Minimal Models in the Theory of Algebraic Surfaces. By Oscar Zariski. 5. Zur Reduktionstheorie Quadratischer Formen. Von Carl Ludwig Siegel. 6. Complex Multiplication of Abelian Varieties and its Applications to Number Theory. By Goro Shimura and Yutaka Taniyama. 7. Equations Diifdrentielles Ordinaires du Premier Ordre dans Ie Champ Complexe. Par Masuo Hukuhara, Tosihusa Kimura et Mme Tizuko Matuda. 8. Theory of Q_-varieties. By Teruhisa Matsusaka. 9. Stability Theory by Liapunov's Second Method. By Taro Yoshizawa. 10. FonctionsEntieresetTransformdesdeFourier. Application. Par Szolem Mandelbrojt. 11. Introduction to the Arithmetic Theory of Automorphic Functions. By Goro Shimura. (Kano Memorial Lectures 1) 12. Introductory Lectures on Automorphic Forms. By Walter L. Baily, Jr. (Kano Memorial Lectures 2) 13. Two Applications of Logic to Mathematics. By Gaisi Takeuti. (Kano Memorial Lectures 3) 14. Algebraic Structures of Symmetric Domains. By Ichiro Satake. (Kano Memorial Lectures 4)

PUBLICATIONS OF THE MATHEMATICAL SOCIETY OF JAPAN 14

ALGEBRAIC STRUCTURES OF SYMMETRIC DOMAINS by

Ichiro Satake ΚΑΝ0 MEMORIAL LECTURES 4

Iwanami Shoten, Publishers and Princeton University Press 1980

©The Mathematical Society of Japan 1980 All rights reserved

Kano Memorial Lectures In 1969, the Mathematical Society of Japan received an anonymous donation to en­ courage the publication of lectures in mathematics of distinguished quality in com­ memoration of the late Kokichi Kano (1865-1942). K. Kano was a remarkable scholar who lived through an era when Western mathe­ matics and philosophy were first introduced to Japan. He began his career as a schol­ ar by studying mathematics and remained a rationalist for his entire life, but enor­ mously enlarged the domain of his interest to include philosophy and history. In appreciating the sincere intentions of the donor, our Society has decided to pub­ lish a series of "Kano Memorial Lectures" as a part of our Publications. This is the fourth volume in the series.

Publications of the Mathematical Society of Japan, volumes 1 through 10, should be ordered directly from the Mathematical Society of Japan. Volume 11 and subsequent volumes should be ordered from Princeton University Press, except in Japan, where they should be ordered from Iwanami Shoten, Publishers.

Printed in the United States of America

Dedicated to S. Iyanaga and K. Iwasawa

Preface

MM

The symmetric domain is rich in algebraic structures. It provides an important common ground for various branches of mathematics, not only for function theory and geometry, but also for number theory and algebraic geometry. However, until recent­ ly, the structures associated with symmetric domains (Lie groups, Jordan algebras, Siegel domains, etc.) have been studied independently from separate view-points with little mutual recognition. It is one of our aims to give a unified treatment, clarifying the relationship between these structures. Our main theme in this book is a study of "morphisms", i. e., equivariant holomorphic maps, of symmetric domains. This enables us to establish, on the one hand, the equivalences between various categories related to symmetric domains (I, §9; II, §8; V, § 7). On the other hand, we study in Chapters III and IV two important special cases of the morphisms, i. e., (1) the case of (H,)-homomorphisms of Sl2(H) into g = Lie Hol (S), which leads to the theory of Wolf and Koranyi on the Siegel domain real­ izations of symmetric domains; (2) the case of symplectic representations g —>gp2n(Jl) sat­ isfying (H1), which leads to the analytic construction of Kuga's fiber varieties. In addi­ tion to these, we will, in Chapter V, study infinitesimal automorphisms of Siegel domains in general, give a characterization of symmetric Siegel domains, and classify an intermedi­ ate class of Siegel domains, called "quasi-symmetric". This portion contains detailed proofs of the results announced in earlier papers of the author ([17], [18]). This book is basically designed for graduate students or non-specialists who have a sound background in the theory of Lie groups. In the hope of making the book more self-contained, we included two introductory chapters I and II, in which a summary of basic results on related topics : algebraic groups, Jordan triple systems and symmetric spaces, is given. The Appendix contains a brief account on classical groups. For more detailed instructions on the reading of this book, the reader should consult the Instructions to the reader.

Some materials of this book have been lectured in various occasions at various places: at Chicago (1966), Tokyo (1968), Berkeley (1972), and Nancy (1977-78). Especially, a set of lecture notes at University of California, Berkeley, prepared by H. Yamaguchi in 1972, served as a base of Chapter III. I wish to express my thanks to Yamaguchi here. I would also like to acknowledge gratefully the helps given by many colleagues and friends, especially by M. Takeuchi, T. Ochiai, and Zelow (Lundquist), who read a large portion of the manuscript and gave me useful suggestions. In preparing Chapter IV, occasional conversations with M. Kuga have been most helpful. I am also indebted to several institutions, especially to the University of California at Berkeley for secretari-

viii

Preface

al assistance (and for comfortable climate there), to the UniversityofNancy for render­ ing facilities while the final version of the manuscript was being prepared, and to the National Science Foundation for continuous financial support. Finally, I am happy to express my thanks to Professor S. Iyanaga who invited me to write a volume in this series of the Publications of Mathematical Society of Japan, and to Mr. H. Arai of Iwanami whose professional cooperation in the process of printing has been invaluable. This book is respectfully dedicated to my two eminent teachers, Professors S. Iyanaga and K. Iwasawa. Berkeley, December 1979

I. Satake

Contents Preface .................................................................................... Vll Instructions to the reader ............................................................ Xl List of symbols........................................................................... xv Chapter I.

Algebraic preliminaries ............................................... .

Linear algebraic groups ................................................ Tori and unipotent algebraic groups .......... ....................... Semi-simple and reductive algebraic groups........................ Cartan involutions of reductive R-groups ........................... Relative roots and parabolic subgroups.............................. The structure group of a (non-degenerate) JTS...... ............... The symmetric Lie algebra associated with aJTS.................. Formally real Jordan algebras and self-dual homogeneous cones ...................................................... § 9. Morphisms of JTS's and self-dual homogeneous cones............

§ 1. § 2. § 3. § 4. § 5. § 6. § 7. § 8.

Chapter II. § 1. § 2. § 3. § 4. § 5. § 6. § 7. § 8.

1 6 8 13 18 21 27 31 38

Basic concepts on symmetric domains ........................... 44

Riemannian symmetric spaces.......... .................... ............ Equivariant maps of Riemannian symmetric spaces............... Hermitian symmetric spaces and hermitian JTS's.................. The Harish-Chandra embedding of a hermitian symmetric space... ............................. ................ ............ The canonical automorphy factors and kernel functions......... The Bergman metric of a symmetric domain.......... .............. The Siegel space .... ............. ....... ............... .................. Equivariant holomorphic maps of symmetric domains............

44 47 52 56 64 71 76 83

Chapter III. Unbounded realizations of symmetric domains (Theory of Wolf-Koranyi)............................................. 89

§ 1. The (Hr)-homomorphisms " of ~r2(R) into g ........................ 89 § 2. The parabolic subalgebra attached to a homomorphism" ..................................................................... The representations of G~i) (i = 1, 2) on V............................ The root structure of g and the determination of adv IG~i) ......... Groups of Harish-Chandra type ....................................... The notion of Siegel domains .......................................... The realization of a symmetric domain as a Siegel domain of the third kind ................................................ § 8. Boundary components of a symmetric domain .....................

§ 3. § 4. § 5. § 6. § 7.

95 100 107 118 127 135 140

x

Contents

§ 9. The relations between two Siegel domain realizations § 10. The correspondence of boundary components under equivariant holomorphic maps Chapter IV. Equivariant holomorphic maps of a symmetric domain into a Siegel space §1. §2. § 3. § 4. § 5. § 6. § 7. § 8.

Fullyreduciblerepresentations Invariantalternatingbilinearforms Scalar extensions Symplectic representations giving rise to equivariant holomorphic maps Further reductions and the determination of solutions The solutions over Q Analytic construction of Kuga's fiber varieties The algebraicity of Kuga's fiber varieties

Chapter V. Infinitesimal automorphisms of symmetric Siegel domains § 1. §2. § 3. § 4. § 5. § 6. § 7.

Holomorphic vector fields on a Siegel domain Explicit determinations of gl/2 and g, The symmetric case The sufficiency of the conditions (i) —(iii) Classification of quasi-symmetric Siegel domains Peirce decompositions of a JTS Morphisms of symmetric and quasi-symmetric Siegel domains

Appendix. § 1. § 2. § 3. § 4. § 5. § 6.

Classical domains Classical groups Quaternion algebras Classical domains of types (/)~(///) Clifford algebras and spin groups The spin representations satisfying (H2) The domains of type ( I V )

References Index

152 158 165 165 170 175 179 185 189 195 202 209 209 213 219 225 233 242 258 267 267 270 273 279 282 285 291 319

Instructions to the reader

1.

The logical interdependence of the chapters is as follows : I

ι-

IIV