Algebraic Structures of Symmetric Domains

342 Pages·1980·18.678 MB·English

by  Satake, Ichiro

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ALGEBRAIC STRUCTURES
OF SYMMETRIC DOMAINS
ICHIRO
SA TAKE
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ALGEBRAIC STRUCTURES OF
SYMMETRIC DOMAINS
PUBLICATIONS OF THE MATHEMATICAL
SOCIETY OF JAPAN
I. 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 Differentielles Ordinaires du Premier Ordre clans le
Champ Complexe. Par Masuo Hukuhara, Tosihusa Kimura et
Mm• Tizuko Matuda.
8. Theory of Q-varieties. By Teruhisa Matsusaka.
9. Stability Theory by Liapunov's Second Method. By Taro Yoshi
zawa.
IO. Fonctions Entieres et Transformees de Fourier. 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
lchiro Satake
KANO 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 ofJ apan 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. lyanaga and K. Iwasawa

Preface
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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 holomor
phic 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., (!) the case of (H1)-homomorphisms of s[2(R) into g=
Lie Ho! (cZ>), 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->Sj:J,n(R) sat
isfying (H which leads to the analytic construction of Kuga's fiber varieties. In addi
1),
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 University of Nancy 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. lyanaga who invited me to write a volume in this
series of the Publications of Mathematical Society of Japan, and to Mr. H. Arai oflwa
nami 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. lwasawa.
Berkeley, December 1979
I. Satake