**Translations of Mathematical Monographs**

2002;
288 pp;
Hardcover

MSC: Primary 11;
Secondary 20

**Print ISBN: 978-0-8218-2767-3
Product Code: MMONO/215**

List Price: $128.00

AMS Member Price: $102.40

MAA Member Price: $115.20

**Electronic ISBN: 978-1-4704-4640-6
Product Code: MMONO/215.E**

List Price: $125.00

AMS Member Price: $100.00

MAA Member Price: $112.50

# Introduction to Prehomogeneous Vector Spaces

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*Tatsuo Kimura*

This is the first introductory book on the theory of prehomogeneous
vector spaces, introduced in the 1970s by Mikio Sato. The author was an
early and important developer of the theory and continues to be active
in the field.

The subject combines elements of several areas of mathematics, such as
algebraic geometry, Lie groups, analysis, number theory, and invariant
theory. An important objective is to create applications to number
theory. For example, one of the key topics is that of zeta functions
attached to prehomogeneous vector spaces; these are generalizations of
the Riemann zeta function, a cornerstone of analytic number theory.
Prehomogeneous vector spaces are also of use in representation theory,
algebraic geometry and invariant theory.

This book explains the basic concepts of prehomogeneous vector
spaces, the fundamental theorem, the zeta functions associated with
prehomogeneous vector spaces, and a classification theory of
irreducible prehomogeneous vector spaces. It strives, and to a large
extent succeeds, in making this content, which is by its nature fairly
technical, self-contained and accessible. The first section of the
book, "Overview of the theory and contents of this book," is
particularly noteworthy as an excellent introduction to the
subject.

#### Readership

This book is most appropriate for second-year graduate students and above, but may be accessible to advanced undergraduate or beginning graduate students; it is also useful to working mathematicians who want to learn about prehomogeneous vector spaces.

#### Reviews & Endorsements

The book will serve as a useful reference for specialists, but its true audience is graduate students and mathematicians who are specialists in other fields, but wish to learn something about prehomogeneous vector spaces … The first and third chapters are elegant and concise overviews of background material from algebra … Kimura is currently one of the most senior figures in the theory of prehomogeneous vector spaces and he writes with great authority about the subject. He has been well served by his translators … who write clear and reasonably idiomatic English, and have preserved the direct and straightforward style that is familiar to readers of Kimura's English papers. He has written an excellent and timely introduction to what is, in the reviewer's opinion, an attractive and significant area of mathematics.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Introduction to Prehomogeneous Vector Spaces

- Cover Cover11
- Title page iii4
- Contents v6
- Preface to the Japanese edition vii8
- Preface to the English edition ix10
- Overview of the theory and contents of this book xi12
- Algebraic preliminaries 124
- Relative invariants of prehomogeneous vector spaces 2346
- Analytic preliminaries 7396
- The fundamental theorem of prehomogeneous vector spaces 113136
- The zeta functions of prehomogeneous vector spaces 157180
- Convergence of zeta functions of prehomogeneous vector spaces 191214
- Classification of prehomogeneous vector spaces 223246
- Appendix: Table of irreducible reduced prehomogeneous vector spaces 261284
- Bibliography 271294
- Index of symbols 281304
- Index 285308
- Back Cover Back Cover1314