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

"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

• Algebraic preliminaries
• Relative invariants of prehomogeneous vector spaces
• Analytic preliminaries
• The fundamental theorem of prehomogeneous vector spaces
• The zeta functions of prehomogeneous vector spaces
• Convergence of zeta functions of prehomogeneous vector spaces
• Classification of prehomogeneous vector spaces
• Appendix: Table of irreducible reduced prehomogeneous vector spaces
• Bibliography
• Index of symbols
• Index