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Introduction to prehomogeneous vector spaces. Translated from the Japanese by Makoto Nagura and Tsuyoshi Niitani. (English) Zbl 1035.11060
Translations of Mathematical Monographs 215. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2767-7/hbk). xxii, 288 p. (2003).
Prehomogeneous vector spaces were introduced by Mikio Sato in connection with a desire to understand when the Fourier transform of a complex power of a polynomial \(f\) is essentially again a complex power of some polynomial. The answer was found in terms of a big action of some group, with respect to which \(f\) is a relative invariant.
More specifically, a triplet \((G,\rho ,V)\) is called a prehomogeneous vector space over \(\mathbb C\) if \(\rho : G\to \text{GL}(V)\) is a rational representation of a connected algebraic group \(G\) on a finite-dimensional complex vector space \(V\), such that \(V\) has an orbit \(\rho (G)v_0\) dense in the Zariski topology (so that the orbit is a homogeneous space, and \(V\) is an “almost” homogeneous one). A relative invariant is a rational function \(f\) on \(V\), such that \(f(\rho (g)x)=\chi (g)f(x)\) for all \(x\in V,g\in G\), where \(\chi\) is a character of \(G\).
These notions are general enough to cover many applications, such as the derivation of functional equations and the study of special values for various zeta and \(L\)-functions in number theory. There are also connections with representation theory. The theory has been extended to the case of a non-Archimedean base field and was applied recently to \(p\)-adic Green functions [F. Sato, Comment. Math. Univ. St. Pauli 51, 79–97 (2002; Zbl 1004.11067)] etc. On the other hand, there exists a deep classification theory of prehomogeneous vector spaces.
The book is intended as an introduction to the theory of prehomogeneous vector spaces containing all the algebraic and analytic preliminaries (Chapters 1 and 3), basic notions (Chapter 2), as well as a number of examples. However it reaches many deep results, such as the fundamental theorem on the Fourier transforms (in a rather general setting; Chapter 4), an extensive study of zeta functions on prehomogeneous vector spaces (Chapters 5 and 6) including the \(p\)-adic and adelic cases, and the classification of prehomogeneous vector spaces over \(\mathbb C\). An annotated bibliography provides instructions for further reading.
The book will surely have a wide readership among students and specialists in number theory, analysis, and representation theory.

11S90 Prehomogeneous vector spaces
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11M41 Other Dirichlet series and zeta functions
20G05 Representation theory for linear algebraic groups
11S40 Zeta functions and \(L\)-functions