Introduction to prehomogeneous vector spaces / Tatsuo Kimura ; translated by Makoto Nagura, Tsuyoshi Niitani.

Format:

Book

Author/Creator:

Kimura, Tatsuo, 1947-

Series:

Translations of mathematical monographs 0065-9282 ; v. 215.

Translations of mathematical monographs, 0065-9282 ; v. 215

Standardized Title:

Gaikinshitsu bekutoru kūkan. English

Language:

English

Japanese

Subjects (All):

Vector spaces.

Physical Description:

xxii, 288 pages : illustrations ; 27 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2003]

Contents:

Chapter 1. Algebraic Preliminaries 1

1.1. Groups, rings and fields 1

1.2. Topological spaces 4

1.3. Algebraic varieties 6

1.4. Algebraic groups 9

1.5. Tangent spaces of algebraic varieties 12

1.6. Lie algebras of algebraic groups 15

Chapter 2. Relative Invariants of Prehomogeneous Vector Spaces 23

2.1. Definition of prehomogeneous vector space 23

2.2. Relative invariants 24

2.3. Reductive prehomogeneous vector spaces 37

2.4. Examples of prehomogeneous vector spaces 44

Chapter 3. Analytic Preliminaries 73

3.1. Review of integral theory 73

3.2. Fourier transforms of rapidly decreasing functions 76

3.3. Distributions 82

3.4. The gamma function 86

3.5. Differential forms with compact supports 92

3.6. Invariant differential forms 99

3.7. Invariant distributions 106

Chapter 4. The Fundamental Theorem of Prehomogeneous Vector Spaces 113

4.1. Proof of the fundamental theorem 113

4.2. Examples of the fundamental theorem 126

4.3. Complement to the fundamental theorem 136

4.4. Poisson's summation formula 148

4.5. Zeta distributions 152

Chapter 5. The Zeta Functions of Prehomogeneous Vector Spaces 157

5.1. Group theoretical preliminaries 157

5.2. Definition of the zeta function 161

5.3. Zeta integrals 167

5.4. Analytic continuation and the functional equation of the zeta function 171

5.5. Residues of zeta functions 172

5.6. Examples of zeta functions of prehomogeneous vector spaces 179

Chapter 6. Convergence of Zeta Functions of Prehomogeneous Vector Spaces 191

6.1. Theorems on the convergence of zeta functions 191

6.2. Equivalent conditions for convergence 194

6.3. The p-adic field 197

6.4. Adeles of the rational number field 199

6.5. Estimation of A(t) 204

6.6. Estimation of the integral 209

6.7. Convergence of multiplicative adelic zeta functions 215

6.8. Convergence of additive adelic zeta functions 218

Chapter 7. Classification of Prehomogeneous Vector Spaces 223

7.1. Castling transforms 223

7.2. Irreducible representations 231

7.3. Irreducible representations and the classification of simple Lie algebras over C 237

7.4. Classification of irreducible prehomogeneous vector spaces and b-functions 245

7.5. Classification of simple prehomogeneous vector spaces 255

7.6. Weakly spherical homogeneous spaces 257

Appendix. Table of irreducible reduced prehomogeneous vector spaces 261.

Notes:

Includes bibliographical references (pages 271-274) and indexes.

ISBN:

0821827677

OCLC:

50479324