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E-raamat: Symmetric Spaces and the Kashiwara-Vergne Method

Franēois Rouvičre

Formaat: PDF+DRM
Sari: Lecture Notes in Mathematics 2115
Ilmumisaeg: 14-Oct-2014
Kirjastus: Springer International Publishing AG
Keel: eng
ISBN-13: 9783319097732

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Formaat: PDF+DRM
Sari: Lecture Notes in Mathematics 2115
Ilmumisaeg: 14-Oct-2014
Kirjastus: Springer International Publishing AG
Keel: eng
ISBN-13: 9783319097732

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Gathering and updating results scattered in journal articles over thirty years, this self-contained monograph gives a comprehensive introduction to the subject. Its goal is to: - motivate and explain the method for general Lie groups, reducing the proof of deep results in invariant analysis to the verification of two formal Lie bracket identities related to the Campbell-Hausdorff formula (the "Kashiwara-Vergne conjecture"); - give a detailed proof of the conjecture for quadratic and solvable Lie algebras, which is relatively elementary; - extend the method to symmetric spaces; here an obstruction appears, embodied in a single remarkable object called an "e-function"; - explain the role of this function in invariant analysis on symmetric spaces, its relation to invariant differential operators, mean value operators and spherical functions; - give an explicit e-function for rank one spaces (the hyperbolic spaces); - construct an e-function for general symmetric spaces, in the spirit

of Kashiwara and Vergne"s original work for Lie groups. The book includes a complete rewriting of several articles by the author, updated and improved following Alekseev, Meinrenken and Torossian"s recent proofs of the conjecture. The chapters are largely independent of each other. Some open problems are suggested to encourage future research. It is aimed at graduate students and researchers with a basic knowledge of Lie theory.

Introduction.- Notation.- The Kashiwara-Vergne method for Lie groups.- Convolution on homogeneous spaces.- The role of e-functions.- e-functions and the Campbell Hausdorff formula.- Bibliography.

Notation

xiii

Introduction

xix

1 The Kashiwara-Vergne Method for Lie Groups

(50)

1.1 A General Convolution Problem

(2)

1.2 The Kashiwara-Vergne Method

(1)

1.3 The Equation (KV1)

(4)

1.4 The Equation (KV2)

(2)

1.5 The Kashiwara-Vergne Conjecture

(6)

1.5.1 The Conjecture

(1)

1.5.2 Remarks on the Conjecture

(4)

1.5.3 Non-uniqueness

(1)

1.6 An Elementary Proof for sl(2, R)

(6)

1.6.1 Choice of F and G

(1)

1.6.2 A Variant of (KV2)

(1)

1.6.3 A Trace Formula for sl(2, R)

(1)

1.6.4 Proof of (KV2)

(3)

1.7 Two Symmetries of the Kashiwara-Vergne Problem

(6)

1.7.1 The Symmetry α

(1)

1.7.2 The Symmetry β

(4)

1.7.3 Symmetry Invariant Solutions

(1)

1.8 From (KV1) Towards (KV2)

(14)

1.8.1 Tangential Derivations and Automorphisms

(2)

1.8.2 Behavior Under Scaling Maps

(1)

1.8.3 Using Associativity

(1)

1.8.4 A “r;Divergence”r; and a “r;Jacobian”r;

(3)

1.8.5 A Cohomology Argument

(2)

1.8.6 An Expression of the Divergence

(1)

1.8.7 Conclusion

(3)

1.9 The Case of Quadratic or Solvable Lie Algebras

(4)

1.10 The General Case

(5)

1.10.1 An Abstract Formulation of the Kashiwara-Vergne Problem

(1)

1.10.2 Drinfeld's Associators and the Kashiwara-Vergne Conjecture

(1)

Notes

(3)

2 Convolution on Homogeneous Spaces

(6)

2.1 Definition and Examples

(3)

2.2 Extension to Line Bundles

(3)

2.2.1 The Line Bundle Lχ

(1)

2.2.2 Convolution of Distributions on Lχ

(1)

Notes

(1)

3 The Role of e-Functions

(62)

3.1 Transferring Convolutions

(5)

3.1.1 The Transfer Map

(2)

3.1.2 e-Functions

(3)

3.2 Special Symmetric Spaces

(5)

3.3 e-Functions and Invariant Differential Operators

(11)

3.3.1 The Fischer Product

(1)

3.3.2 Explicit Transfer of Invariant Differential Operators

(4)

3.3.3 Example: The Laplace-Beltrami Operator

(4)

3.3.4 Towards a Generalized Duflo Isomorphism

(2)

3.4 e-Functions, Mean Values and Spherical Functions

(7)

3.4.1 The Case of an Isotropic Riemannian Symmetric Space

(3)

3.4.2 Expansion of Mean Value Operators and Spherical Functions

(4)

3.5 e-Functions and Integral Formulas

(5)

3.6 e-Functions and Noncompact Symmetric Spaces

(14)

3.6.1 The Function E ∞

(5)

3.6.2 Link with the Iwasawa Projection

(3)

3.6.3 Extension of Duflo's Isomorphism

(6)

3.7 An e-Function for Symmetric Spaces of Rank One

105

(8)

3.7.1 Outline of the Proof

107

(1)

3.7.2 Properties of the Rank One e-Function

108

(4)

3.7.3 Application to Spherical Functions

112

(1)

3.8 Extension to Line Bundles

113

(3)

3.9 Open Problems

116

(3)

Notes

116

(3)

4 e-Functions and the Campbell-Hausdorff Formula

119

(58)

4.1 Contractions of Symmetric Spaces

120

(2)

4.2 The Diffeomorphism Φ

122

(15)

4.2.1 Lie Series

122

(1)

4.2.2 Z, h

123

(4)

4.2.3 A, B, C

127

(4)

4.2.4 a, B and Φ

131

(2)

4.2.5 c, H and Φ

133

(4)

4.3 Campbell-Hausdorff e-Functions

137

(8)

4.3.1 An Auxiliary Lemma

137

(1)

4.3.2 Construction of an e-Function

138

(4)

4.3.3 Application to Invariant Differential Operators

142

(3)

4.4 Properties of Campbell-Hausdorff e-Functions

145

(12)

4.4.1 e and c

146

(1)

4.4.2 Some Deeper Properties of e

147

(10)

4.5 Extension to Line Bundles

157

(3)

4.6 Taylor Expansions

160

(14)

4.6.1 Expansions of Z, h, A, B, C

161

(1)

4.6.2 Expansion of Φ

162

(1)

4.6.3 Expansions of E and ex

163

(3)

4.6.4 Expansions of Z, h. A, C Modulo Y2

166

(5)

4.6.5 Expansion of Φ Modulo Y2

171

(1)

4.6.6 Expansions of E and ex Modulo Y2

172

(2)

4.7 Open Problems

174

(3)

Notes

175

(2)

A Proof of Lemma 3.22

177

(4)

B Proof of Theorem 3.23

181

(10)

Bibliography

191

(4)

Index

195

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