Representation Theory and Complex Analysis : Lectures Given at the C. I. M. E. Summer School Held in Venice, Italy, June 10-17 2004

by Cowling, M.; Frenkel, Edward; Kashiwara, Masaki; Valette, Alain; Vogan, David A.

• ISBN13: 9783540768913
• ISBN10: 3540768912
• Format: Paperback
• Copyright: 2008-04-03
• Publisher: Springer Verlag

Summary

Six leading experts lecture on a wide spectrum of recent results on the subject of the title. They present a survey of various interactions between representation theory and harmonic analysis on semisimple groups and symmetric spaces, and recall the concept of amenability. They further illustrate how representation theory is related to quantum computing; and much more. Taken together, this volume provides both a solid reference and deep insights on current research activity.

Table of Contents

Applications of Representation Theory to Harmonic Analysis of Lie Groups (and Vice Versa)	p. 1
Basic Facts of Harmonic Analysis on Semisimple Groups and Symmetric Spaces	p. 2
Structure of Semisimple Lie Algebras	p. 2
Decompositions of Semisimple Lie Groups	p. 4
Parabolic Subgroups	p. 5
Spaces of Homogeneous Functions on G	p. 6
The Plancherel Formula	p. 8
The Equations of Mathematical Physics on Symmetric Spaces	p. 10
Spherical Analysis on Symmetric Spaces	p. 10
Harmonic Analysis on Semisimple Groups and Symmetric Spaces	p. 12
Regularity of the Laplace-Beltrami Operator	p. 16
Approaches to the Heat Equation	p. 18
Estimates for the Heat and Laplace Equations	p. 18
Approaches to the Wave and Schrodinger Equations	p. 20
Further Results	p. 21
The Vanishing of Matrix Coefficients	p. 22
Some Examples in Representation Theory	p. 22
Matrix Coefficients of Representations of Semisimple Groups	p. 24
The Kunze-Stein Phenomenon	p. 27
Property T	p. 28
The Generalised Ramanujan-Selberg Property	p. 29
More General Semisimple Groups	p. 31
Graph Theory and its Riemannian Connection	p. 31
Cayley Graphs	p. 32
An Example Involving Cayley Graphs	p. 33
The Field of p-adic Numbers	p. 34
Lattices in Vector Spaces over Local Fields	p. 35
Adeles	p. 36
Further Results	p. 37
Carnot-Caratheodory Geometry and Group Representations	p. 38
A Decomposition for Real Rank One Groups	p. 38
The Conformal Group of the Sphere in R[superscript n]	p. 38
The Groups SU(1, n + 1) and Sp(1, n + 1)	p. 41
References	p. 46
Ramifications of the Geometric Langlands Program	p. 51
Introduction	p. 51
The Unramified Global Langlands Correspondence	p. 56
Classical Local Langlands Correspondence	p. 61
Langlands Parameters	p. 61
The Local Langlands Correspondence for GL[subscript n]	p. 62
Generalization to Other Reductive Groups	p. 63
Geometric Local Langlands Correspondence over C	p. 64
Geometric Langlands Parameters	p. 64
Representations of the Loop Group	p. 65
From Functions to Sheaves	p. 66
A Toy Model	p. 68
Back to Loop Groups	p. 70
Center and Opers	p. 71
Center of an Abelian Category	p. 71
Opers	p. 73
Canonical Representatives	p. 75
Description of the Center	p. 76
Opers vs. Local Systems	p. 77
Harish-Chandra Categories	p. 81
Spaces of K-Invariant Vectors	p. 81
Equivariant Modules	p. 82
Categorical Hecke Algebras	p. 83
Local Langlands Correspondence: Unramified Case	p. 85
Unramified Representations of G(F)	p. 85
Unramified Categories [characters not reproducible]-Modules	p. 87
Categories of G[[t]]-Equivariant Modules	p. 88
The Action of the Spherical Hecke Algebra	p. 90
Categories of Representations and D-Modules	p. 92
Equivalences Between Categories of Modules	p. 96
Generalization to other Dominant Integral Weights	p. 98
Local Langlands Correspondence: Tamely Ramified Case	p. 99
Tamely Ramified Representations	p. 99
Categories Admitting ([characters not reproducible], I) Harish-Chandra Modules	p. 103
Conjectural Description of the Categories of ([characters not reproducible], I) Harish-Chandra Modules	p. 105
Connection between the Classical and the Geometric Settings	p. 109
Evidence for the Conjecture	p. 115
Ramified Global Langlands Correspondence	p. 117
The Classical Setting	p. 117
The Unramified Case, Revisited	p. 120
Classical Langlands Correspondence with Ramification	p. 122
Geometric Langlands Correspondence in the Tamely Ramified Case	p. 122
Connections with Regular Singularities	p. 126
Irregular Connections	p. 130
References	p. 132
Equivariant Derived Category and Representation of Real Semisimple Lie Groups	p. 137
Introduction	p. 137
Harish-Chandra Correspondence	p. 138
Beilinson-Bernstein Correspondence	p. 140
Riemann-Hilbert Correspondence	p. 141
Matsuki Correspondence	p. 142
Construction of Representations of G[subscript R]	p. 143
Integral Transforms	p. 146
Commutativity of Fig. 1	p. 147
Example	p. 148
Organization of the Note	p. 151
Derived Categories of Quasi-abelian Categories	p. 152
Quasi-abelian Categories	p. 152
Derived Categories	p. 154
t-Structure	p. 156
Quasi-equivariant D-Modules	p. 158
Definition	p. 158
Derived Categories	p. 162
Sumihiro's Result	p. 163
Pull-back Functors	p. 167
Push-forward Functors	p. 168
External and Internal Tensor Products	p. 170
Semi-outer Hom	p. 171
Relations of Push-forward and Pull-back Functors	p. 172
Flag Manifold Case	p. 175
Equivariant Derived Category	p. 176
Introduction	p. 176
Sheaf Case	p. 176
Induction Functor	p. 179
Constructible Sheaves	p. 179
D-module Case	p. 180
Equivariant Riemann-Hilbert Correspondence	p. 181
Holomorphic Solution Spaces	p. 182
Introduction	p. 182
Countable Sheaves	p. 183
C[superscript infinity]-Solutions	p. 185
Definition of RHom[superscript top]	p. 186
DFN Version	p. 189
Functorial Properties of RHom[superscript top]	p. 190
Relation with the de Rham Functor	p. 192
Whitney Functor	p. 194
Whitney Functor	p. 194
The Functor RHom[characters not reproducible]	p. 195
Elliptic Case	p. 196
Twisted Sheaves	p. 197
Twisting Data	p. 197
Twisted Sheaf	p. 198
Morphism of Twisting Data	p. 199
Tensor Product	p. 200
Inverse and Direct Images	p. 200
Twisted Modules	p. 201
Equivariant Twisting Data	p. 201
Character Local System	p. 202
Twisted Equivariance	p. 202
Twisting Data Associated with Principal Bundles	p. 203
Twisting (D-module Case)	p. 204
Ring of Twisted Differential Operators	p. 205
Equivariance of Twisted Sheaves and Twisted D-modules	p. 207
Riemann-Hilbert Correspondence	p. 207
Integral Transforms	p. 208
Convolutions	p. 208
Integral Transform Formula	p. 209
Application to the Representation Theory	p. 210
Notations	p. 210
Beilinson-Bernstein Correspondence	p. 212
Quasi-equivariant D-modules on the Symmetric Space	p. 214
Matsuki Correspondence	p. 216
Construction of Representations	p. 217
Integral Transformation Formula	p. 219
Vanishing Theorems	p. 221
Preliminary	p. 221
Calculation (I)	p. 222
Calculation (II)	p. 224
Vanishing Theorem	p. 226
References	p. 229
List of Notations	p. 231
Index	p. 233
Amenability and Margulis Super-Rigidity	p. 235
Introduction	p. 235
Amenability for Locally Compact Groups	p. 236
Definition, Examples, and First Characterizations	p. 236
Stability Properties	p. 239
Lattices in Locally Compact Groups	p. 240
Reiter's Property (P[subscript 1])	p. 241
Reiter's Property (P[subscript 2])	p. 242
Amenability in Riemannian Geometry	p. 244
Measurable Ergodic Theory	p. 244
Definitions and Examples	p. 244
Moore's Ergodicity Theorem	p. 247
The Howe-Moore Vanishing Theorem	p. 249
Margulis' Super-rigidity Theorem	p. 252
Statement	p. 252
Mostow Rigidity	p. 252
Ideas to Prove Super-rigidity, k = R	p. 253
Proof of Furstenberg's Proposition 4.1 - Use of Amenability	p. 255
Margulis' Arithmeticity Theorem	p. 256
References	p. 257
Unitary Representations and Complex Analysis	p. 259
Introduction	p. 259
Compact Groups and the Borel-Weil Theorem	p. 264
Examples for SL(2, R)	p. 272
Harish-Chandra Modules and Globalization	p. 274
Real Parabolic Induction and the Globalization Functors	p. 284
Examples of Complex Homogeneous Spaces	p. 294
Dolbeault Cohomology and Maximal Globalizations	p. 302
Compact Supports and Minimal Globalizations	p. 318
Invariant Bilinear Forms and Maps between Representations	p. 327
Open Questions	p. 341
References	p. 343
Quantum Computing and Entanglement for Mathematicians	p. 345
The Basics	p. 346
Basic Quantum Mechanics	p. 346
Bits	p. 348
Qubits	p. 349
References	p. 350
Quantum Algorithms	p. 351
Quantum Parallelism	p. 351
The Tensor Product Structure of n-qubit Space	p. 352
Grover's Algorithm	p. 353
The Quantum Fourier Transform	p. 354
References	p. 355
Factorization and Error Correction	p. 355
The Complexity of the Quantum Fourier Transform	p. 356
Reduction of Factorization to Period Search	p. 359
Error Correction	p. 360
References	p. 362
Entanglement	p. 362
Measures of Entanglement	p. 363
Three Qubits	p. 365
Measures of Entanglement for Two and Three Qubits	p. 367
References	p. 368
Four and More Qubits	p. 369
Four Qubits	p. 369
Some Hilbert Series of Measures of Entanglement	p. 374
A Measure of Entanglement for n Qubits	p. 374
References	p. 376
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