This volume carries the same title as that of an international conference held at the National University of Singapore, 9-11 January 2006 on the occasion of Roger E. Howe's 60th birthday. Authored by leading members of the Lie theory community, these contributions, expanded from invited lectures given at the conference, are a fitting tribute to the originality, depth and influence of Howe's mathematical work. The range and diversity of the topics will appeal to a broad audience of research mathematicians and graduate students interested in symmetry and its profound applications.
Author(s): Jian-shu Li, Jian-shu Li, Eng-Chye Tan, Nolan Wallach, Chen-Bo Zhu
Series: Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore 12
Publisher: World Scientific
Year: 2007
Language: English
Pages: 446
City: Singapore; Hackensack, N.J
CONTENTS......Page 6
Foreword......Page 8
Preface......Page 10
2. Fock Model: Complex Lie Algebra......Page 13
4. Fock Model: Real Lie Algebra......Page 19
5. Duality......Page 24
6. Compact Dual Pairs......Page 27
7. Joint Harmonics......Page 32
8. Induction Principle......Page 37
9. Examples......Page 41
References......Page 47
The Heisenberg Group, SL(3;R), and Rigidity Andreas Cap, Michael G. Cowling, Filippo De Mari, Michael Eastwood and Rupert McCallum......Page 53
1. Introduction......Page 54
2. An Example......Page 55
3. Related Questions in Two Dimensions......Page 57
4. Proof of Theorem 2.1......Page 61
5. Final Remarks......Page 63
References......Page 64
1. Introduction......Page 65
2. Strategies for the Multivariate Game......Page 68
3. Strategies for the Single Variable Game......Page 71
4. Strategies for Some Constricted Multivariate Games......Page 73
5. Appendix: Pfa ans Associated with Payo Matrices......Page 76
References......Page 84
Introduction......Page 85
1. The Classical Method......Page 88
2. The Rankin-Selberg Generalization of de la Vall ee Poussin......Page 89
3. An Approach Using Eisenstein Series on SL(2; R)......Page 91
4. The General Method......Page 94
References......Page 98
Introduction......Page 101
Errors and Misprints in [H4]......Page 105
0. Preliminary Notation......Page 110
1. Eisenstein Series on Unitary Similitude Groups......Page 111
2. The Local Theta Correspondence......Page 115
Appendix. Generic Calculation of the Unrami ed Correspondence......Page 133
3. Applications to Special Values of L-Functions......Page 137
4. Applications to Period Relations......Page 154
Contents......Page 163
1.1. Semigroup generated by a differential operator D......Page 165
1.2. Comparison with the Hermite operator D......Page 167
1.3. The action of SL(2;R) O(m)......Page 168
1.4. Minimal representation as hidden symmetry......Page 169
2.1. Maximal parabolic subgroup of the conformal group......Page 173
2.2. L2-model of the minimal representation......Page 175
2.3. K-type decomposition......Page 176
3. Branching Law of +......Page 177
3.2. K- nite functions on the forward light cone C+......Page 178
3.3. Description of in nitesimal generators of sl(2;R)......Page 180
3.4. Central element Z of kC......Page 183
3.5. Proof of Proposition 3.2.1......Page 185
3.6. One parameter holomorphic semigroup (etZ )......Page 186
4.1. Result of the section......Page 187
4.2. Upper estimate of the kernel function......Page 189
4.3. Proof of Theorem 4.1.1 (Case Re t > 0)......Page 191
4.4. Proof of Theorem 4.1.1 (Case Re t = 0)......Page 194
4.6. Dirac sequence operators......Page 195
5.1. Result of the section......Page 196
5.2. Upper estimates of the kernel function......Page 198
5.3. Proof of Theorem 5.1.1 (Case Re t > 0)......Page 199
5.4. Proof of Theorem 5.1.1 (Case Re t = 0)......Page 200
5.5. Spectra of an O(m)-invariant operator......Page 201
5.7. Expansion formulas......Page 203
6.1. Result of the section......Page 205
6.2. Inversion and Plancherel formula......Page 207
6.3. The Hankel transform......Page 208
7. Explicit Actions of the Whole Group on L2(C)......Page 209
7.1. Bruhat decomposition of O(m + 1; 2)......Page 210
7.2. Explicit action of the whole group......Page 211
8.1. Laguerre polynomials......Page 212
8.2. Hermite polynomials......Page 213
8.3. Gegenbauer polynomials......Page 214
8.5. Bessel functions......Page 215
References......Page 218
Classification des Series Discretes pour Certains Groupes Classiques p-Adiques Colette MEglin......Page 221
1. Support Cuspidal des S eries Discr etes......Page 227
2. Morphismes Associes aux Representations Cuspidales de G(n) et Points de Reductibilite des Induites de Cuspidales......Page 236
3. Classification et Paquet de Langlands......Page 239
4. Classification a la Langlands des Series Discretes de SO(2n + 1; F)......Page 240
5. Le cas des Groupes Orthogonaux Impairs non Deployes......Page 254
References......Page 257
1. Introduction......Page 259
2. The Convolution Algebras H(G)b and U(G)......Page 262
3. Some Properties of the Convolution Algebras H(G)b and U(G)......Page 269
4. Some Explicit G-invariant Essentially Compact Distributions......Page 281
Acknowledgment......Page 286
References......Page 287
1. Introduction......Page 289
2. Minimal Polynomials......Page 293
3. Projection to the Cartan Subalgebra......Page 298
4. Generalized Verma Modules......Page 308
5. Integral Transforms on Generalized Flag Manifolds......Page 323
6. Closure of Ideals......Page 327
References......Page 330
Contents......Page 333
1. General Introduction......Page 334
2. Technical Introduction......Page 341
3. Highest Weights for K......Page 344
4. The R-Group of K and Irreducible Representations of the Large Cartan......Page 351
5. Fundamental Series, Limits, and Continuations......Page 359
6. Characters of Compact Tori......Page 364
7. Split Tori and Representations of K......Page 365
8. Parametrizing Extended Weights......Page 370
9. Proof of Lemma 8.13......Page 376
10. Highest Weights for K and -Stable Parabolic Subalgebras......Page 378
11. Standard Representations, Limits, and Continuations......Page 381
12. More Constructions of Standard Representations......Page 389
13. From Highest Weights to Discrete Final Limit Parameters......Page 398
14. Algorithm for Projecting a Weight on the Dominant Weyl Chamber......Page 400
15. Making a List of Representations of K......Page 403
16. G-Spherical Representations as Sums of Standard Representations......Page 408
References......Page 413
1. Introduction......Page 415
2. Invariant Theory......Page 419
3. Representation Theory......Page 422
4. A Proof of a Theorem in Penkov-Zuckerman......Page 433
5. Example: G2......Page 434
6. Tables for Maximal Parabolic Subalgebras......Page 438
References......Page 439
