The Institute for Mathematical Sciences at the National University of Singapore hosted a research program on "Representation Theory of Lie Groups" from July 2002 to January 2003. As part of the program, tutorials for graduate students and junior researchers were given by leading experts in the field.
This invaluable volume collects the expanded lecture notes of those tutorials. The topics covered include uncertainty principles for locally compact abelian groups, fundamentals of representations of p-adic groups, the Harish–Chandra–Howe local character expansion, classification of the square-integrable representations modulo cuspidal data, Dirac cohomology and Vogan’s conjecture, multiplicity-free actions and Schur–Weyl–Howe duality.
The lecturers include Tomasz Przebinda from the University of Oklahoma, USA; Gordan Savin from the University of Utah, USA; Stephen DeBacker from Harvard University, USA; Marko Tadiæ from the University of Zagreb, Croatia; Jing-Song Huang from The Hong Kong University of Science and Technology, Hong Kong; Pavle Pandžiæ from the University of Zagreb, Croatia; Chal Benson and Gail Ratcliff from East Carolina University, USA; and Roe Goodman from Rutgers University, USA.
Author(s): Eng-Chye Tan, Chen-Bo Zhu
Series: Lecture notes series / Institute for Mathematical Sciences, National University of Singapore 2
Publisher: Singapore University Press; World Scientific
Year: 2004
Language: English
Pages: 425
City: Singapore :, River Edge, NJ
Title......Page 1
Series......Page 2
Title page......Page 3
Date-line......Page 4
Contents......Page 5
Foreword......Page 7
Preface......Page 9
Three Uncertainty Principles for an Abelian Locally Compact Group. Tomasz Przebinda......Page 11
Lectures on Representations of $p$-adic Groups. Gordan Savin......Page 29
Lectures on Harmonic Analysis for Reductive $p$-adic Groups. Stephen DeBacker......Page 57
On Classification of Some Classes of Irreducible Representations of Classical Groups. Marko Tadic......Page 105
1 Harmonic analysis and unitary duals......Page 107
2 Non-discrete locally compact fields, classical groups, reductive groups......Page 109
3 $K_0$-finite vectors......Page 113
4 Smooth representations......Page 114
5 Parabolically induced representations......Page 115
6 Jacquet modules......Page 119
7 Filtrations of Jacquet modules......Page 121
8 Square integrable and tempered representations......Page 122
9 Langlands classification......Page 124
10 Geometric lemma and algebraic structures......Page 130
11 Square integrable representations of $p$-adic general linear groups......Page 133
12 Two simple examples of square integrable representations of classical $p$-adic groups......Page 135
13 Invariants of square integrable representations of classical $p$-adic groups......Page 137
14 Reduction to cuspidal lines......Page 145
15 Parameters of $\\mathcal{D}(\\rho;\\sigma)$......Page 147
16 Integral case......Page 148
17 Non-integral case......Page 151
18 Local Langlands correspondences......Page 153
19 Non-unitary duals of classical $p$-adic groups......Page 155
20 Unitary duals of general linear groups over local fields......Page 157
21 On the unitarizability problem for classical $p$-adic groups......Page 164
References......Page 168
Dirac Operators in Representation Theory. Jing-Song Huang and Pavle Pandzic......Page 173
On Multiplicity Free Actions. Chal Benson and Gail Ratcliff......Page 231
1.2 Regular functions......Page 233
1.4 Structure theory......Page 234
1.5 Rational representations......Page 236
1.6 Highest weight theory......Page 238
1.8 Decompositions and multiplicities......Page 239
1.9 Group actions......Page 240
2 Multiplicity free actions......Page 241
2.1 Borel orbits......Page 242
2.2 Quasi-regular representations......Page 243
2.3 Maximal unipotent subgroups......Page 245
2.4 $S$-varieties......Page 246
2.5 Spherical pairs......Page 247
2.6 Section 2 notes......Page 248
3.1 Connectivity of $G$......Page 249
3.2 Borel orbits......Page 251
3.3 Fundamental highest weights for a multiplicity free action......Page 255
3.4 Section 3 notes......Page 257
4.1 $\\mathbf{GL(n)} \\otimes \\mathbf{GL(m)}$......Page 258
4.2 $\\mathbf{S^2(GL(n))}$......Page 262
4.3 $\\mathbf{\\Lambda^2(GL(n))}$......Page 264
4.4 $\\mathbf{SO(n)} \\times \\mathbf{C}^\\times$......Page 267
4.5 $\\mathbf{GL(n)} \\oplus_{\\mathbf{GL(n)}} \\mathbf{\\Lambda^2(GL(n))}$......Page 268
4.6 Section 4 notes......Page 270
5 A recursive criterion for multiplicity free actions......Page 272
5.1 $\\mathbf{GL(n)}$......Page 273
5.3 $\\mathbf{GL(n) \\oplus_{\\mathbf{GL(n)}} \\mathbf{\\Lambda^2(GL(n))}$......Page 274
6.1 Irreducible multiplicity free actions......Page 276
6.3 Saturated indecomposable multiplicity free actions......Page 278
6.5 Completing the classification......Page 280
6.6 Proof outline......Page 282
7.1 Polynomial coefficient differential operators......Page 283
7.2 Invariants in $\\mathcal{PD}(V)$......Page 286
7.3 A canonical basis for the invariants......Page 288
7.4 The fundamental invariants......Page 289
7.5 The algebra $\\mathbb{C}[V_\\mathbb{R}]^K$......Page 291
7.6 Section 7 notes......Page 293
8.1 The polynomials $q_\\lambda$......Page 294
8.2 The generalized binomial coefficients......Page 295
8.3 Eigenvalues for operators in $\\mathcal{PD}(V)^G$......Page 298
8.4 Examples......Page 299
8.5 Section 8 notes......Page 303
9.1 Eigenvalue polynomials......Page 304
9.2 A Harish-Chandra homomorphism for multiplicity free actions......Page 306
9.3 Characterizing the eigenvalue polynomials......Page 308
9.4 $\\mathbf{GL(n)} \\otimes \\mathbf{GL(n)}$ yet again......Page 309
References......Page 311
Multiplicity-Free Spaces and Schur-Weyl-Howe Duality. Roe Goodman......Page 315
1.1. Representations of algebraic groups......Page 317
1.2. Examples......Page 318
1.3. Reductive groups and isotypic decompositions......Page 319
1.4. Multiplicities and duality......Page 321
2.1. Density lemmas......Page 323
2.2. Proof of duality theorem......Page 325
2.3. Examples......Page 326
3.1. Commutant of GL($n$) action on tensors......Page 330
3.2. Highest weight theory......Page 331
3.3. Duality and $N$-fixed vectors......Page 336
4.2. Frobenius and determinant character formulas......Page 339
4.3. Proof of Frobenius character formula......Page 340
4.4. Proof of determinant character formula......Page 341
5.1. Frobenius formula for $\\mathfrak{S}_k$ characters......Page 342
5.2. Determinant formula for $\\mathfrak{S}_k$ characters......Page 344
5.3. Schur-Weyl duality and GL($k$)-GL($n$) duality......Page 346
6.1. Invariant polynomials......Page 348
6.2. Invariants of vectors and covectors......Page 349
6.4. Polynomial FFT for the orthogonal group......Page 351
6.5. Polynomial FFT for the symplectic group......Page 353
7.1. Tensor invariants for GL($V$)......Page 354
7.2. Proof of polynomial FFT for GL($V$)......Page 355
7.3. Tensor invariants for orthogonal and symplectic groups......Page 357
7.4. Proof of polynomial FFT for orthogonal and symplectic groups......Page 360
8.1. Duality in the Weyl algebra......Page 361
8.2. Howe duality for orthogonal/symplectic groups......Page 365
8.3. Howe duality for GL($k$)......Page 367
9.1. Harmonic polynomials......Page 368
9.2. Main theorem......Page 370
10. Decomposition of harmonic polynomials......Page 374
10.1. O($k$) Harmonics ($k$ odd)......Page 375
10.2. O($k$) Harmonics ($k$ even)......Page 381
10.3. Examples of harmonic decompositions......Page 384
11. Symplectic group and oscillator representation......Page 385
11.1. Real symplectic group......Page 386
11.2. Holomorphic (coherent-state) model for oscillator representation......Page 393
11.3. Bargmann-Segal transform......Page 397
11.4. Real (oscillatory-wave) model for oscillator representation......Page 399
11.5. Analytic vectors......Page 401
12.1. Decomposition of $\\mathrm{H}^2(M_{n\\times k})$ under $\\mathrm{Mp}(n,\\mathbb{R}) x \\mathrm{O}(k)$......Page 404
12.2. Square-integrable representations of $\\mathrm{Sp}(n,\\mathbb{R})$......Page 409
13.1. Centralizer algebra and Brauer diagrams......Page 412
13.2. Generators for the centralizer algebra......Page 415
13.3. Relations in the centralizer algebra......Page 417
13.4. Harmonic tensors......Page 419
13.5. Decomposition of harmonic tensors for Sp($V$)......Page 420
References......Page 424
