Studies in Lie Theory (eBook)

Contents 8
Preface 10
Publications of Anthony Joseph 14
Students of Anthony Joseph 22
List of Summer Students 22
From Denise Joseph 24
From Jacques Dixmier: A Recollection of Tony Joseph 25
Part I Survey and Review 27
The work of Anthony Joseph in classical representation theory 29
Quantized representation theory following Joseph 35
1 Local Finiteness 36
2 Geometry 38
3 Trickle Down Economics 41
References 42
Part II Research Articles 45
Opérateurs différentiels invariants et problème de Noether de Noether 47
Introduction 47
1. Une extension du problème de Noether pour les algèbres de Weyl 49
2. Cas d’une somme directe de repr ´ esentations de dimension 1 54
3. Cas d’une repr ´ esentation de dimension 2 58
Bibliographie 75
Langlands parameters for Heisenberg modules 77
Introduction 77
1. The space of T .-local systems 78
2. The Heisenberg modules and the spectral decomposition 81
References 86
Instanton counting via affine Lie algebras II: From Whittaker vectors to the Seiberg–Witten prepotential 87
1. Introduction 88
2. Schrödinger operators and the prepotential: the one-dimensional case 91
3. Schrödinger operators in higher dimensions and integrable systems 97
4. Proof of Nekrasov’s conjecture 101
References 103
Irreducibility of perfect representations of double affine Hecke algebras 105
1. Af.ne Weyl groups 107
2. Double Hecke algebras 109
3. Macdonald polynomials 112
4. The radical 115
5. The irreducibility 116
6. A non-semisimple example 118
References 121
Algebraic groups over a 2-dimensional local field: Some further constructions 123
Introduction 123
1. The pro-vector space of distributions 126
2. Existence of certain left adjoint functors 131
3. The functor of coinvariants 135
4. The functor of semi-invariants 137
5. Proof of Proposition 4.2 141
6. Distributions on a stack 145
7. Induction via the moduli stack of bundles 148
8. Proof of Theorem 7.9 151
References 156
Modules with a Demazure flag 157
1. Introduction 157
2. Notation and background 159
3. Properties of the global basis 162
4. The combinatorics of Demazure crystals 171
5. Demazure .ags 175
6. The PRV theorem 191
Index of Notation 192
References 194
Microlocalization of ind-sheaves 197
0. Introduction 198
1. Microlocal kernels 199
2. Microlocalization of ind-sheaves 220
Acknowledgments 246
References 247
Endoscopic decomposition of certain depth zero representations 249
0. Introduction 249
1. Basic de.nitions and constructions 252
2. Endoscopic decomposition 286
Appendix A. Springer Hypothesis 309
Appendix B. 311
List of main terms and symbols 324
References 325
Odd family algebras 329
1. Generalities about odd family algebras 329
2. The character of g-module (g) 331
3. Structure of odd family algebras 335
4. Odd family algebras for standard representations of Classical Lie algebras (types A, B, C) 339
5. Other examples 343
References 344
Gelfand–Zeitlin theory from the perspective of classical mechanics. I 345
0. Introduction 346
1. Preliminaries 352
2. Commuting vector .elds arising from Gelfand–Zeitlin theory 356
3. The group A and its orbit structure on M(n) 370
References 389
Extensions of algebraic groups 391
Introduction 391
1. Extensions of Algebraic Groups 392
2. Analogue of Van-Est Theorem for algebraic group cohomology 397
Acknowledgments 402
References 402
Differential operators and cohomology groups on the basic affine space 403
1. Introduction 403
2. Preliminaries 406
3. The structure of 410
4. The D(X)-module H.(X,OX) 414
5. Differential operators on S-varieties 419
6. Exotic differential operators 422
Acknowledgement 428
References 428
A q-analogue of an identity of N. Wallach 431
Centralizers in the quantum plane algebra 437
Introduction 437
Centralizers in Cq 438
Centralizers in quantum spaces 440
Conclusion and remarks 441
References 442
Centralizer construction of the Yangian of the queer Lie superalgebra 443
1. Main results 443
2. Proof of Theorem 1.2 450
3. Proof of Theorem 1.5 459
References 466
Definitio nova algebroidis verticiani 469
Prooemium 469
Caput primum. Structurae Calabi–Yautianae 472
1. Revocatio 472
2. Complexus Hochschild–De Rhamianus 472
3. Cocyclus canonicus 475
Caput secundum. Structurae verticianae 475
1. Koszul et de Rham 475
2. Pede plana 483
3. Tabulatum primum 489
4. Tabulatum secundum 496
1. Pede plana 499
2. Tabulatum primum 511
Pars tertia. Finale 516
1. Cocyclus canonicus 516
2. De.nitio altera 518
3. Complexus de Rham–Koszul–Hochschildianus 519

Irreducibility of perfect representations of double affine Hecke algebras (p. 79)

Ivan Cherednik.
Department of Mathematics
UNC Chapel Hill
Chapel Hill, North Carolina 27599
USA

Dedicated to A. Joseph on his 60th birthday

Summary. It is proved that the quotient of the polynomial representation of the double af.ne Hecke algebra by the radical of the duality pairing is always irreducible apart from the roots of unity provided that it is .nite dimensional. We also .nd necessary and suf.cient conditions for the radical to be zero, a generalization of Opdam’s formula for the singular parameters such that the corresponding Dunkl operators have multiple zero-eigenvalues.

Subject Classification: 20C08

In the paper we prove that the quotient of the polynomial representation of the double affine Hecke algebra (DAHA) by the radical of the duality pairing is always irreducible (apart from the roots of unity) provided that it is finite dimensional. We also find necessary and suf.cient conditions for the radical to be zero, which is a qgeneralization of Opdam’s formula for the singular k-parameters with the multiple zero-eigenvalue of the corresponding Dunkl operators.

Concerning the terminology, perfect modules in the paper are finite dimensional possessing a non-degenerate duality pairing. The latter induces the canonical duality anti-involution of DAHA. Actually, it suf.ces to assume that the pairing is perfect, i.e., identi.es the module with its dual as a vector space, but we will stick to the finitedimensional case.

We also assume that perfect modules are spherical, i.e., quotients of the polynomial representation of DAHA, and invariant under the projective action of PSL(2,Z). We do not impose the semisimplicity in contrast to [C3]. The irreducibility theorem in this paper is stronger and at the same time the proof is simpler than that in [C3].