Bernstein / Hinich / Melnikov Studies in Lie Theory

Dedicated to A. Joseph on his Sixtieth Birthday
2006
ISBN: 978-0-8176-4478-9
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 243, 494 Seiten

Reihe: Progress in Mathematics

Contains new results on different aspects of Lie theory, including Lie superalgebras, quantum groups, crystal bases, representations of reductive groups in finite characteristic, and the geometric Langlands program

Autoren/Hrsg.

Bernstein, Joseph

Hinich, Vladimir

Melnikov, Anna

Inhaltsverzeichnis

1;Contents;8
2;Preface;10
3;Publications of Anthony Joseph;14
4;Students of Anthony Joseph;22
4.1;List of Summer Students;22
5;From Denise Joseph;24
6;From Jacques Dixmier: A Recollection of Tony Joseph;25
7;Part I Survey and Review;27
7.1;The work of Anthony Joseph in classical representation theory;29
7.2;Quantized representation theory following Joseph;35
7.2.1;1 Local Finiteness;36
7.2.2;2 Geometry;38
7.2.3;3 Trickle Down Economics;41
7.2.4;References;42
8;Part II Research Articles;45
8.1;Opérateurs différentiels invariants et problème de Noether de Noether;47
8.1.1;Introduction;47
8.1.2;1. Une extension du problème de Noether pour les algèbres de Weyl;49
8.1.3;2. Cas d’une somme directe de repr ´ esentations de dimension 1;54
8.1.4;3. Cas d’une repr ´ esentation de dimension 2;58
8.1.5;Bibliographie;75
8.2;Langlands parameters for Heisenberg modules;77
8.2.1;Introduction;77
8.2.2;1. The space of T .-local systems;78
8.2.3;2. The Heisenberg modules and the spectral decomposition;81
8.2.4;References;86
8.3;Instanton counting via affine Lie algebras II: From Whittaker vectors to the Seiberg–Witten prepotential;87
8.3.1;1. Introduction;88
8.3.2;2. Schrödinger operators and the prepotential: the one-dimensional case;91
8.3.3;3. Schrödinger operators in higher dimensions and integrable systems;97
8.3.4;4. Proof of Nekrasov’s conjecture;101
8.3.5;References;103
8.4;Irreducibility of perfect representations of double affine Hecke algebras;105
8.4.1;1. Af.ne Weyl groups;107
8.4.2;2. Double Hecke algebras;109
8.4.3;3. Macdonald polynomials;112
8.4.4;4. The radical;115
8.4.5;5. The irreducibility;116
8.4.6;6. A non-semisimple example;118
8.4.7;References;121
8.5;Algebraic groups over a 2-dimensional local field: Some further constructions;123
8.5.1;Introduction;123
8.5.2;1. The pro-vector space of distributions;126
8.5.3;2. Existence of certain left adjoint functors;131
8.5.4;3. The functor of coinvariants;135
8.5.5;4. The functor of semi-invariants;137
8.5.6;5. Proof of Proposition 4.2;141
8.5.7;6. Distributions on a stack;145
8.5.8;7. Induction via the moduli stack of bundles;148
8.5.9;8. Proof of Theorem 7.9;151
8.5.10;References;156
8.6;Modules with a Demazure flag;157
8.6.1;1. Introduction;157
8.6.2;2. Notation and background;159
8.6.3;3. Properties of the global basis;162
8.6.4;4. The combinatorics of Demazure crystals;171
8.6.5;5. Demazure .ags;175
8.6.6;6. The PRV theorem;191
8.6.7;Index of Notation;192
8.6.8;References;194
8.7;Microlocalization of ind-sheaves;197
8.7.1;0. Introduction;198
8.7.2;1. Microlocal kernels;199
8.7.3;2. Microlocalization of ind-sheaves;220
8.7.4;Acknowledgments;246
8.7.5;References;247
8.8;Endoscopic decomposition of certain depth zero representations;249
8.8.1;0. Introduction;249
8.8.2;1. Basic de.nitions and constructions;252
8.8.3;2. Endoscopic decomposition;286
8.8.4;Appendix A. Springer Hypothesis;309
8.8.5;Appendix B.;311
8.8.6;List of main terms and symbols;324
8.8.7;References;325
8.9;Odd family algebras;329
8.9.1;1. Generalities about odd family algebras;329
8.9.2;2. The character of g-module (g);331
8.9.3;3. Structure of odd family algebras;335
8.9.4;4. Odd family algebras for standard representations of Classical Lie algebras (types A, B, C);339
8.9.5;5. Other examples;343
8.9.6;References;344
8.10;Gelfand–Zeitlin theory from the perspective of classical mechanics. I;345
8.10.1;0. Introduction;346
8.10.2;1. Preliminaries;352
8.10.3;2. Commuting vector .elds arising from Gelfand–Zeitlin theory;356
8.10.4;3. The group A and its orbit structure on M(n);370
8.10.5;References;389
8.11;Extensions of algebraic groups;391
8.11.1;Introduction;391
8.11.2;1. Extensions of Algebraic Groups;392
8.11.3;2. Analogue of Van-Est Theorem for algebraic group cohomology;397
8.11.4;Acknowledgments;402
8.11.5;References;402
8.12;Differential operators and cohomology groups on the basic affine space;403
8.12.1;1. Introduction;403
8.12.2;2. Preliminaries;406
8.12.3;3. The structure of;410
8.12.4;4. The D(X)-module H.(X,OX);414
8.12.5;5. Differential operators on S-varieties;419
8.12.6;6. Exotic differential operators;422
8.12.7;Acknowledgement;428
8.12.8;References;428
8.13;A q-analogue of an identity of N. Wallach;431
8.14;Centralizers in the quantum plane algebra;437
8.14.1;Introduction;437
8.14.2;Centralizers in Cq;438
8.14.3;Centralizers in quantum spaces;440
8.14.4;Conclusion and remarks;441
8.14.5;References;442
8.15;Centralizer construction of the Yangian of the queer Lie superalgebra;443
8.15.1;1. Main results;443
8.15.2;2. Proof of Theorem 1.2;450
8.15.3;3. Proof of Theorem 1.5;459
8.15.4;References;466
8.16;Definitio nova algebroidis verticiani;469
8.16.1;Prooemium;469
8.16.2;Caput primum. Structurae Calabi–Yautianae;472
8.16.2.1;1. Revocatio;472
8.16.2.2;2. Complexus Hochschild–De Rhamianus;472
8.16.2.3;3. Cocyclus canonicus;475
8.16.3;Caput secundum. Structurae verticianae;475
8.16.3.1;1. Koszul et de Rham;475
8.16.3.2;2. Pede plana;483
8.16.3.3;3. Tabulatum primum;489
8.16.3.4;4. Tabulatum secundum;496
8.16.4;1. Pede plana;499
8.16.5;2. Tabulatum primum;511
8.16.6;Pars tertia. Finale;516
8.16.6.1;1. Cocyclus canonicus;516
8.16.6.2;2. De.nitio altera;518
8.16.6.3;3. Complexus de Rham–Koszul–Hochschildianus;519

Leseproben

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].