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Studies in Lie Theory Dedicated to A. Joseph on his Sixtieth Birthday

:	Joseph Bernstein, Vladimir Hinich, Anna Melnikov
:	Studies in Lie Theory Dedicated to A. Joseph on his Sixtieth Birthday
:	Birkhäuser Basel
:	9780817644789
:	1
:	CHF 150.40
:

:	Arithmetik, Algebra
:	English

:	494
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

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

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].

	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