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, eBook

Reihe: Progress in Mathematics

Dedicated to Anthony Joseph, this volume contains surveys and invited articles by leading specialists in representation theory. The focus here is on semisimple Lie algebras and quantum groups, where the impact of Joseph's work has been seminal and has changed the face of the subject.Two introductory biographical overviews of Joseph's contributions in classical representation theory (the theory of primitive ideals in semisimple Lie algebras) and quantized representation theory (the study of the quantized enveloping algebra) are followed by 16 research articles covering a number of varied and interesting topics in representation theory.

Zielgruppe

Research

Autoren/Hrsg.

Bernstein, Joseph

Hinich, Vladimir

Melnikov, Anna

Inhaltsverzeichnis

Survey and Review.- The work of Anthony Joseph in classical representation theory.- Quantized representation theory following Joseph.- Research Articles.- Opérateurs différentiels invariants et problème de Noether.- Langlands parameters for Heisenberg modules.- Instanton counting via affine Lie algebras II: From Whittaker vectors to the Seiberg-Witten prepotential.- Irreducibility of perfect representations of double affine Hecke algebras.- Algebraic groups over a 2-dimensional local field: Some further constructions.- Modules with a Demazure flag.- Microlocalization of ind-sheaves.- Endoscopic decomposition of certain depth zero representations.- Odd family algebras.- Gelfand-Zeitlin theory from the perspective of classical mechanics. I.- Extensions of algebraic groups.- Differential operators and cohomology groups on the basic affine space.- A q-analogue of an identity of N. Wallach.- Centralizers in the quantum plane algebra.- Centralizer construction of the Yangian of the queer Lie superalgebra.- Definitio nova algebroidis verticiani.

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].