Eric M. Opdam

Harmonic analysis for certain representations of the graded Hecke algebra

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 1. Notations and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 78 2. Cheredniks operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3. The Knizhnik-Zamolodchikov connection . . . . . . . . . . . . . . . . 85 4. Invariant Hermitean structures . . . . . . . . . . . . . . . . . . . . . . 90 5. Harmonic analysis on T . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6. Asymptotic expansions and growth estimates . . . . . . . . . . . . . 99 7. The Cherednik transform . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8. The Paley-Wiener theorem . . . . . . . . . . . . . . . . . . . . . . . . 109 9. Inversion formulas and the Plancherel formula . . . . . . . . . . . . . 113

On the category O for rational Cherednik algebras

Abstract We study the category 𝒪 of representations of the rational Cherednik algebra AW attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor: 𝒪→ℋW-mod, where ℋW is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between 𝒪/𝒪tor, the quotient of 𝒪 by the subcategory of AW-modules supported on the discriminant, and the category of finite-dimensional ℋW-modules. The standard AW-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of “cells”, provided W is a Weyl group and the Hecke algebra ℋW has equal parameters. We prove that the category 𝒪 is equivalent to the module category over a finite dimensional algebra, a generalized “q-Schur algebra” associated to W.

Some applications of hypergeometric shift operators

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Dunkl operators for complex reflection groups

Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameterized family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the ?rational Cherednik algebra?, and a natural contravariant form on this module. In the case of the imprimitive complex reflection groups

Singular polynomials for finite reflection groups

G(m, p, N)

The Schwartz algebra of an affine Hecke algebra

, the set of singular parameters in the parameterized family of these structures is described explicitly, using the theory of non-symmetric Jack polynomials.

Periodic Integrable Systems with Delta-Potentials

The Dunkl operators involve a multiplicity function as parameter. For generic values of this function the simultaneous kernel of these operators, acting on polynomials, is equal to the constants. For special values, however, this kernel is larger. We determine these singular values completely and give partial results on the representations of G that occur in this kernel.

Trigonometric Cherednik algebra at critical level and quantum many-body problems

For a general affine Hecke algebra H we study its Schwartz completion S. The main theorem is an exact description of the image of S under the Fourier isomorphism. An important ingredient in the proof of this result is the definition and computation of the constant terms of a coefficient of a generalized principal series representation. Finally we discuss some consequences of the main theorem for the theory of tempered representations of H.

On the category for rational Cherednik algebras

In this paper we study root system generalizations of the quantum Bose-gas on the circle with pair-wise delta-function interactions. The underlying symmetry structures are shown to be governed by the associated graded algebra of Cheredniks (suitably filtered) degenerate double affine Hecke algebra, acting by Dunkl-type differential-reflection operators. We use Gutkins generalization of the equivalence between the impenetrable Bose-gas and the free Fermi-gas to derive the Bethe ansatz equations and the Bethe ansatz eigenfunctions.

A remark on the irreducible characters and fake degrees of finite real reflection groups

Abstract.For any module over the affine Weyl group we construct a representation of the associated trigonometric Cherednik algebra A(k) at critical level in terms of Dunkl type operators. Under this representation the center of A(k) produces quantum conserved integrals for root system generalizations of quantum spin-particle systems on the circle with delta function interactions. This enables us to translate the spectral problem of such a quantum spin-particle system to questions in the representation theory of A(k). We use this approach to derive the associated Bethe ansatz equations. They are expressed in terms of the normalized intertwiners of A(k).