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Laguerre semigroup and Dunkl operators

Part of: Integral transforms, operational calculus Abstract harmonic analysis Groups and semigroups of linear operators, their generalizations and applications Lie groups

Published online by Cambridge University Press:  18 May 2012

Salem Ben Saïd ,
Toshiyuki Kobayashi  and
Bent Ørsted
Salem Ben Saïd
Affiliation:
Université Henri Poincaré Nancy 1, Institut Elie Cartan, B.P. 239, 54506 Vandoeuvre-Les-Nancy, France (email: salem.bensaid@iecn.u-nancy.fr)
Toshiyuki Kobayashi
Affiliation:
The University of Tokyo, IPMU, Graduate School of Mathematical Sciences, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan (email: toshi@ms.u-tokyo.ac.jp)
Bent Ørsted
Affiliation:
University of Aarhus, Department of Mathematical Sciences, Building 530, Ny Munkegade, DK 8000, Aarhus C, Denmark (email: orsted@imf.au.dk)
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Abstract

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We construct a two-parameter family of actions ωk,a of the Lie algebra 𝔰𝔩(2,ℝ) by differential–difference operators on ℝN∖{0}. Here k is a multiplicity function for the Dunkl operators, and a>0 arises from the interpolation of the two 𝔰𝔩(2,ℝ) actions on the Weil representation of Mp(N,ℝ) and the minimal unitary representation of O(N+1,2). We prove that this action ωk,a lifts to a unitary representation of the universal covering of SL (2,ℝ) , and can even be extended to a holomorphic semigroup Ωk,a. In the k≡0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe (a=2) and the Laguerre semigroup studied by the second author with G. Mano (a=1) . One boundary value of our semigroup Ωk,a provides us with (k,a) -generalized Fourier transforms ℱk,a, which include the Dunkl transform 𝒟k (a=2) and a new unitary operator ℋk  (a=1) , namely a Dunkl–Hankel transform. We establish the inversion formula, a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty relation for ℱk,a. We also find kernel functions for Ωk,a and ℱk,a for a=1,2 in terms of Bessel functions and the Dunkl intertwining operator.

Keywords

Dunkl operatorsgeneralized Fourier transformCoxeter groupsSchrödinger modelholomorphic semigroupWeil representationHermite semigroupHankel transformsHeisenberg inequalityrational Cherednik algebraminimal representation

MSC classification

Secondary: 33C52: Orthogonal polynomials and functions associated with root systems 22E46: Semisimple Lie groups and their representations 43A32: Other transforms and operators of Fourier type 44A15: Special transforms (Legendre, Hilbert, etc.) 47D03: Groups and semigroups of linear operators
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 4 , July 2012 , pp. 1265 - 1336
Copyright
Copyright © Foundation Compositio Mathematica 2012

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