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Comptes Rendus Mathématique
Volume 347, n° 19-20
pages 1119-1124 (octobre 2009)
Doi : 10.1016/j.crma.2009.07.015
Received : 2 June 2009 ;  accepted : 17 July 2009
Generalized Fourier transforms  
Transformation de Fourier généralisée  
 

Salem Ben Saïd a , Toshiyuki Kobayashi b, 1 , Bent Ørsted c
a Université Henri-Poincaré-Nancy 1, Institut Elie-Cartan, B.P. 239, 54506 Vandoeuvre-Les-Nancy, France 
b The University of Tokyo, Graduate School of Mathematical Sciences, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan 
c University of Aarhus, Department of Mathematical Sciences, Ny Munkegade, DK 8000, Aarhus C, Denmark 

Abstract

We construct a two-parameter family of actions   of the Lie algebra   by differential-difference operators on  . Here, k is a multiplicity-function for the Dunkl operators, and   arises from the interpolation of the Weil representation and the minimal unitary representation of the conformal group. The action   lifts to a unitary representation of the universal covering of  , and can even be extended to a holomorphic semigroup  . Our semigroup generalizes the Hermite semigroup studied by R. Howe ( ,  ) and the Laguerre semigroup by T. Kobayashi and G. Mano ( ,  ). The boundary value of our semigroup   provides us with  -generalized Fourier transforms  , which includes the Dunkl transform   ( ) and a new unitary operator   ( ) as a Dunkl-type generalization of the classical Hankel transform. To cite this article: S. Ben Saïd et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.
Résumé

À l’aide des opérateurs différentiels et aux différences de Dunkl sur  , on construit une famille d’actions   de l’algèbre de Lie   dépendant de deux paramètres k et a . Ici k est une fonction de multiplicité associée aux opérateurs de Dunkl, et   un paramètre d’interpolation entre la représentation de Weil et la représentation minimale du groupe conforme. On montre que   s’intègre à une représentation unitaire du revêtement universel du groupe  , et se prolonge à un semi-groupe holomorphe  . Notre semi-groupe généralise le semi-groupe de Hermite, étudié par R. Howe ( ,  ), ainsi que le semi-groupe de Laguerre dû à T. Kobayashi et G. Mano ( ,  ). La valeur au bord de notre semi-groupe   donne une transformation de Fourier  -généralisée   qui correspond à la transformation de Dunkl pour  , et à une nouvelle transformation   pour   qui généralise la transformation de Hankel classique. Pour citer cet article : S. Ben Saïd et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.
1  Partially supported by Grant-in-Aid for Scientific Research (B) (18340037), Japan Society for the Promotion of Science, Max Planck Institute at Bonn, and the Alexander Humboldt Foundation.


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