A Laplace-Type Representation of the Generalized Spherical Functions Associated with the Root Systems of Type A

作者：Sawyer Patrice^*

来源：Mediterranean Journal of Mathematics, 2017, 14(4): UNSP 147.

DOI：10.1007/s00009-017-0948-0

• Generalized spherical function
• Dunkl setting
• root system
• intertwining operator
• Abel transform

In this paper, we extend the iterative expression for the generalized spherical functions associated with the root systems of type A previously obtained (Sawyer in Trans Am Math Soc 349(9): 3569-3584, 1997; Sawyer in Q J Math Oxf Ser (2) 50(197): 71-86, 1999) beyond regular elements. We also provide a similar expression in the corresponding flat case. From there, we derive a Laplace-type representation for the generalized spherical functions associated with the root systems of type A in the Dunkl setting as well as in the trigonometric Dunkl setting. This representation leads us to describe precisely the support of the generalized Abel transform. Thanks to a recent result of Gallardo and Rejeb (Support properties of the intertwining and the mean value operators in Dunkls analysis. Preprint [hal01331693], pp 1-10, 2016) and Rejeb (Harmonic and subharmonic functions associated with root systems. Mathematics, Universite Francois-Rabelais de Tours, Universit'e de Tunis El Manar, 2015), which allows us to give the support for the Dunkl intertwining operator.

• 出版日期2017-8