Abstract
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, Universite de Tunis El Manar, 2015), which allows us to give the support for the Dunkl intertwining operator.
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