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

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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, Université François-Rabelais de Tours, Université de Tunis El Manar, 2015), which allows us to give the support for the Dunkl intertwining operator.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Laurentian University, Sudbury, ON, P3E 2C6, Canada

    Patrice Sawyer

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  1. Patrice Sawyer
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Correspondence to Patrice Sawyer.

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This research is supported by funding from Laurentian University. The author is thankful to the Institut für Mathematik at the Universiät Paderborn for their hospitality in July 2013 during which this work was started and to Professor Margit Rösler for helpful conversations. The author is grateful to the anonymous referee for many helpful comments.

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Sawyer, P. A Laplace-Type Representation of the Generalized Spherical Functions Associated with the Root Systems of Type A . Mediterr. J. Math. 14, 147 (2017). https://doi.org/10.1007/s00009-017-0948-0

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  • DOI: https://doi.org/10.1007/s00009-017-0948-0

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