Mediterr. J. Math.
Springer International Publishing AG 2017
A Laplace-Type Representation of the
Generalized Spherical Functions Associated
with the Root Systems of Type A
Abstract. In this paper, we extend the iterative expression for the gen-
eralized 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 reg-
ular 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 op-
erators in Dunkls analysis. Preprint [hal01331693], pp 1–10, 2016)and
Rejeb (Harmonic and subharmonic functions associated with root sys-
tems. Mathematics, Universit´eFran¸cois-Rabelais de Tours, Universit´e
de Tunis El Manar, 2015), which allows us to give the support for the
Dunkl intertwining operator.
Mathematics Subject Classiﬁcation. 33C67, 43A90, 33C80, 43A85.
Keywords. Generalized spherical function, Dunkl setting, root system,
intertwining operator, Abel transform.
We start by providing some background. We refer the reader to [11,15]for
a more complete exposition on the Dunkl and trigonometric Dunkl settings.
Given a root system R and a Cartan subalgebra a, for every α ∈ R,let
This research is supported by funding from Laurentian University. The author is thankful to
the Institut f¨ur Mathematik at the Universi¨at Paderborn for their hospitality in July 2013
during which this work was started and to Professor Margit R¨osler for helpful conversations.
The author is grateful to the anonymous referee for many helpful comments.