Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications

A series expansion for Heckman-Opdam hypergeometric functions φλ is obtained for all λ ∈ a∗ C . As a consequence, estimates for φλ away from the walls of a Weyl chamber are established. We also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of Helgason and Johnson on the bounded spherical functions on a Riemannian symmetric space of the noncompact type. The Ltheory for the hypergeometric Fourier transform is developed for 0 < p < 2. In particular, an inversion formula is proved when 1 ≤ p < 2. Introduction A natural extension of Harish-Chandra’s theory of spherical functions on Riemannian symmetric spaces of the noncompact type was introduced by Heckman and Opdam in the late eighties ([14], [12], [21]). In this theory, the symmetric space is replaced by a triple (a,Σ,m) consisting of a finite dimensional real Euclidean vector space a, a root system Σ in the dual a of a, and a positive multiplicity function m on Σ. A commuting family D = D(a,Σ,m) of differential operators on a is associated with this triple. The hypergeometric functions of Heckman and Opdam are joint eigenfunctions of D. For certain values of the multiplicity function, the triple (a,Σ,m) indeed arises from a Riemannian symmetric space of the noncompact type G/K. In this case, D coincides with the algebra of radial components of the G-invariant differential operators on G/K, and Heckman-Opdam’s hypergeometric functions are the restrictions to a of Harish-Chandra’s elementary spherical functions on G/K. Heckman-Opdam’s theory of hypergeometric functions associated with root systems underwent an important development with the discovery of Cherednik operators (see [3], [22], [23] and references therein). The Cherednik operators (also called Dunkl-Cherednik operators or trigonometric Dunkl operators, as they are the curved analogue of the Dunkl operators on R) are a commuting family of first order differential-reflection operators. They allow to construct algebraically all elements of D. Let W denote the Weyl group of Σ, and let C c (a) W be the space of compactly-supported W -invariant smooth functions on a. The spectral decomposition of D on C c (a) W is obtained by means of the hypergeometric Fourier transform. Let a C be the complexified dual of a, and 2010 Mathematics Subject Classification. Primary: 33C67; secondary: 43A32, 43A90.