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Hankel transforms associated to finite reflection groups. (English) Zbl 0789.33008
Richards, Donald St. P. (ed.), Hypergeometric functions on domains of positivity, Jack polynomials, and applications. Proceedings of an AMS special session held March 22-23, 1991 in Tampa, FL, USA. Providence, RI: American Mathematical Society. Contemp. Math. 138, 123-138 (1992).

Root systems provide a rich framework for the study of special functions and integral transforms in several variables. Finite reflection groups are subgroups of the orthogonal group which are generated by reflections. The set of suitably normalized vectors normal to the hyperplanes corresponding to the reflections in the group is called a root system. This paper concerns weight functions which are products of powers of the linear functions vanishing on the reflecting hyperplanes and which are invariant under the group. These functions are specified by as many parameters as there are conjugacy classes of reflections.

There is an analogue of the exponential function $exp\left({\sum }_{j=1}^{n}{x}_{j}{y}_{j}\right)$, $x,y\in {ℂ}^{N}$, for the weight functions, which allows the definition of a transform generalizing the Fourier transform. The one-dimensional specialization is the classical Hankel transform. There is a Plancherel-type result. Its proof depends on an orthogonal basis of rapidly decreasing functions involving Laguerre polynomials and polynomials harmonic for the differential-difference Laplacian associated to the group.

The author has previously developed the theory of a commutative algebra of differential-difference operators which generalize partial differentiation. The transform maps these onto multiplication operators.

The weight functions under discussion are related to the Macdonald-Mehta integrals. Since the publication of this paper, M. F. E. de Jeu [Invent. Math. 113, 147-162 (1993; reviewed above)) has proven that the transform is ${L}^{1}$-bounded.

Reviewer: Charles F.Dunkl

##### MSC:
 33C80 Connections of hypergeometric functions with groups and algebras 44A15 Special transforms (Legendre, Hilbert, etc.) 42B10 Fourier type transforms, several variables
##### Keywords:
root systems; reflection groups