Paley-Wiener theorems for the Dunkl transform

TL;DR

This paper explores a conjectured geometric Paley-Wiener theorem for the Dunkl transform, proves specific cases, and links Dunkl operators to the Cartan motion group, revealing new insights into invariant differential operators and spherical functions.

Contribution

It introduces a conjectured geometric Paley-Wiener theorem for the Dunkl transform, proves three instances, and connects Dunkl operators with the Cartan motion group and invariant differential operators.

Findings

Proved three instances of the conjectured Paley-Wiener theorem for Dunkl transform.

Established the connection between Dunkl operators and the Cartan motion group.

Showed how the Abel transform can be inverted by a differential operator for certain multiplicities.

Abstract

We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, one of which involves a limit transition from Opdam's results for the graded Hecke algebra. Furthermore, the connection between Dunkl operators and the Cartan motion group is established. It is shown how the algebra of radial parts of invariant differential operators can be described explicitly in terms of Dunkl operators, which implies that the generalized Bessel functions coincide with the spherical functions. In this context, the restriction of Dunkl's intertwining operator to the invariants can be interpreted in terms of the Abel transform. Using shift operators we also show that, for certain values of the multiplicities of the restricted roots, the Abel transform is essentially inverted by a differential operator.