An Uncertainty Principle for the Dunkl Transform

The Dunkl transform is an integral transform on R" which generalises the classical Fourier transform. On suitable function spaces, it establishes a natural correspondence between the action of multiplication operators on one hand and so-called Dunkl operators on the other. These are differential-difference operators, generalising the usual partial derivatives, which are associated with a finite reflection group on some Euclidean space. They play, for example, a useful role in the algebraic description of exactly solvable quantum many body systems of Calogero-Moser-Sutherland type; among the broad literature in this context, we refer to [1], [9], and [11]. In his paper [8], de Jeu proved a quite general uncertainty principle for integral operators with bounded kernel which applies to the Dunkl transform; this result has the form of an e — ^-concentration principle as first stated in [4] for the Fourier transform. Analogues of the classical variance-based Weyl-Heisenberg uncertainty principle for the Dunkl transform have up to now only been given in the one-dimensional case ([14] and [15]). It is the aim of this note to present an extension to general Dunkl transforms in arbitrary dimensions. Our setting, which is described in more detail in section 2, is as follows: Let R be a finite (reduced) root system on R" and k : R -> [0, oo] a nonnegative multiplicity function on R. Let wk denote the weight function