On the Dunkl intertwining operator

Dunkl operators are differential-difference operators parametrized by a finite reflection group and a weight function. The commutative algebra generated by these operators generalizes the algebra of standard differential operators and intertwines with this latter by the so-called intertwining operator. In this paper, we give an integral representation for the operator Vk∘eΔ/2 for an arbitrary Weyl group and a large class of regular weights k containing those of nonnegative real parts. Our representing measures are absolutely continuous with respect the Lebesgue measure in Rd, which allows us to derive out new results about the intertwining operator Vk and the Dunkl kernel Ek. We show in particular that the operator Vk∘eΔ/2 extends uniquely as a bounded operator to a large class of functions which are not necessarily differentiables. In the case of nonnegative weights, this operator is shown to be positivity-preserving.