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 V k ∘ e Δ / 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 R d , which allows us to derive out new results about the intertwining operator V k and the Dunkl kernel E k . We show in particular that the operator V k ∘ e Δ / 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. [ABSTRACT FROM AUTHOR]