On The Dunkl Intertwining Opereator

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 $V_k\circ e^{\mathrm{\Delta }\mathrm{/}2}$ for an arbitrary Weyl group and a large class of regular weights $k$ containing those of non negative real parts. Our representing measures are absolute continuous with respect the Lebesgue measure in $\text{Rd}$, 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 $V_k\circ e^{\mathrm{\Delta }\mathrm{/}2}$ extends uniquely as a bounded operator to a large class of functions which are not necessarily differentiables. In the case of non negative weights, this operator is shown to be positivity-preserving.