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Dunkl operators associated with finite reflection groups generate acommutative algebra of differential-difference operators. There exists a uniquelinear operator called intertwining operator which intertwines between thisalgebra and the algebra of standard differential operators. There also exists ageneralization of the Fourier transform in this context called Dunkl transform.In this paper, we determine an integral expression for the Dunkl kernel, whichis the integral kernel of the Dunkl transform, for all dihedral groups. We alsodetermine an integral expression for the intertwining operator in the case ofdihedral groups, based on observations valid for all reflection groups. As aspecial case, we recover the result of [Xu, Intertwining operators associatedto dihedral groups. Constr. Approx. 2019]. Crucial in our approach is asystematic use of the link between both integral kernels and the simplex in asuitable high dimensional space.

The Dunkl kernel and intertwining operator for dihedral groups