Dunkl kernel associated with dihedral group

Abstract: In this paper, we pursue the investigations started in \cite{Mas-You} where the authors provide a construction of the Dunkl intertwining operator for a large subset of the set of regular multiplicity values. More precisely, we make concrete the action of this operator on homogeneous polynomials when the root system is of dihedral type and under a mild assumption on the multiplicity function. In particular, we obtain a formula for the corresponding Dunkl kernel and another representation of the generalized Bessel function already derived in \cite{Demni0}. When the multiplicity function is everywhere constant, our computations give a solution to the problem of counting the number of decompositions of an element from a dihedral group into a fixed number of (non necessarily simple) reflections. In the remainder of the paper, we supply another method to derive the Dunkl kernel associated with dihedral systems from the corresponding generalized Bessel function. This time, we use the shift principle together with multiple combinations of Dunkl operators corresponding to the vectors of the canonical basis of $\mathbb{R}^2$. When the dihedral system is of order six and only in this case, a single combination suffices to get the Dunkl kernel and agrees up to an isomorphism with the formula recently obtained by Amri \cite[Lemma1]{Amri} in the case of a root system of type $A_2$. We finally derive an integral representation for the Dunkl kernel associated with the dihedral system of order eight.