The Dunkl kernel and intertwining operator for dihedral groups

Abstract

Dunkl operators associated with finite reflection groups generate a commutative algebra of differential-difference operators. There exists a unique linear operator called intertwining operator which intertwines between this algebra and the algebra of standard differential operators. There also exists a generalization of the Fourier transform in this context called Dunkl transform.In this paper, we determine an integral expression for the Dunkl kernel, which is the integral kernel of the Dunkl transform, for all dihedral groups. We also determine an integral expression for the intertwining operator in the case of dihedral groups, based on observations valid for all reflection groups. As a special case, we recover the result of [Xu, Intertwining operators associated to dihedral groups. Constr. Approx. 2019]. Crucial in our approach is a systematic use of the link between both integral kernels and the simplex in a suitable high dimensional space. Contents 1. Introduction 1 2. Preliminaries 3 2.1. Basics of Dunkl theory 3 2.2. Humbert function of several variables 5 3. New formulas for the intertwining operator and its inverse 6 4. The case of dihedral groups 11 4.1. The generalized Bessel function 11 4.2. The intertwining operator for invariant polynomials 15 4.3. The Dunkl kernel 18 4.4. The intertwining operator for polynomials 21 4.5. New proof of Xu's result 28 5. Conclusions 29 Acknowledgements 29 References 29

Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl--Dirac operator

“…However, there are many more possible families of irreducible representations available, as readily seen from Theorems 4.1 and 4.2, but we do not know of concrete useful examples of them. As the values where irreducibility and unitarity fail resemble conditions appearing in related work, for example in [4,8], it seems interesting to link them. It would also serve as a motivation to study the structure of the representations when they are reducible.…”

“…They are often the first non-trivial examples one can hope to consider completely. Recent investigations on the dihedral case include: closed formulas for intertwining operators [8,29], geometric properties of the Calogero-Moser space associated with dihedral groups [2] and the complete descriptions of the deformed unitary Howe dual pairs [5].…”

Dunkl intertwining operator for symmetric groups

“…. , νx n−1 , νx n ] where the last identity is obtained by the Laplace transform of Φ (n) 2 , see [6,9]. If we take ν = 1, we get the following corollary.…”

“…G = Z n 2 and S 3 , see [12]. Some partial progress for the dihedral group was obtained recently in [20] and a full expression was recently obtained in [6].…”

“…[21]. However, explicit expressions are only obtained in a few cases, for example Z n 2 , the dihedral groups and the symmetric group S 3 , we refer to [6,12,20] and the references therein.…”