Integral Kernels with Reflection Group Invariance

Abstract

Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group-invariant measure. The theory has been developed in several papers of the author [4,5,6,7]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.

A Dirac–Dunkl Equation on S 2 and the Bannai–Ito Algebra

“…The Bannai-Ito algebra also arises in representation theoretic problems [17] and in superintegrable systems [16]; see [2] for a recent overview. Following their introduction in [8,9,10], Dunkl operators have appeared in various areas. They enter the study of Calogero-Moser-Sutherland models [28], they play a central role in the theory of multivariate orthogonal polynomials associated to reflection groups [11], they give rise to families of stochastic processes [20,25], and they can be used to construct quantum superintegrable systems involving reflections [13,14].…”

Generalized Dunkl-Lipschitz spaces

“…In this section we state some definitions and results which are useful in the sequel and we refer for more details to the articles [7,10,11,12], [9], [24] and [26]. We first begin by some notations.…”

“…For k = 0, D 0 reduces to the usual derivative which will be denoted by D. The Dunkl intertwining operator V k is defined in [11] on polynomials f by…”

Reproducing inversion formulas for the Dunkl-Wigner transforms

“…For y ∈ R d , the initial problem D j u(., y)(x) = y j u(x, y), j = 1, ..., d, with u(0, y) = 1 admits a unique analytic solution on R d , which will be denoted by E k (x, y) and called Dunkl kernel [2,4]. This kernel has a unique analytic extension to C d × C d (see [7]).…”