Harmonic Polynomials Associated With Reflection Groups

Xu, Yuan

Canadian mathematical bulletin, Tome 43 (2000) no. 4, pp. 496-507

Résumé

We extend Maxwell’s representation of harmonic polynomials to $h$ -harmonics associated to a reflection invariant weight function ${{h}_{k}}$ . Let ${{\mathcal{D}}_{i}},\,1\,\le \,i\,\le \,d$ , be Dunkl’s operators associated with a reflection group. For any homogeneous polynomial $P$ of degree $n$ ,we prove the polynomial ${{\left| x \right|}^{2\gamma +d-2+2n}}P\left( \mathcal{D} \right)\left\{ 1/{{\left| x \right|}^{2\gamma +d-2}} \right\}$ is a $h$ -harmonic polynomial of degree $n$ , where $\gamma \,=\,\sum \,ki$ and $\mathcal{D}\,=\,\left( {{\mathcal{D}}_{1}},\ldots ,{{\mathcal{D}}_{d}} \right)$ . The construction yields a basis for $h$ -harmonics. We also discuss self-adjoint operators acting on the space of $h$ -harmonics.

Détail

DOI : 10.4153/CMB-2000-057-2

Mots-clés : 33C50, 33C45, h-harmonics, reflection group, Dunkl’s operators