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Harmonic Polynomials Associated With Reflection Groups

Published online by Cambridge University Press:  20 November 2018

Yuan Xu
Yuan Xu*
Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222, USA, e-mail: yuan@math.uoregon.edu
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Abstract

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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.

Keywords

33C5033C45h-harmonicsreflection groupDunkl’s operators
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 43 , Issue 4 , 01 December 2000 , pp. 496 - 507
Copyright
Copyright © Canadian Mathematical Society 2000

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