Published online by Cambridge University Press: 20 November 2018
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.