Let $G$ be a finite group acting on a complex vector space $V$. Let $\mathcal{D}$ denote the differential operators with constant coefficients and $\mathcal{D}^{G}$ be the $G$-invariant operators. A polynomial $p$ is called harmonic if $D(p)=0$ for all $D\in\mathcal{D}^G$. Denote the set of harmonic polynomials by $H_G$. In the case of the symmetric group these polynomials are spanned by the Vandermonde determinant and all its partial derivatives. This result seems to generalize to all finite reflection groups. For a general finite group, it is clear that $H_G$ is a finite dimensional vector space, but is there a similar construction to obtain a basis?