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?
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1$\begingroup$ It would help if you could tell us what the definition is of a harmonic polynomial associated to a finite group. $\endgroup$– Tom De MedtsCommented Dec 3, 2015 at 8:36
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$\begingroup$ Also, it's important here to specify the field over which you work, since that affects notions such as "finite reflection group". What source material are you starting with? $\endgroup$– Jim HumphreysCommented Dec 4, 2015 at 15:02
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$\begingroup$ If I understand the Theorem on page 263 in Kane's book correctly, Steinberg in fact proved that only in the case of finte reflection groups, the harmonic polynomials are generated by one polynomial and its partial derivatives?! $\endgroup$– SeppoCommented Dec 4, 2015 at 20:55
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Maybe you need to be more clear about the word "similar". For example, the celebrated case of "diagonal harmonics" corresponding to the diagonal action of $S_n$ on $(\mathbb{C}^{2})^n$ there is some result of the same flavour, which is however much more complex (Section 4 in https://math.berkeley.edu/~mhaiman/ftp/vanishing/van.pdf), but in general, e.g. for $(\mathbb{C}^3)^n$ no such statement seems to be available.
answered Dec 4, 2015 at 9:45