Action of Coxeter groups on $m$-harmonic polynomials and Knizhnik–Zamolodchikov equations

The Matsuo–Cherednik correspondence is an isomorphism from solutions of Knizhnik–Zamolodchikov equations to eigenfunctions of generalized Calogero–Moser systems associated to Coxeter groups $G$ and a multiplicity function m on their root systems. We apply a version of this correspondence to the most degenerate case of zero spectral parameters. The space of eigenfunctions is then the space Hm of $m$-harmonic polynomials. We compute the Poincaré polynomials for the space Hm and for its isotypical components corresponding to each irreducible representation of the group $G$. We also give an explicit formula for m-harmonic polynomials of lowest positive degree in the $S_n$ case.