ACTION OF COXETER GROUPS ON m-HARMONIC POLYNOMIALS AND KZ 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. It is valid for generic values of the spectral parameters, in the complement of the discriminant locus. We extend the Matsuo–Cherednik correspondence correspondence to all values of the spectral parameters. We apply our result to the most degenerate case of zero spectral parameters. The space of eigenfunctions is then the space Hm of m-harmonic polynomials, recently introduced in [8]. We compute the Poincaré polynomials for the space Hm and of 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 Sn case.