Rings of quantum integrals for generalised Calogero–Moser problems

The rings of quantum integrals for generalised Calogero–Moser problems are studied in the special case when all the parameters are integers. The problem is reduced to the description of the rings of polynomials satisfying a certain quasi-invariance property (quasi-invariants). The quasi-invariants of dihedral groups are fully described. It is shown that they form a free module over invariants generated by m-harmonic polynomials. The m-harmonic polynomials for general Coxeter group are introduced and investigated. For the non-Coxeter generalisations of Calogero–Moser problems related to the systems An(m), Cn+1(m, l), the rings of quantum integrals are considered. The Poincare series for the quasi-invariants of two-dimensional deformations are computed. It is shown that the rings of quasi-invariants are Gorenstein like in the Coxeter case.