Singular polynomials from orbit spaces

Abstract

Abstract We consider the polynomial representation S ( V * ) of the rational Cherednik algebra H c ( W ) associated to a finite Coxeter group W at constant parameter c . We show that for any degree d of W and m ∈ℕ the space S ( V * ) contains a single copy of the reflection representation V of W spanned by the homogeneous singular polynomials of degree d −1+ hm , where h is the Coxeter number of W ; these polynomials generate an H c ( W ) submodule with the parameter c =( d −1)/ h + m . We express these singular polynomials through the Saito polynomials which are flat coordinates of the Saito metric on the orbit space V / W . We also show that this exhausts all the singular polynomials in the isotypic component of the reflection representation V for any constant parameter c .