Singular polynomials from orbit spaces

TL;DR

This paper constructs and characterizes singular polynomials in the polynomial representation of the rational Cherednik algebra for Coxeter groups, linking them to Saito polynomials and providing a complete classification.

Contribution

It explicitly constructs singular polynomials for all degrees and parameters, connecting them to flat coordinates on the orbit space and proving completeness.

Findings

Singular polynomials form a single copy of the reflection representation in S(V*)

Explicit formulas relate singular polynomials to Saito polynomials

All singular polynomials in the reflection isotypic component are classified

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 nonnegative integer m 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 that 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.