Singular polynomials from orbit spaces

E-ISSN： 1570-5846|148|6|1867-1879

ISSN： 0010-437x

Source： Compositio Mathematica, Vol.148, Iss.6, 2012-11, pp. : 1867-1879

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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.