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Singular polynomials from orbit spaces

Part of: Symplectic geometry, contact geometry Representation theory of rings and algebras

Published online by Cambridge University Press:  15 October 2012

Misha Feigin  and
Alexey Silantyev
Misha Feigin
Affiliation:
School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, UK (email: misha.feigin@glasgow.ac.uk)
Alexey Silantyev
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan (email: aleksejsilantjev@gmail.com)
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Abstract

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We consider the polynomial representation S(V*) of the rational Cherednik algebra Hc(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 Hc (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.

Keywords

Dunkl operatorsrational Cherednik algebrapolynomial representationFrobenius manifoldSaito metric

MSC classification

Secondary: 16G99: None of the above, but in this section 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1867 - 1879
Copyright
Copyright © Foundation Compositio Mathematica 2012

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