Johnston, David
(2012)
Quasi-invariants of hyperplane arrangements.
PhD thesis, University of Glasgow.
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Abstract
The ring of quasi-invariants $Q_m$ can be associated with the root system $R$ and multiplicity function $m$. It first appeared in the work of Chalykh and Veselov in the context of quantum Calogero-Moser systems. One can define an analogue $Q_{\mathcal{A}}$ of this ring for a collection $\mathcal{A}$ of vectors with multiplicities. We study the algebraic properties of these rings. For the class of arrangements on the plane with at most one multiplicity greater than one we show that the Gorenstein property for $Q_{\mathcal{A}}$ is equivalent to the existence of the Baker-Akhiezer function, thus suggesting a new perspective on systems of Calogero-Moser type.
The rings of quasi-invariants $Q_m$ have a well known interpretation as modules for the spherical subalgebra of the rational Cherednik algebra with integer valued multiplicity function. We explicitly construct the anti-invariant quasi-invariant polynomials corresponding to the root system $A_n$ as certain representations of the spherical subalgebra of the Cherednik algebra $H_{1/m}(S_{mn})$. We also study the relation of the algebra $\Lambda_{n,1,k}$ introduced by Sergeev and Veselov to the ring of quasi-invariants for the deformed root system $\mathcal{A}_n(k)$. We find the Poincar\'e series for a `symmetric part' of $Q_{\mathcal{A}_n(k)}$ for positive integer values of $k$.
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