Quasi-invariants of special arrangements.

David Johnstone (Glasgow)

Tuesday 1st November, 2011 15:00-16:00 Mathematics Building, room 515


'The ring of \emph{quasi-invariants} $Q_m$ can be associated with the root system $R$ and integer-valued multiplicity function $m$. It first appeared in the work of Chalykh and Veselov in the context of quantum Calogero-Moser systems. The ring $Q_m$ consists of all quantum integrals for the corresponding Calogero-Moser operator. These rings have nice algebraic properties. In particular they are Gorenstein, as was established by Feign and Veselov in the dihedral case and Etingof and Ginzburg in general. One can define an analogue $Q_{\mathcal{A}}$ of this ring for any arrangement $\mathcal{A}$ of vectors with multiplicities. In general these rings do not have the nice algebraic properties apparent in the Coxeter case. However for certain special arrangements (in particular the deformed root systems $\mathcal{A}_n(m)$ introduced by Chalykh, Feigin and Veselov) some of these properties are apparent. These arrangements are significant from a physical standpoint as integrability of the corresponding generalized Calogero-Moser operator can be established via the existence of the Baker-Akhiezer function. For the class of arrangements on the plane with at most one multiplicity greater than one we investigate when the Gorenstein property holds for $Q_{\mathcal{A}}$. We show that this property is intimately connected with the existence of the Baker-Akhiezer function for arrangements within this class. Time permitting we will also discuss work contributing to the problem of establishing the Gorenstein property for the ring of quasi-invariants for the deformed root system $\mathcal{A}_n(m)$. We explain how to find the Poincar\'e series for a `symmetric part' of $Q_{\mathcal{A}_n(m)}$.

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