Deformed Calogero--Moser operators and ideals of rational Cherednik algebras

Abstract

We consider a class of hyperplane arrangements $\mathcal{A}$ in ${\mathbb{C}}^n$ that generalise the locus configurations of \cite{CFV}. To such an arrangement we associate a second order partial differential operator of Calogero-Moser type, and prove that this operator is completely integrable (in the sense that its centraliser in $\mathcal{D}({\mathbb{C}}^n\setminus \mathcal{A})$ contains a maximal commutative subalgebra of Krull dimension $n$). Our approach is based on the study of shift operators and associated ideals in the spherical Cherednik algebra that may be of independent interest. The examples include all known families of deformed (rational) Calogero-Moser systems that appeared in the literature; we also construct some new examples, including a BC-type analogues of completely integrable operators recently found by D. Gaiotto and M. Rapčák in \cite{GR}. We describe these examples in a general framework of rational Cherednik algebras close in spirit to \cite{BEG} and \cite{BC}.

v2: 41 pages, substantially rewritten, new material and results added. Section 4 is entirely new. New examples added in subsections 6.4, 6.5

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