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

TL;DR

This paper introduces a class of integrable Calogero-Moser type operators associated with hyperplane arrangements, expanding the known examples and establishing their complete integrability through Cherednik algebra techniques.

Contribution

It constructs new integrable Calogero-Moser operators linked to hyperplane arrangements and explores their algebraic properties within rational Cherednik algebras.

Findings

Proves complete integrability of the operators.

Includes all known deformed Calogero-Moser systems.

Constructs new BC-type integrable operators.

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. Rap\v{c}\'ak in \cite{GR}. We describe these examples in a general framework of rational Cherednik algebras close in spirit to \cite{BEG} and \cite{BC}.