Deformed Calogero–Moser Operators and Ideals of Rational Cherednik Algebras

We introduce a class of hyperplane arrangements A in Cn that generalise the locus configurations of Chalykh, Feigin and Veselov. 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 D(Cn∖A) contains a maximal commutative subalgebra of Krull dimension n). Our approach is based on the study of shift operators and associated ideals in spherical Cherednik algebras that may be of independent interest. Examples include all known completely integrable deformations of Calogero–Moser operators with rational potentials. In addition, we construct new families of examples, including a BC-type generalisation of the deformed Calogero-Moser operators recently found by Gaiotto and Rapčák. We describe these examples in a unified representation-theoretic framework of rational Cherednik algebras.