Homogeneous symmetry operators in Kerr-NUT-AdS spacetimes

It is well known that the Kerr-Newmann-Unti-Tamburino-anti-de Sitter spacetimes possess hidden symmetries encoded in the so-called principal Killing-Yano tensor. In this paper, focusing on the four-dimensional case, we obtain a number of symmetry operators for scalar, vector, and tensor perturbations, that are of degree 2 (to be defined below) and homogeneous in the principal tensor. In particular, by considering homogeneous operators that are linear, quadratic, and cubic in the principal tensor, we recover a complete set of four mutually commuting operators for scalar perturbations, underlying the separability of (massive) scalar wave equation. Proceeding to vector and tensor perturbations of the Kerr-Newmann-Unti-Tamburino-anti-de Sitter spacetimes, we find a set of seven and eight commuting operators, respectively. It remains to be seen whether such operators can be used to separate the corresponding spin 1 and spin 2 test field equations in these spacetimes.