General Kerr-NUT-AdS metrics in all dimensions

The Kerr-AdS metric in dimension D has cohomogeneity [D/2]; the metric components depend on the radial coordinate r and [D/2] latitude variables mu(i) that are subject to the constraint Sigma(i)mu(2)(i) = 1. We find a coordinate reparametrization in which the mu(i) variables are replaced by [D/2] - 1 unconstrained coordinates y(alpha), and having the remarkable property that the Kerr-AdS metric becomes diagonal in the coordinate differentials dy(alpha). The coordinates r and ya now appear in a very symmetrical way in the metric, leading to an immediate generalization in which we can introduce [D/2] - 1 NUT parameters. We find that (D - 5)/2 are non-trivial in odd dimensions whilst (D - 2)/2 are non-trivial in even dimensions. This gives the most general Kerr-NUT-AdS metric in D dimensions. We find that in all dimensions D >= 4, there exist discrete symmetries that involve inverting a rotation parameter through the AdS radius. These symmetries imply that Kerr-NUT-AdS metrics with over-rotating parameters are equivalent to under-rotating metrics. We also consider the BPS limit of the Kerr-NUT-AdS metrics, and thereby obtain, in odd dimensions and after Euclideanization, new families of Einstein-Sasaki metrics.