  Assessing the stability of higher-dimensional rotating black holes requires a
study of linearized gravitational perturbations around such backgrounds. We
study perturbations of Myers-Perry black holes with equal angular momenta in an
odd number of dimensions (greater than five), allowing for a cosmological
constant. We find a class of perturbations for which the equations of motion
reduce to a single radial equation. In the asymptotically flat case we find no
evidence of any instability. In the asymptotically anti-de Sitter case, we
demonstrate the existence of a superradiant instability that sets in precisely
when the angular velocity of the black hole exceeds the speed of light from the
point of view of the conformal boundary. We suggest that the endpoint of the
instability may be a stationary, nonaxisymmetric black hole.
