  A massive vector boson field in the vicinity of a rotating black hole is
known to suffer an instability, due to the exponential amplification of
(co-rotating, low-frequency) bound states by black hole superradiance. Here we
calculate the bound state spectrum by exploiting the separation of variables
recently achieved by Frolov, Krtous, Kubiznak and Santos (FKKS) for the Proca
field on Kerr-(A)dS-NUT spacetimes of arbitrary dimension. Restricting to the
4D Kerr case, we first establish the relationship between the FKKS and
Teukolsky variables in the massless case; obtain exact results for the angular
eigenvalues in the marginally-bound case; and present a spectral method for
solving the angular equation in the general case. We demonstrate that all three
physical polarizations can be recovered from the FKKS ansatz, resolving an open
question. We present numerical results for the instability growth rate for a
selection of modes of all three polarizations, and discuss physical
implications.
