  The Teukolsky master equation and its associated spin-weighted spheroidal
harmonic decomposition simplify considerably the study of linear gravitational
perturbations of the Kerr(-AdS) black hole. However, the formulation of the
problem is not complete before we assign the physically relevant boundary
conditions. We find a set of two Robin boundary conditions (BCs) that must be
imposed on the Teukolsky master variables to get perturbations that are
asymptotically global AdS, i.e. that asymptotes to the Einstein Static
Universe. In the context of the AdS/CFT correspondence, these BCs allow a
non-zero expectation value for the CFT stress-energy tensor while keeping fixed
the boundary metric. When the rotation vanishes, we also find the gauge
invariant differential map between the Teukolsky and the Kodama-Ishisbashi
(Regge-Wheeler-Zerilli) formalisms. One of our Robin BCs maps to the scalar
sector and the other to the vector sector of the Kodama-Ishisbashi
decomposition. The Robin BCs on the Teukolsky variables will allow for a
quantitative study of instability timescales and quasinormal mode spectrum of
the Kerr-AdS black hole. As a warm-up for this programme, we use the Teukolsky
formalism to recover the quasinormal mode spectrum of global AdS-Schwarzschild,
complementing previous analysis in the literature.
