  Scattering data can be generically described in terms of monodromies. Here we
obtain scattering amplitudes for conformally coupled scalar fields in Kerr-de
Sitter black holes using this monodromy technique. The only non-trivial
parameter, the composite monodromy parameter $\sigma_{ij}$ between two regular
singular points, can be solved implicitily in terms of the Painlev\'e VI
$\tau$-function. The application of the Virasoro conformal blocks to solve the
latter can now be interpreted as a verification of the striking relationship
between conformal symmetry and black holes.
