  In this paper, we study the direct and inverse scattering theory at fixed
energy for massless charged Dirac fields evolving in the exterior region of a
Kerr-Newman-de Sitter black hole. In the first part, we establish the existence
and asymptotic completeness of time-dependent wave operators associated to our
Dirac fields. This leads to the definition of the time-dependent scattering
operator that encodes the far-field behavior (with respect to a stationary
observer) in the asymptotic regions of the black hole: the event and
cosmological horizons. We also use the miraculous property (quoting
Chandrasekhar) - that the Dirac equation can be separated into radial and
angular ordinary differential equations - to make the link between the
time-dependent scattering operator and its stationary counterpart. This leads
to a nice expression of the scattering matrix at fixed energy in terms of
stationary solutions of the system of separated equations. In a second part, we
use this expression of the scattering matrix to study the uniqueness property
in the associated inverse scattering problem at fixed energy. Using essentially
the particular form of the angular equation (that can be solved explicitely by
Frobenius method) and the Complex Angular Momentum technique on the radial
equation, we are finally able to determine uniquely the metric of the black
hole from the knowledge of the scattering matrix at a fixed energy.
