  We have studied extreme mass-ratio inspirals (EMRIs) in spacetimes containing
a rotating black hole and a non self-gravitating torus with constant specific
angular momentum. We have found that the effect of the hydrodynamic drag
exerted by the torus on the satellite is much smaller than the corresponding
one due to radiation reaction, for systems such as those generically expected
in AGNs and at distances from the SMBH which can be probed with LISA. However,
given the uncertainty on the parameters of these systems, there exist
configurations in which the effect of the hydrodynamic drag can be comparable
to the radiation-reaction one in phases of the inspiral which are detectable by
LISA. This is the case, for instance, for a 10^6 M_sun SMBH surrounded by a
corotating torus of comparable mass and with radius of 10^3-10^4 gravitational
radii, or for a 10^5 M_sun SMBH surrounded by a corotating 10^4 M_sun torus
with radius of 10^5 gravitational radii. Should these conditions be met in
astrophysical systems, EMRI-gravitational waves could provide a characteristic
signature of the presence of the torus. In fact, while radiation reaction
always increases the inclination of the orbit with respect to the equatorial
plane, the hydrodynamic drag from a torus corotating with the SMBH always
decreases it. However, even when initially dominating over radiation reaction,
the influence of the hydrodynamic drag decays very rapidly as the satellite
moves into the very strong-field region of the SMBH (i.e., p <~ 5M). Although
our results have been obtained for a specific class of tori, we argue that they
will be qualitatively valid also for more generic distributions of the specific
angular momentum.
