  We derive the finite temperature oscillation modes of a harmonically confined
Bose-Einstein condensed gas undergoing rigid body rotation supported by a
vortex lattice in the condensate. The hydrodynamic modes separate into two
classes corresponding to in-phase (center-of-mass) and counter-phase (relative)
oscillations of the thermal cloud and the condensate. The in- and counter-phase
oscillations are independent of each other in the case where the thermal cloud
is inviscid for all modes studied, except the radial pulsations which couple
them because the pressure perturbations of the condensate and the thermal cloud
are governed by different adiabatic indices. If the thermal cloud is viscous,
the two classes of oscillations are coupled, i.e. each type of motion involves
simultaneously mass and entropy currents. The counter-phase oscillations are
damped by the mutual friction between the condensate and the thermal cloud
mediated by the vortex lattice. The damping is large for the values of the
drag-to-lift ratio of the order of unity and becomes increasingly ineffective
in either limit of small or large friction. An experimental measurement of a
subset of these oscillation modes and their damping rates can provide
information on the values of the phenomenological mutual friction coefficients,
and hence the quasiparticle-vortex scattering processes in dilute atomic Bose
gases.
