  We analyze theoretically and experimentally the wake behind a horizontal
cylinder of diameter $d$ horizontally translated at constant velocity $U$ in a
fluid rotating about the vertical axis at a rate $\Omega$. Using particle image
velocimetry measurements in the rotating frame, we show that the wake is
stabilized by rotation for Reynolds number ${\rm Re}=Ud/\nu$ much larger than
in a non-rotating fluid. Over the explored range of parameters, the limit of
stability is ${\rm Re} \simeq (275 \pm 25) / {\rm Ro}$, with ${\rm
Ro}=U/2\Omega d$ the Rossby number, indicating that the stabilizing process is
governed by the Ekman pumping in the boundary layer. At low Rossby number, the
wake takes the form of a stationary pattern of inertial waves, similar to the
wake of surface gravity waves behind a ship. We compare this steady wake
pattern to a model, originally developed by [Johnson, J. Fluid Mech. 120, 359
(1982)], assuming a free-slip boundary condition and a weak streamwise
perturbation. Our measurements show a quantitative agreement with this model
for ${\rm Ro}\lesssim 0.3$. At larger Rossby number, the phase pattern of the
wake is close to the prediction for an infinitely small line object. However,
the wake amplitude and phase origin are not correctly described by the
weak-streamwise-perturbation model, calling for an alternative model for the
boundary condition at moderate rotation rate.
