  Star-planet tidal interactions may result in the excitation of inertial waves
in the convective region of stars. Their dissipation plays a prominent role in
the long-term orbital evolution of short-period planets. If the star is assumed
to be rotating as a solid-body, the waves' Doppler-shifted frequency is
restricted to $[-2 \Omega, 2 \Omega]$ ($\Omega$ being the angular velocity of
the star) and they can propagate in the entire convective region. However,
turbulent convection can sustain differential rotation with an equatorial
acceleration (as in the Sun) or deceleration that may modify waves propagation.
We thus explore the properties of inertial modes of oscillation in a conically
differentially rotating background flow whose angular velocity depends on the
latitudinal coordinate only, close to what is expected in the external
convective envelope of low-mass stars. We find that their frequency range is
broadened by differential rotation, and that they may propagate only in a
restricted part of the envelope. In some cases, inertial waves form shear
layers around short-period attractor cycles. In others, they exhibit a
remarkable behavior when a turning surface or a corotation layer exists in the
star. We discuss how all these cases can impact tidal dissipation in stars.
