  Discontinuity gravity modes may arise in perturbed quark-hadron hybrid stars
when a sharp density jump exists in the stellar interior and are a potential
fingerprint to infer the existence of quark matter cores in compact objects.
When a hybrid star is perturbed, conversion reactions may occur at the
quark-hadron interface and may have a key role in global stellar properties
such as the dynamic stability and the quasi-normal mode spectrum. In this work
we study the role of the conversion rate at the interface. To this end, we
first derive the junction conditions that hold at the sharp interface of a
non-radially perturbed hybrid star in the case of slow and rapid conversions.
Then, we analyse the discontinuity $g$-mode in both cases. For rapid
conversions, the discontinuity $g$-mode has zero frequency because a displaced
fluid element near the phase splitting surface adjusts almost immediately its
composition to its surroundings and gravity cannot provide a buoyancy force.
For slow conversions, a $g$-mode exists and its properties are analysed here
using modern hadronic and quark equations of state. Moreover, it has been shown
recently that in the case of slow conversions an extended branch of stable
hybrid configurations arises for which $\partial M/ \partial \epsilon_c <0$. We
show that $g$-modes of the standard branch (that is, the one with $\partial M/
\partial \epsilon_c > 0$) have frequencies and damping times in agreement with
previous results in the literature. However, $g$-modes of the extended branch
have significantly larger frequencies (in the range $1-2 \, \mathrm{kHz}$) and
much shorter damping times (few seconds in some cases). We discuss the
detectability of $g$-mode GWs with present and planned GW observatories.
