  We investigated the stellar population properties of a sample of 1858 massive
compact galaxies (MCGs) extracted from the SDSS survey. Motivated by previous
results showing that older compact galaxies tend to have larger velocity
dispersion at fixed stellar mass, we used the distance to the
$\sigma_\mathrm{e}$ vs. $R_\mathrm{e}$ and $M_\star$ vs. $\sigma_\mathrm{e}$
relations as selection criteria. We found that MCGs are old ($\gtrsim 10$ Gyr),
$\alpha$-enhanced ([$\alpha/\mathrm{Fe}] \sim 0.2$) and have solar to
super-solar stellar metallicities. Metallicity increases with
$\sigma_\mathrm{e}$, while age and [$\alpha$/Fe] do not vary significantly.
Moreover, at fixed $\sigma_\mathrm{e}$, metallicity and stellar mass are
correlated. Compared to a control sample of typical quiescent galaxies, MCGs
have, on average, lower metallicities than control sample galaxies (CSGs) of
similar $\sigma_\mathrm{e}$. For $\sigma_\mathrm{e} \lesssim 225$ km/s, MCGs
are older and more $\alpha$-enhanced than CSGs, while for higher
$\sigma_\mathrm{e}$ ages and $\alpha$-enhancement are similar. The differences
in age and $\alpha$-enhancement can be explained by lower-$\sigma_\mathrm{e}$
CSGs being an amalgam of quiescent galaxies with a variety of ages. The origin
of the differences in metallicity, however, is not clear. Lastly, we compared
the stellar mass within the region probed by the SDSS fiber finding that, at
fixed fiber velocity dispersion, MCGs have lower stellar masses on average.
Since the velocity dispersion is a tracer of the dynamical mass, this raises
the possibility that MCGs have, on average, a bottom heavier initial mass
function or a larger dark matter fraction within the inner $\sim 1-2$ kpc.
