  The presence of extreme horizontal branch (EHB) and blue hook stars in some
Galactic globular clusters (GGCs) constitutes one of the remaining mysteries of
stellar evolution. In this paper, we focus on $\omega$ Cen, a peculiar, massive
GGC that hosts multiple stellar populations. We use non-LTE model atmospheres
to derive atmospheric parameters (Teff, log g and N(He)/N(H)) and spectroscopic
masses for 152 EHB stars in the cluster. This constitutes the largest
spectroscopic sample of EHB stars ever analyzed in a GGC and represents ~20% of
the EHB population of $\omega$ Cen. We also search for close binaries among
these stars based on radial velocity variations. Our results show that the EHB
population of $\omega$ Cen is divided into three spectroscopic groups that are
very distinct in the Teff - helium abundance plane. The majority of our sample
consists of sdOB stars that have roughly solar or super-solar atmospheric
helium abundances. It is these objects that constitute the blue hook at $V >$
18.5 mag in the $\omega$ Cen color-magnitude diagram. Interestingly, the
helium-enriched sdOBs do not have a significant counterpart population in the
Galactic field. Another major difference between the EHB stars in $\omega$ Cen
and the field is the fraction of close binaries. From our radial velocity
survey we identify two binary candidates and estimate an EHB close binary
fraction of ~5% in $\omega$ Cen. This low fraction is in line with findings for
other GGCs, but in sharp contrast to the situation in the field, where around
50% of the sdB stars reside in close binaries. Finally, the mass distribution
derived is very similar for all three spectroscopic groups, however the average
mass (0.38 Msun) is lower than that expected from stellar evolution theory.
While this mass conundrum has previously been noted for EHB stars in $\omega$
Cen, it so far appears to be unique to that cluster.
