  We analyze the Illustris-1 hydrodynamical cosmological simulation to explore
the stellar velocity dispersion of quiescent galaxies as an observational probe
of dark matter halo velocity dispersion and mass. Stellar velocity dispersion
is proportional to dark matter halo velocity dispersion for both central and
satellite galaxies. The dark matter halos of central galaxies are in virial
equilibrium and thus the stellar velocity dispersion is also proportional to
dark matter halo mass. This proportionality holds even when a line-of-sight
aperture dispersion is calculated in analogy to observations. In contrast, at a
given stellar velocity dispersion, the dark matter halo mass of satellite
galaxies is smaller than virial equilibrium expectations. This deviation from
virial equilibrium probably results from tidal stripping of the outer dark
matter halo. Stellar velocity dispersion appears insensitive to tidal effects
and thus reflects the correlation between stellar velocity dispersion and dark
matter halo mass prior to infall. There is a tight relation (<0.2 dex scatter)
between line-of-sight aperture stellar velocity dispersion and dark matter halo
mass suggesting that the dark matter halo mass may be estimated from the
measured stellar velocity dispersion for both central and satellite galaxies.
We evaluate the impact of treating all objects as central galaxies if the
relation we derive is applied to a statistical ensemble. A large fraction
(>2/3) of massive quiescent galaxies are central galaxies and systematic
uncertainty in the inferred dark matter halo mass is <0.1 dex thus simplifying
application of the simulation results to currently available observations.
