  We employ a bias-corrected abundance matching technique to investigate the
coevolution of the LCDM dark halo mass function (HMF), the observationally
derived velocity dispersion and stellar mass functions (VDF, SMF) of galaxies
between z=1 and 0. We use for the first time the evolution of the VDF
constrained through strong lensing statistics by Chae (2010) for galaxy-halo
abundance matching studies. As a local benchmark we use a couple of z ~ 0 VDFs
(a Monte-Carlo realised VDF based on SDSS DR5 and a directly measured VDF based
on SDSS DR6). We then focus on connecting the VDF evolution to the HMF
evolution predicted by N-body simulations and the SMF evolution constrained by
galaxy surveys. On the VDF-HMF connection, we find that the local dark halo
virial mass-central stellar velocity dispersion (Mvir-sigma) relation is in
good agreement with the individual properties of well-studied low-redshift dark
haloes, and the VDF evolution closely parallels the HMF evolution meaning
little evolution in the Mvir-sigma relation. On the VDF-SMF connection, it is
also likely that the stellar mass-stellar velocity dispersion (Mstar-sigma)
relation evolves little taking the abundance matching results together with
other independent observational results and hydrodynamic simulation results.
Our results support the simple picture that as the halo grows hierarchically,
the stellar mass and the central stellar velocity dispersion grow in parallel.
We discuss possible implications of this parallel coevolution for galaxy
formation and evolution under the LCDM paradigm.
