  We present deep Keck/LRIS spectroscopy and HST/WFC3 imaging in the rest-frame
optical for a sample of eight galaxies at z~1.5 with high
photometrically-determined stellar masses. The data are combined with
VLT/XShooter spectra of five galaxies from van de Sande et al. (2011, 2012 to
be submitted). We find that these thirteen galaxies have high velocity
dispersions, with a median of sigma=301 km s^{-1}. This high value is
consistent with their relatively high stellar masses and compact sizes. We
study their stellar populations using the strength of Balmer absorption lines,
which are not sensitive to dust absorption. We find a large range in Balmer
absorption strength, with many galaxies showing very strong lines indicating
young ages. The median Hdelta_A equivalent width, determined directly or
inferred from the H10 line, is 5.4 Angstroms, indicating a luminosity-weighted
age of ~1 Gyr. Although this value may be biased towards higher values because
of selection effects,high-dispersion galaxies with such young ages are
extremely rare in the local Universe. Interestingly we do not find a simple
correlation with rest-frame U-V color: some of the reddest galaxies have very
strong Balmer absorption lines. These results demonstrate that many
high-dispersion galaxies at z~1.5 were quenched recently. This implies that
there must be a population of star-forming progenitors at z~2 with high
velocity dispersions or linewidths, which are notoriously absent from CO/Halpha
selected surveys.
