  We present the results from a VLT/SINFONI and Keck/NIRSPEC near-infrared
spectroscopic survey of 16 Lyman-alpha emitters (LAEs) at $z$ = 2.1 - 2.5 in
the COSMOS and GOODS-N fields discovered from the HETDEX Pilot Survey. We
detect rest-frame optical nebular lines (H$\alpha$ and/or [OIII]$\lambda$5007)
for 10 of the LAEs and measure physical properties, including the star
formation rate (SFR), gas-phase metallicity, gas-mass fraction, and Ly$\alpha$
velocity offset. We find that LAEs may lie below the mass-metallicity relation
for continuum-selected star-forming galaxies at the same redshift. The LAEs all
show velocity shifts of Ly$\alpha$ relative to the systemic redshift ranging
between +85 and +296 km s$^{-1}$ with a mean of +180 km s$^{-1}$. This value is
smaller than measured for continuum-selected star-forming galaxies at similar
redshifts. The Ly$\alpha$ velocity offsets show a moderate correlation with the
measured star formation rate (2.5$\sigma$), but no significant correlations are
seen with the SFR surface density, specific SFR, stellar mass, or dynamical
mass ($\lesssim$ 1.5$\sigma$). Exploring the role of dust, kinematics of the
interstellar medium (ISM), and geometry on the escape of Ly$\alpha$ photons, we
find no signature of selective quenching of resonantly scattered Ly$\alpha$
photons. However, we also find no evidence that a clumpy ISM is enhancing the
Ly$\alpha$ equivalent width. Our results suggest that the low metallicity in
LAEs may be responsible for yielding an environment with a low neutral hydrogen
column density as well as less dust, easing the escape of Ly$\alpha$ photons
over that in continuum-selected star-forming galaxies.
