  We present the first spectroscopic measurements of the [OIII] 5007 A line in
two z ~ 3.1 Lyman-alpha emitting galaxies (LAEs) using the new near-infrared
instrument LUCIFER1 on the 8.4m Large Binocular Telescope (LBT). We also
describe the optical imaging and spectroscopic observations used to identify
these Lya emitting galaxies. Using the [OIII] line we have measured accurate
systemic redshifts for these two galaxies, and discovered a velocity offset
between the [OIII] and Ly-alpha lines in both, with the Lya line peaking 342
and 125 km/s redward of the systemic velocity. These velocity offsets imply
that there are powerful outflows in high-redshift LAEs. They also ease the
transmission of Lya photons through the interstellar medium and intergalactic
medium around the galaxies. By measuring these offsets directly, we can refine
both Lya-based tests for reionization, and Lya luminosity function measurements
where the Lya forest affects the blue wing of the line. Our work also provides
the first direct constraints on the strength of the [OIII] line in
high-redshift LAEs. We find [OIII] fluxes of 7 and 36 x 10^-17 erg s^-1 cm^-2
in two z ~ 3.1 LAEs. These lines are strong enough to dominate broad-band flux
measurements that include the line (in thiscase, K_s band photometry). Spectral
energy distribution fits that do not account for the lines would therefore
overestimate the 4000 A (and/or Balmer) break strength in such galaxies, and
hence also the ages and stellar masses of such high-z galaxies.
