  We study the relaxation dynamics of driven, two-dimensional semiconductors,
where itinerant electrons dress optically pumped excitons to form two
Fermi-polaron branches. Repulsive polarons excited around zero momentum quickly
decay to the attractive branch at high momentum. Collisions with electrons
subsequently lead to a slower relaxation of attractive polarons, which
accumulate at the edge of the light-cone around zero momentum where the
radiative loss dominates. The bosonic nature of exciton polarons enables
stimulated scattering, which results in a lasing transition at higher pump
power. The latter is characterized by a superlinear increase of light emission
as well as extended spatiotemporal coherence. As the coherent peak is at the
edge of the light-cone and not at the center, the many-body dressing of
excitons can reduce the linewidth below the limit set by the exciton
nonradiative lifetime.
