  We study the optical absorption spectrum of a doped two-dimensional
semiconductor in the spin-valley polarized limit. In this configuration, the
carriers in the Fermi sea are indistinguishable from one of the two carriers
forming the exciton. Most notably, this indistinguishability requires the
three-body trion state to have p-wave symmetry. To explore the consequences of
this, we evaluate the system's optical properties within a polaron description,
which can interpolate from the low density limit -- where the relevant
excitations are few-body bound states -- to higher density many-body states. In
the parameter regime where the trion is bound, we demonstrate that the spectrum
is characterized by an attractive quasiparticle branch, a repulsive branch, and
a many-body continuum, and we evaluate the doping dependence of the
corresponding energies and spectral weights. In particular, at low doping we
find that the oscillator strength of the attractive branch scales with the
square of the Fermi energy as a result of the trion's p-wave symmetry. Upon
increasing density, we find that both the repulsive and attractive branches
blueshift, and that the orbital character of the states associated with these
branches interchanges. We compare our results with previous investigations of
the scenario where the Fermi sea involves carriers distinguishable from those
in the exciton, for which the trion ground state is s-wave.
