  The coupling between a 2D semiconductor quantum well and an optical cavity
gives rise to combined light-matter excitations, the exciton-polaritons. These
were usually measured when the conduction band is empty, making the single
polariton physics a simple single-body problem. The situation is dramatically
different in the presence of a finite conduction band population, where the
creation or annihilation of a single exciton involves a many-body shakeup of
the Fermi sea. Recent experiments in this regime revealed a strong modification
of the exciton-polariton spectrum. Previous theoretical studies concerned with
nonzero Fermi energy mostly relied on the approximation of an immobile valence
band hole with infinite mass, which is appropriate for low-mobility samples
only; for high-mobility samples, one needs to consider a mobile hole with large
but finite mass. To bridge this gap we present an analytical diagrammatic
approach and tackle a model with short-ranged (screened) electron-hole
interaction, studying it in two complementary regimes. We find that the finite
hole mass has opposite effects on the exciton-polariton spectra in the two
regimes: In the first, where the Fermi energy is much smaller than the exciton
binding energy, excitonic features are enhanced by the finite mass. In the
second regime, where the Fermi energy is much larger than the exciton binding
energy, finite mass effects cut off the excitonic features in the polariton
spectra, in qualitative agreement with recent experiments.
