  Interaction of excitons with resident charge carriers in semiconductors gives
rise to bound three-particle complexes, trions, whose optical response is
conveniently described in the framework of many-body correlated Fermi polaron
states. These states are formed as a result of correlation of photocreated
trion with the Fermi sea hole and possess the angular momentum component of
$\pm 1$ depending on the helicity of the photon. We study theoretically the
energy spectrum fine structure of Fermi polarons in two-dimensional
semiconductors based on transition metal dichalcogenides. We demonstrate both
by the symmetry analysis and microscopic calculation that the Fermi polarons
with nonzero in-plane wavevector $\bf k$ are split, similarly to the neutral
exciton states, into the linearly polarized longitudinal and transverse, with
respect to the $\bf k$, states. The origin of this longitudinal-transverse
splitting is the long-range electron-hole exchange interaction that can be also
described as the interaction of Fermi polarons with their induced
electromagnetic field. The effective Hamiltonian describing the Fermi polaron
fine structure is derived, and its parameters are determined from the
microscopic model.
