  Within the Floquet theory of periodically driven quantum systems, we
developed the theory of light-induced modification of electronic states in
semiconductor materials described by the Luttinger Hamiltonian (the electronic
term $\Gamma_8$). Particularly, exact solutions of the Floquet problem are
found for the band edge in the cases of linearly and circularly polarized
irradiation. It is shown that the irradiation changes electron effective masses
near the band edge, induces anisotropy of the electron dispersion and splits
the bands. It is demonstrated that the light-induced band splitting strongly
depends on the light polarization. Namely, the circularly polarized light acts
similarly to a stationary magnetic field and lifts the spin degeneracy of
electron branches, whereas a linearly polarized light does not affect the spin
degeneracy and only splits the bands in the center of the Brillouin zone. The
present theory can be applied to describe electronic properties of various
semiconductor structures irradiated by an electromagnetic field in the broad
frequency range.
