  We study a hybrid semiconductor-optomechanical system, which consists of a
cavity with an oscillating mirror made by semiconducting materials or with a
semiconducting membrane inside. The cavity photons and the excitons in the
oscillating mirror or semiconducting membrane form into polaritons. And
correspondingly, the optomechanical interaction between the cavity photons and
the mirror or membrane is changed into the polariton-mechanical interaction. We
theoretically study the eigenenergies and eigenfunctions of this tripartite
hybrid system with the generalized rotating-wave approximation. We show that
the emission spectrum of polariton mode is modulated by the mechanical
resonator. We also study the mechanical effect on the statistical properties of
the polariton when the cavity is driven by a weak classical field. This work
provides a detailed description of the rich nonlinearity owing to the
competition between parametric coupling and three-wave mixing interaction
concerning the polariton modes and the phonon mode. It also offers a way to
operate the photons, phonons and excitons, e.g., detect the properties of
mechanical resonator through the fine spectra of the polaritons or control the
transmission of light in the integrated semiconducting-optomechanical platform.
