LAUSR

We explore the optomechanically induced transparency (OMIT) in a parity-time-symmetric ($\mathcal{PT}$-symmetric) optomechanical system (OMS) in which the mechanical oscillator is coupled to another one via the Coulomb interaction. Calculating the transmission rate of the probe field, we find that double optomechanically induced absorption (OMIA) can be observed by modulating the Coulomb coupling strength with zero gain rate of the active cavity. The absorption window for the system in the $\mathcal{PT}$-symmetry-broken phase is deeper than that in other phases due to the strong field localization. When the gain rate is not zero, the output field of the system exhibits double inverted OMIT. Both splits of the double OMIA and double OMIT depend linearly on the strength of the Coulomb interaction. The inverted OMIT can also be changed into an absorption window and the system can achieve perfect absorption by modulating the Coulomb coupling strength. The transmission light at the double inverted OMIT is always fast light and the transmission light at $\ensuremath{\delta}={\ensuremath{\omega}}_{m}$ is always of slow light, independent of the phase the system in (the $\mathcal{PT}$-symmetry phase or the $\mathcal{PT}$-symmetry-broken phase). The absolute value of the group delay at $\ensuremath{\delta}={\ensuremath{\omega}}_{m}$ can be manipulated by controlling the strength of the Coulomb interaction. These observations provide us with a tool to control light propagation which might have potential application in quantum optical devices and quantum information networks.