We theoretically study high-order optomechanically-induced transparency (OMIT) process in a nonlinear Kerr resonator. A frequency shift induced by the Kerr effect, is identified for the optical cavity mode, which results in asymmetric OMIT windows of the signal and its higher-order sidebands. We also find that both the sideband amplitude and its associated group delay sensitively depend on the strength of the Kerr nonlinearity. This indicates the possibility to enhance or steer the performance of OMIT devices with various nonlinear optical cavities. [5][6][7][8]. A recent advance closely related to the present study is optomechanically induced transparency (OMIT) [9][10][11]. As a solid-state analogy to electromagnetically induced transparency (EIT) originally observed in atomic gases [12], the fundamental OMIT features the two-channel destructive interference of the absorptions of the probe photons (by the cavity itself or the mechanical mode). Beyond this picture, high-order OMIT effects also emerge due to the intrinsic nonlinear OM interactions [13][14][15] [26]. In contrast, high-order OMIT sidebands are generally much weaker than the probe signal and thus hard to be detected or utilized. Hence the sideband enhancement becomes important for its potential applications in e.g., precise sensing of charges [27,28] [42][43][44], and nonlinear OM control [45]. In a recent experiment with a high-Q silica microsphere, asymmetric Fano-like OMIT spectrum was observed due to the optical * Electronic address: jinghui73@gmail.com Kerr effect [46], which can be further tuned or compensated by varying the pump power and the optical frequency.In this paper, based on the OMIT experiment in a Kerr cavity [46], we proceed to study high-order OMIT and its associated group delay in such a nonlinear cavity. We find that in the presence of optical Kerr effect, compared to that in a linear resonator, the amplitude of second-order sideband can be significantly enhanced. This sideband amplitude also can be further tuned by the external light, since the Kerr-induced shift can be either compensated or amplified by varying the pump frequency. Moreover, the delay time of second-order sideband sensitively depends on the Kerr nonlinearity. At high pump power, the sideband can be tuned from fast light to slow light, which is potentially useful for optical storage or switch. The enhanced nonlinear OMIT in a Kerr resonator, as revealed here, opens up a promising new way to study other important OM effects, e.g., motion cooling [47] or squeezing [48], lightsound entanglement [49], and photon blockade [50][51][52].As shown in Fig. 1(a), we consider the nonlinear OMIT in a Kerr resonator. A pump laser of frequency ω l and a probe laser of frequency ω p are applied to the system via the evanescent coupling of the optical fiber and the resonator, the field amplitudes are given bywhere P L and P s are the pump and probe powers, respectively. In the rotating frame at the pump frequency ω l , the Hamiltonian of this OM system is given by (for = 1): ...