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Strong gravitational lensing by a rotating non-Kerr compact object

We study the strong gravitational lensing in the background of a rotating non-Kerr compact object with a deformed parameter $ε$ and an unbound rotation parameter $a$. We find that the photon sphere radius and the deflection angle depend sharply on the parameters $ε$ and $a$. For the case in which the black hole is more prolate than a Kerr black hole, the photon sphere exists only in the regime $ε\leqε_{max}$ for prograde photon. The upper limit $ε_{max}$ is a function of the rotation parameter $a$. As $ε>ε_{max}$, the deflection angle of the light ray closing very to the naked singularity is a positive finite value, which is different from those in both the usual Kerr black hole spacetime and in the rotating naked singularity described by Janis-Newman-Winicour metric. For the oblate black hole and the retrograde photon, there does not exist such a threshold value. Modelling the supermassive central object of the Galaxy as a rotating non-Kerr compact object, we estimated the numerical values of the coefficients and observables for gravitational lensing in the strong field limit.