Determinantal structure of the conditional expectation of the overlaps for the induced Ginibre unitary ensemble

As is widely known, a non-Hermitian matrix exhibits distinct left and right eigenvectors, which form a bi-orthogonal system. Chalker and Mehling initiated the study of the joint statistics of the eigenvalues and the overlaps defined by the left and right eigenvectors of the Ginibre unitary ensemble. Later, Akemann et al. continued their investigation by studying the $k$-th correlation function weighted by the on- and off-overlaps of the Ginibre unitary ensemble. In this paper, as a natural extension of their work, we investigate the $k$-th correlation function weighted by the on- and off-diagonal overlaps of the induced Ginibre unitary ensemble. Similar to the Ginibre unitary ensemble case, we will demonstrate the determinantal structure. As a result, we will confirm the universality of the $k$-th correlation function weighted by the on- and off-diagonal overlaps in both the bulk and edge scaling limits in the strongly non-unitary regime. Furthermore, in the weakly non-unitary regime and at the singular origin, we will report new relationships between the overlap and such spectral regimes.