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As is widely known, a non-Hermitian matrix exhibits distinct left and righteigenvectors, which form a bi-orthogonal system. Chalker and Mehling initiatedthe study of the joint statistics of the eigenvalues and the overlaps definedby the left and right eigenvectors of the Ginibre unitary ensemble. Later,Akemann et al. continued their investigation by studying the $k$-th correlationfunction 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 $k$-thcorrelation function weighted by the on- and off-diagonal overlaps of theinduced Ginibre unitary ensemble. Similar to the Ginibre unitary ensemble case,we will demonstrate the determinantal structure. As a result, we will confirmthe universality of the $k$-th correlation function weighted by the on- andoff-diagonal overlaps in both the bulk and edge scaling limits in the stronglynon-unitary regime. Furthermore, in the weakly non-unitary regime and at thesingular origin, we will report new relationships between the overlap and suchspectral regimes.

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