We study the integrable structure and scaling limits of the conditioned eigenvector overlap of the symplectic Ginibre ensemble of Gaussian non-Hermitian random matrices with independent quaternion elements. The average of the overlap matrix elements constructed from left and right eigenvectors, conditioned to x, are derived in terms of a Pfaffian determinant. Regarded as a two-dimensional Coulomb gas with the Neumann boundary condition along the real axis, it contains a kernel of skew-orthogonal polynomials with respect to the weight function ^ (over) (z) =|z-x|² (1+|z-x|²) e^-2|z|², including a non-trivial insertion of a point charge. The mean off-diagonal overlap is related to the diagonal (self-) overlap by a transposition, in analogy to the complex Ginibre ensemble. For x conditioned to the real line, extending previous results at x=0, we determine the skew-orthogonal polynomials and their skew-kernel with respect to ^ (over) (z). This is done in two steps and involves a Christoffel perturbation of the weight ^ (over) (z) =|z-x|²^ (pre) (z), by computing first the corresponding quantities for the unperturbed weight ^ (pre) (z). Its kernel is shown to satisfy a differential equation at finite matrix size N. This allows us to take different large-N limits, where we distinguish bulk and edge regime along the real axis. The limiting mean diagonal overlaps and corresponding eigenvalue correlation functions of the point processes with respect to ^ (over) (z) are determined. We also examine the effect on the planar orthogonal polynomials when changing the variance in ^ (pre) (z), as this appears in the eigenvector statistics of the complex Ginibre ensemble.
Akemann et al. (Thu,) studied this question.