Derivation of an eigenvalue probability density function relating to the Poincaré disk

A result of Zyczkowski and Sommers [J. Phys. A 33, 2045–2057 (2000)] gives the eigenvalue probability density function for the top N ×N sub-block of a Haar distributed matrix from U(N + n). In the case n ≥ N , we rederive this result, starting from knowledge of the distribution of the sub-blocks, introducing the Schur decomposition, and integrating over all variables except the eigenvalues. The integration is done by identifying a recursive structure which reduces the dimension. This approach is inspired by an analogous approach which has been recently applied to determine the eigenvalue probability density function for random matrices AB, where A and B are random matrices with entries standard complex normals. We relate the eigenvalue distribution of the sub-blocks to a many body quantum state, and to the one-component plasma, on the pseudosphere.