analysis methods by which the density functions for various types of ensembles were derived case by case. But a systematical method to compute the density functions was desired.

0.3cm The first achievement in this direction was made by Dyson [@Dy], who introduced an idea of expressing various kinds of circular ensemble in terms of symmetric spaces with invariant probability measures. From then on, guided by Dyson’s idea, many authors observed new random matrix ensembles in terms of Cartan’s classification of Riemannian symmetric spaces, and obtained the joint density functions for such ensembles using the integration formula on symmetric space (see, for example, [@AZ; @Ca1; @Ca2; @Du; @Iv; @TBFM; @Zi]). Here we mention the recent work of Dueñez [@Du] briefly. Dueñez explored the random matrix ensembles which correspond to infinite families of compact irreducible Riemannian symmetric spaces of type I, including circular orthogonal and symplectic ensembles and various kinds of Jacobi ensembles. Using an integration formula associated with the $KAK$ decomposition of compact groups, he obtained the induced measure on the space of eigenvalues associated with the underlying symmetric space, and then derived the eigenvalue distribution of the corresponding random matrix ensemble. These methods of deriving the eigenvalue distributions of random 