measure $d\nu$ and the function $\mathcal{P}(y)$ on $Y$ are called *generalized eigenvalue distribution* and *generalized joint density function*, respectively. Using Formula , one can derive the joint density function for Gaussian ensemble, chiral ensemble, new transfer matrix ensembles, circular ensemble, Jacobi ensemble, and some other new generalized ensembles. The precise deriving process will be the content of a sequel paper [@AWY]. Here we should point out that the proof of Formula is not difficult, but this formula is very effective and available. The derivation of all concrete examples in [@AWY], including all classical random matrix ensembles, will be based on it.

0.3cm Once the eigenvalue distribution $d\nu$ is derived by Formula , under a covering condition, we can get the associated integration formula. The Weyl integration formula for compact Lie groups, the Harish-Chandra’s integration formula for complex semisimple Lie groups and real reductive groups, the integration formulae on Riemannian symmetric spaces of noncompact and compact types which were appeared in [@He], as well as their Lie algebra versions are all of particular cases of it (see [@AWY]).

0.3cm Now let us give a sketch of each section of this paper. In §2 we will develop some geometrical preliminaries on the 