matrix ensembles by means of Riemannian symmetric spaces were summarized by the excellent review article of Caselle and Magnea [@CM].

0.3cm In this paper we provide a generalization of the random matrix ensembles, including all classical ensembles, and then give an unified way to derive the joint density function for the eigenvalue distribution by one simple formula. The proof of this formula make no use of integration formula. In fact, the corresponding integration formula can be derived from this formula as corollary. We also give a classification scheme of the generalized random matrix ensembles, which will include all classical ensembles and some new ensembles which were not considered before.

0.3cm More precisely, Let $\sigma:G\times X\rightarrow X$ be a smooth action of a Lie group $G$ on a Riemannian manifold $X$, preserving the induced Riemannian measure $dx$. Let $p(x)$ be a $G$-invariant smooth function on $X$, and consider the measure $p(x)dx$ on $X$, which is not necessary a finite measure. We choose a closed submanifold $Y$ of $X$ consisting of representation points for almost all $G$-orbits in $X$. The Riemannian structure on $X$ induces a Riemannian measure $dy$ on $Y$. Let $K$ be the closed subgroup of $G$ which fixes all 