${\mathfrak{g}}$. Let $K=\{g\in G:\Theta(g)=g\}$, and let ${\mathfrak{p}}$ be the eigenspace of $\theta$ corresponding the eigenvalue $-1$. Let $P=\exp({\mathfrak{p}})$, then $P$ is invariant under the conjugate action $\sigma$ of $K$. It was proved in [@AW] that $P$ is a closed submanifold of $G$ satisfies $T_eP={\mathfrak{p}}$, which is just the identity component of the set $\{g\in G:\Theta(g)=g^{-1}\}$. Let ${\mathfrak{a}}$ be a maximal abelian subspace of ${\mathfrak{p}}$, and let $A$ be the torus with Lie algebra ${\mathfrak{a}}$. There is a Riemannian structure on $P$ induces a $K$-invariant Riemannian measure $dx$ on $P$ and a Riemannian measure $da$ on $A$. Let $M=\{k\in K:\sigma_k(a)=a,
\forall a\in A\}$, then there is a $K$-invariant smooth measure $d\mu$ on $K/M$. Let $p(x)$ be a $K$-invariant positive smooth function on $P$, then it can be proved that the system $(K,\sigma,P,p(x)dx,A,da)$ is a generalized random matrix ensemble, which we call it *compact ensemble*. It can be shown that the circular ensemble and the Jacobi ensembles are particular examples of compact ensemble.

0.3cm Let $G$ be an unimodular Lie group $G$ with Lie algebra ${\mathfrak{g}}$. Then there are Riemannian structures on $G$ and ${\mathfrak{g}}$ inducing a $\sigma$-invariant Riemannian measure $dg$ on $G$ and an ${\mathrm{Ad}}$-invariant Riemannian measure $dX$ on ${\mathfrak{g}}$, where 