spaces ${\mathfrak{p}}$ and $P$ are invariant under the adjoint action ${\mathrm{A}}={\mathrm{Ad}}|_K$ and the conjugate action $\sigma$ of $K$, respectively. Let ${\mathfrak{a}}$ be a maximal abelian subspace of ${\mathfrak{p}}$, and let $A$ be the connected subgroup of $G$ with Lie algebra ${\mathfrak{a}}$. Let $M=\{k\in
K:{\mathrm{A}}_k(\eta)=\eta, \forall \eta\in{\mathfrak{a}}\}=\{k\in K:\sigma_k(a)=a,
\forall a\in A\}$. It can be shown that there are Riemannian structures on ${\mathfrak{p}}$ and $P$ inducing $K$-invariant Riemannian measures $dX$ on ${\mathfrak{p}}$ and $dx$ on $P$. They also induce Riemannian measures $dY$ on ${\mathfrak{a}}$ and $da$ on $A$. There is also a $K$-invariant smooth measure $d\mu$ on $K/M$. Let $p_1(\xi)$ and $p_2(x)$ be $K$-invariant positive smooth functions on ${\mathfrak{p}}$ and $P$, then it can be proved that the systems $(K,{\mathrm{A}},{\mathfrak{p}},p_1(\xi)dX(\xi),{\mathfrak{a}},dY)$ and $(K,\sigma,P,p_2(x)dx,A,da)$ are generalized random matrix ensembles, which we called *linear ensemble* and *nonlinear noncompact ensemble*, respectively. It can be shown that the Gaussian ensemble and the chiral ensemble are particular examples of linear ensemble, and the new transfer matrix ensembles are particular examples of nonlinear noncompact ensemble.

0.3cm Next we define the compact ensemble. Let $G$ be a connected compact Lie group $G$ with Lie algebra ${\mathfrak{g}}$. Suppose $\Theta$ is a global involutive of $G$ with the induced involution $\theta=d\Theta$ of 