an open problem.

0.5cm

Generalized random matrix ensembles
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0.5cm Now we are prepared to establish the generalized random matrix ensembles.

0.3cm Let $G$ be a Lie group which acts on an $n$-dimensional smooth manifold $X$ by $\sigma: G\times
X\rightarrow X$. For the convenience, we suppose $X$ is a Riemannian manifold. Suppose the induced Riemannian measure $dx$ is $G$-invariant (note that we do not require the Riemannian structure on $X$ to be $G$-invarinant). Let $Y$ be a closed submanifold of $X$ which is endowed the induced Riemannian measure $dy$, and let $K=\{g\in G:\sigma_g(y)=y, \forall y\in Y\}$. As in §2, we form the map $\varphi: G/K\times Y\rightarrow X$ by $\varphi([g],y)=\sigma_g(y)$. Let $X_{\mathrm{z}}\subset X$, $Y_{\mathrm{z}}\subset
Y$ be closed zero measure subsets of $X$ and $Y$, respectively. Denote $X' = X \setminus X_{\mathrm{z}}$, $Y' = Y \setminus Y_{\mathrm{z}}$. We suppose the conditions (a), (b), (c), and (d) of §2 hold. For the reader’s convenience, we list them below.

0.3cm (*invariance condition*) $X' = {\displaystyle}\bigcup_{y\in Y'} O_y$.

0.3cm (*transversality condition*) $T_y X=T_y O_y\oplus T_y Y$, $\forall y\in
Y'$.

0.3cm (*dimension condition*) $\mathrm{dim}G_y=\mathrm{dim}K, \quad \forall
y\in Y'$.

0.3cm (*orthogonality condition*) $T_yY\perp T_yO_y$, $\forall y\in Y'$.

0.3cm [Suppose]{} $d\mu$ is a $G$-invariant smooth measure on $G/K$, 