formulae. In this section we give some preliminaries on integrations. The reader who has more interest in the eigenvalue distributions of the generalized random matrix ensembles may skip this section and go to §4 directly.

0.3cm The following proposition generalizes the change of variables formula for multiple integration.

\[P:jifenbianhuan\] Let $\varphi: M\rightarrow N$ be a smooth map between two $n$-dimensional smooth manifolds $M$ and $N$, $dy$ a smooth measure on $N$. If $\varphi$ is a local diffeomorphism and is a $d$-sheeted covering map, then for any $f\in C^\infty(N)$ with $f\geq0$ or with $f\in L^1(N,dy)$, we have $$\label{F:jifenbianhuan}
\int_N f(y) dy=\frac{1}{d}\int_M f(\varphi(x)) \varphi^*(dy).$$

It is a standard argument using partition of unity, the details is omitted here.

Formula seems like a formula which relates degree of a map and integration of volume forms on manifold. When $M, N$ are compact and oriented, then under the conditions of Proposition \[P:jifenbianhuan\], up to a “$\pm$" sign, formula says nothing but of this. But, in general, the integration of differential forms is not suitable for us. What we will need is a change of variables formula which should ignore the negative sign.

As in the previous section, Let $X$ be a $G$-space, where $X$ 