TITLE: Integral over space of matrices
QUESTION [7 upvotes]: Let $\Omega_n=\{Y\in GL_n(\mathbb{R}):Y^t=Y>0\}$ be the set of symmetric positive definite real square matrices of size $n$, and let $dY=\prod_{1\le i\le j\le n}dY_{ij}$. Let $F:\Omega_n\to\mathbb{R}$ be any function such that $F(BAB^t)=F(A)$ for any orthogonal $B\in O_n(\mathbb{R})$, of which I have to compute the integral
$$I:=\int_{\Omega_n}F(Y)dY$$
(assuming such integral makes sense) and I want to make a change of variables.
Let $\Gamma_n=\{diag(\lambda_1,\dots,\lambda_n):0<\lambda_1<\dots<\lambda_n\}$. Most of the $Y\in\Omega_n$ (exactly those with distinct eigenvalues) can be diagonalized in a unique way to a $\gamma\in\Gamma$.

Is it true that the subset of $\Omega_n$ consisting of matrices with at least two coincident eigenvalues has measure $0$ (with respect to $dY$)? I think so, since the condition forces such subset -informally speaking- to be of lower dimension than $\Omega_n$. If that's the case, what is the correct formal argument?

Provided this is the case, let $\Omega_n^+$ be the subset of matrices with distinct eigenvalues, so that we can integrate over $\Omega_n^+$ rather than $\Omega_n$. Every $Y\in\Omega_n^+$ can be diagonalized as $Y=U\gamma U^t$ for a unique $\gamma\in\Gamma_n$ and some orthogonal $U\in O_n(\mathbb{R})$. Further, $U$ is unique up to right multiplication by a matrix of the form $diag(\pm 1,\dots,\pm 1)$ (of which we have exactly $2^n$ choices).

By changing variables, I ideally want to express my original integral $I$ (using the hypothesis $F(BAB^t)=F(A)$)as
  $$I=\int_{\Omega_n^+}F(Y)dY=\int_{O_n(\mathbb{R})/diag(\pm 1,\dots,\pm 1)}\int_{\Gamma_n}F(\gamma)|J(\gamma,U)|dU d\gamma$$
  where $J(\gamma,U)$ is the Jacobian determinant. What are the correct measures $dU$ and $d\gamma$? I think that $d\gamma=d\lambda_1\cdots d\lambda_n$, while I have no idea about $dU$. I have read on a paper that $J(\gamma,U)$ turns out to depend only on $\gamma$ (no explanation why), in which case I can further simplify 
  $$I=\int_{O_n(\mathbb{R})/diag(\pm 1,\dots,\pm 1)}dU \cdot \int_{\Gamma_n}F(\gamma)|J(\gamma)|d\gamma$$
  Is this correct? In this case, is the first integral convergent?

REPLY [2 votes]: There is indeed an integral formula of this type, see this Mathoverflow question as well as Theorem 5.8 on page 186 in 
S.Helgason "Groups and geometric analysis:  Integral geometry  invariant differential operators and spherical functions", Academic Press 1984.
As for your 1st question, yes, indeed, the measure is zero since this is a proper algebraic subvariety in the space of symmetric matrices, see here.