Integration on the space of symmetric matrices

Question

Let $\mu$ be a Lebesgue measure on the space $G$ of real symmetric $n \times n$ matrices (the Haar measure on the additive group of such matrices). For any $A \in G$ let $\chi_{A}(x)$ be its characteristic polynomial:

$$\chi_{A}(-x) = \det (A+xI) = x^n + a_{1}x^{n-1}+\ldots+a_{n-1}x + a_{n}.$$

Let $g(A) = ( a_1, \ldots, a_n )$ for any $A \in G$.

Consider a function $f \colon \mathbb{R}^n \to \mathbb{R}$ and define an integral

$$I =\int\limits_{H} f(g(A)) \, \mu(dA),$$

where $H$ is some subset of $G$ well characterized by $a_1,\dots,a_n$, for example the set of all positively defined matrices.

My question is: how to reduce the integration with respect to $\mu$ to the integration with respect to the Lebesgue measure on the space of eigenvalues? I did't find an easy way. Maybe I have to use some generalized version of coarea formula to split the integration on the integration with respect to the Haar measure on $O(n)$ plus the integration on the space of eigenvalues?

Answer 1

There is a Weyl integration formula that deals with this problem. $\newcommand{\bR}{\mathbb{R}}$ Denote by $G_n$ the space of symmetric $n\times n$ real matrices. It states that for any $O(n)$-invariant $h:G_n\to \bR$ we have

$$ (2\pi)^{-\frac{n(n+1)}{4}} \int_{G_n} h(A) e^{-\frac{{\rm tr} A^2}{4}} dA $$

$$ =\frac{1}{Z_n}\int_{\mathbb{R}^n} h(\lambda_1,\dotsc, \lambda_n) e^{-\frac{1}{4} \sum_{j=1}^n \lambda_j^2} \; {\prod_{j<k}} |\lambda_j-\lambda_k| \; d\lambda_1\cdots d\lambda_n$$

where

$$Z_n =2^{\frac{n}{2}}n! \prod_{j=1}^n \Gamma(\frac{j}{2}).$$

Now, use this formula for

$$ h(A)= f(g(A))\, e^{\frac{{\rm tr} A^2}{4}}. $$

For more details see Appendix C of this paper and the references therein.

Answer 2

I think you are looking for the Haar measure of $SL(n,\mathbb{R})$ in $K A K$ coordinates. You can find it for example in Knapp's book "Representation Theory of Semisimple Groups: An Overview Based on Examples".

The short answer: the measure you want is the Haar measure on $O(n) \times O(n)$ times Haar measure on the space of diagonal matrices (viewed as a group under multiplication) times the product of the hyperbolic sines of the positive roots.