Haar measure on the groups SO(n) and SO(n,m) Would you please give me some information about Haar measure on special orthogonal group SO(n) and indefinite special orthogonal group SO(n,m)?
Thank you so much!
 A: The Haar measure on $SO(n)$ can be described inductively as follows.
Let $\{e_1,\ldots,e_n\}$ be an orthonormal basis for $\mathbb{R}^n$, and let $H$ be the stabsubgroup of $SO(n)$ that fixes $e_n$.  Note that $H$ is isomorphic to $SO(n-1)$, so it has a Haar measure, which we already know.
Now, if $g_1,g_2\in SO(n)$, then $g_1H = g_2H$ if and only if $g_1(e_n)=g_2(e_n)$.  Thus, the coset space $SO(n)/H$ is isomorphic to the sphere $S^{n-1}$.  This sphere also has a nice, well-known measure.
Now, let $f\colon S^{n-1}\to SO(n)$ be any measurable function that picks a transversal for each coset.  That is, for any point $p$ in the sphere, $f(p)$ should be an element of $SO(n)$ that maps $e_n$ to $p$.  This transversal need not be continuous---only measurable.  Then the measure of a measurable subset $X\subseteq SO(n)$ is given by the formula
$$
\mu_{SO(n)}(X) \;=\; \int_{S^{n-1}} \mu_{H}\bigl(f(p)^{-1}(X\cap f(p)H)\bigr)\; d\mu_{S^{n-1}}(p)
$$
where $\mu_H$ denotes the Haar measure on $H$, and $\mu_{S^{n-1}}$ denotes the standard measure on $S^{n-1}$.
Is that helpful?  There is (presumably) an alternative description involving the Jacobian of the exponential map $\mathfrak{so}(n) \to SO(n)$, if that would be better.
I don't know much about the Haar measure for $SO(n,m)$ off the top of my head, but there should be an inductive description for it similar to the description above.
