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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!

Farzaneh
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  • What sort of information are you looking for? Do you want to integrate functions on these groups, or do you need to understand these measures for some theoretical purpose? – Jim Belk Nov 21 '13 at 19:53
  • I need to understand these measures for proof of admissibility condition for n-dimensional wavelet. – Farzaneh Nov 23 '13 at 07:41

1 Answers1

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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.

Jim Belk
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  • Note: I give an explicit description of the Haar measure on $SO(n)$ using hyperspherical coordinates in my answer to this question. – Jim Belk Jul 18 '15 at 00:40
  • for the nice integration formula displayed in the middle, is there a rigorous proof for it? It might be argued by using the uniqueness of Haar measure. It's certainly true that the RHS is left and right invariant. Normalization is not a problem. But how about regularity? – henryforever14 Mar 04 '16 at 14:33