## volume of compact simple Lie groups under the natural Euclidean embedding

I am looking for a quick reference for the volume formula for all the compact simple Lie groups embedded as matrix groups in the natural way. The one I care most for are the real orthogonal groups. I don't see how to deduce them from the Weyl integration formula. I believe they are of the order diameter raised to the power of the dimension. So for instance for $SO(n)$ the volume should be of order $n^{\Theta(n^2)}$.

-
For the orthogonal group $SO(n)$, isn't it just the product of the volumes of the $k$-spheres in their natural embeddings, as $k$ ranges from 1 to $n{-}1$? Similarly, for $U(n)$, it should be the product of the $2k{-}1$-spheres as $k$ ranges from $1$ to $n$, and so on. – Robert Bryant Jan 4 2012 at 1:26
Oh you are right. Thanks for the observation! – John Jiang Jan 4 2012 at 1:29
This is not quite true: $SO(2)$ as a circle in the vector space of $2\times 2$ matrices has radius $\sqrt 2$, in the metric that I am thinking of. – Tom Goodwillie Jan 4 2012 at 1:38
@Tom: You are quite right. I forgot about the factor of $(\sqrt{2})^{k-1}$ that you have to put in at each level because the natural map $\pi:SO(k)\to S^{k-1}$ is not a Riemannian submersion, but is a Riemannian submersion scaled by a factor of $1/\sqrt{2}$, i.e., each horizontal vector for this bundle is shrunk by a factor of $\sqrt{2}$ by the differential of $\pi$. Thus, the overall factor you need to multiply the answer I gave by is $(\sqrt{2})^{n(n-1)/2}$. The recipe I gave for $U(n)$ is not right either, as the map $\pi:U(k)\to S^{2k-1}$ shrinks horizontal volumes by $2^{k-1}$. Sorry. – Robert Bryant Jan 4 2012 at 2:21
I guess in general even when $\pi$ is not a Riemannian submersion one could integrate the inverse of the generalized jacobian map along the preimage of $\pi$, and then integrate that over the base manifold? – John Jiang Jan 4 2012 at 2:54
show 1 more comment

 I have actually run into this article before but the formula is a bit hard for me to parse. My interest comes from analyzing the Kac random walk on $SO(n)$. – John Jiang Jan 4 2012 at 8:57