On the Volume of Compact matrix Lie groups

Question

When we define the volume of a compact matrix lie group (subgroup of $M_n(C)$) by viewing it as a subspace of $R^{n^{2}}$ and applying the usual Lebesgue measure, what's the volume of SO(n), SU(n), Sp(n) and...?

Answer 1

SO(n) is a compact manifold of dimension $n(n-1)/2$ in $\mathbb R^{n^2}$. It follows that its $n^2$-dimensional Lebesgue measure is $0$. The same argument applies to the other examples.

I think the right question would be what is the volume of these groups with respect to the Lebesgue measure on the manifold itself. Computing the volume of things like SO(2) is straight forward, since we have a good parametrization of this group.

The volume of SO(n) is computed here: http://arxiv.org/pdf/0809.0808.pdf

Answer 2

One book in which most of these are computed is Robb Muirhead: "Aspects of multivariate Statistical Analysis"