TITLE: On the Volume of Compact matrix Lie groups
QUESTION [6 upvotes]: 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...?

REPLY [3 votes]: I think the following paper of Mkrtchyan and Veselovmay be relevant to your question:
http://arxiv.org/abs/1304.3031
It computes the volume of a compact Lie group when normalized using the Cartan-Killing form of its Lie algebra. Now if your group is a compact real form of a classical Lie group then, up to a constant, the Cartan-Killing form is equivalent to the form $(X,Y)=Tr X^{\ast}Y$ where $X$ and $Y$ are elements of your Lie algebra. This latest form, of course, allows you to compute the volume of your group as a subset of the space of matrices. If you compute the constant relating the Cartan-Killing form to the form I defined above, then you can read the answer using the paper in the link.