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 $n^{\Theta(n^2)}$. 
 A: As far as I know, the most standard volume formula was obtained by I.G. Macdonald in a very short 1980 paper here.   Since the volume depends on the choice of Haar measure, Macdonald starts with the (complexified) Lie algebra of the compact Lie group, fixing a Lebesgue measure there along with a fixed lattice such as a Chevalley $\mathbb{Z}$-form.   The second ingredient in his formula comes from the standard invariants in the cohomology calculation for the group.
There is another approach to Macdonald's formula in a later paper by Y. Hashimoto:
On Macdonald’s formula for the volume of a compact Lie group., Comment. Math. Helv. 72 (1997), no. 4, 660–662.
A: Explicit formulas for the volumes of compact Lie groups, with respect to their Haar measures, given in terms of their root data are given in: M. S. Marinov: Invariant volumes of J. Phys A: Math. Gen. 13  (1980) 3357-3376. Availabe in Prof. Marinov's memorial site.
The final formula is given in equation 19, which is also tabulated for the various simple types. The article compares the general formula with the sphere based computations for some of the classical groups.
The method of computation in the article is based on the Weyl's integration formula, however, the volume of the flag manifold (called the orbit space in the article) was read from the evolution operator of the Laplacian on the group manifold. Prof. Marinov's interest in this subject was due to his work in quantum mechanics and path integrals on group manifolds.
A: The volume of $O(n)$ is worked out in Muirhead's book `Aspects of multivariate statistical theory' (Wiley, 1982). See Corollary 2.1.16.
