Date: Tue, 3 Oct 2000 16:58:23 +0200 (MET DST)
From: Hans-Werner Henn <henn@irmast1.u-strasbg.fr>
Subject: Re: 3 more on Grassmanians

> Date: Mon, 2 Oct 2000 18:57:08 -0400 (EDT)
> From: "Douglas C. Ravenel" <drav@math.rochester.edu>
> Subject: Third question on Grassmanians

> Assuming this method is valid, one could study the fibrations
> $$
>         SO(n-1) ----> SO(n) ----> S^{n-1}
> $$
>
> by induction on $n$ and arrive at volumes for $SO(n)$ (the
> sepcial orthogonal groups) from those of the the spheres.  (A
> formula for the volume of $S^{n-1}$ can be found in Coxeter's
> book on regular polytopes.)
>
> For example we could deduce that $SO(3)$ has volume $8\pi^2$, so
> $Spin(3)$ is isomteric to a 3-sphere of radius 2.


Doug,

there are formulas of the type you are speculating on in the following
survey article:

V.E. Voskresenski,  Adele groups and Siegel Tamagawa numbers,
Journal of Math. Sciences, Vol. 73, (1995), 47-113

In section 14 he discusses measures and corresponding volumes for SO(n).
You find formulas there, 14.6 for example, which express the volume in
terms of values of Gamma functions and powers of pi.
There should be much more elementary and thorough treatments
in the literature, but section 14 is pretty much independant
of the rest of the article.

Hans-Werner Henn