I am learning basic theory of automorphic forms, and I met such a very interesting topic in Goldfeld's nice book <Automorphic forms and L functions for the group $GL(n,\mathbb{R})$>, section 1.6:
The volume (in the sense of a suitable invariant measure) of $SL(n,\mathbb{Z})\backslash SL(n,\mathbb{R})/SO(n,\mathbb{R})$ is $\zeta(2)\zeta(3)...\zeta(n)$.
He mentioned that this is a classical problem (and some friend told me that was the so-called "Tamagawa number" in the modern setting) first computed by Siegel in 1936. I followed his reference and found some materials on Paul Garrett's homepage, and here is the link:
http://www-users.math.umn.edu/~garrett/m/v/volumes.pdf
In this notes he wrote how to compute similar problem for symplectic groups. So I want to ask, (since I'm a total beginner in automorphic forms), is there any reference dealing with this interesting problem on other classical groups? For example, $GL(n)$ or some groups over $\mathbb{C}$ and $p$-adic fields? I guess there're some systematic approaches to this topic, but I don't know how to search for them.
Thanks in advance for any recommendation or suggestion!
Yes. See: http://sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/chev-ps.pdf
The paper is not easy to follow for a beginner, but you can get the answer in the first page. It is a product of zeta values determined by the topology of the underlying classical group (the Poincare polynomial).