Let $F$ be a global field. What is the measure of $PGL_2(F) \backslash PGL_2(\mathbb{A})$?

This depends of course on the normalizations of the Haar measures on $PGL_2(F)$ and $PGL_2(\mathbb{A})$. Probably its most likely available in the literature for $PGL_2(F)$ admits the discrete measure and $PGL_2(\mathbb{A})$ the Tamagawa measure, but I couldn't find!?

I remember that there was a question about the measure of $SL_n(\mathbb{Z}) \backslash SL_n(\mathbb{R})$ here in the past, but couldn't find it.

It should be related to special values of the Dedekind zeta function.

Adeles and algebraic groups, chapter 3. – ACL Mar 14 '13 at 15:57naturalmeasure gives that quotient measure essentially $\zeta_F(2)$. The renormalization as in Siegel, Weil and elsewhere will make it $1$ or a power of $2$, probably, as usual. For $PGL_n$, it's similarly $\zeta_F(2)\zeta_F(3)...\zeta_F(n)$. A classical argument over $\mathbb Q$ (which obviously generalizes), also for $Sp(n)$, is at math.umn.edu/~garrett/m/v/volumes.pdf – paul garrett Mar 14 '13 at 16:04simply connectedgroup here. Clozel has a useful Sem. Bourbaki expose 702, following the work by Kottwitz and others/ – Jim Humphreys Mar 14 '13 at 17:22