# Local factors of Tamagawa measure

This is a reference request to some computations which I hope can be found in the literature somewhere.

Let $G\subset GL_n$ be a semisimple linear algebraic group over $\mathbb Q$. The Tamagawa measure $\mu$ on the group of adelic points $G({\mathbb A})$ is uniquely determined by the product formula.
Nevertheless, it can be written as a product of local factors $\mu=\prod_{p\le\infty}\ \mu_p$ in a non-unique way.
For each place $p<\infty$ choose a special maximal compact open subgroup $K_p\subset G({\mathbb Q}_p)$ in a way that at almost all places these subgroups are hyper-special and coincide with $G({\mathbb Z}_p)=G({\mathbb Q}_p)\cap GL_n({\mathbb Z}_p)$. Then normalise the local measures $\mu_p$ so as to satisfy $\mu_p(K_P)=1$ for $p<\infty$. For $p=\infty$ choose $m$ to be the measure which comes from the canonical Riemannian metric on the symmetric space $X=G({\mathbb R})/K$, where $K$ is a maximal compact subgroup of $G({\mathbb R})$, where the metric is given by the Killing form.
Then $\mu_\infty=cm$ for some $c>0$ and I would like to know this number $c$.

The $K_p$ are not unique which results in $c$ not being unique up to a rational number and I think not all rational numbers are possible. I expect that $c$ is a rational number times a power of $\pi$ and it would already help to know this power.

If no general result is known, it would help to have these numbers in special cases like $SL_n$, $PGL_n$, $Sp_{2n}$ and so on.

## Answer

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