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I want to compare the two canonical volume forms on a noncompact symmetric space fo real rank one. The first one is the volume form induced by the Riemannian structure given by the Killing form restricted on $\mathfrak p$. The second one is $d\bar g$ induced by $dg =\gamma(a_t)\ dk_1 \ da \ dk_2$. It is known that $$ dx=c \ d\bar g,\qquad \text{with $c\in\mathbb R$.} $$ What is $c$?

Now, a more detailed exposition. Let $G$ be a connected semisimple Lie group of real rank one and finite center. Let:

  • $\mathfrak g=\mathfrak k \oplus \mathfrak p$ a Cartan decomposition;
  • $G=NAK$ be an Iwasawa decomposition of $G$;
  • $\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak n$ the corresponding decomposition of $\mathfrak g$;
  • $B$ Killing form on $\mathfrak g$,
  • $\alpha$ the simple root ($G$ has real rank one);
  • $p=\dim \mathfrak n_\alpha$, $q=\dim \mathfrak n_{2\alpha}$;
  • $H_\alpha\in\mathfrak a$ with $\alpha(H_\alpha)=1$;
  • $A^+=\{ a_t:=\exp(t H_\alpha) : t>0 \}$,
  • $M$ the centralizer of $A$ in $K$.

Let $X=G/K$ be the symmetric space with the Riemannian structure induced by $B$ over $\mathfrak p$. Let $dx$ the volume form induced by this Riemannian structure.

Let $dg$ (resp. $d\bar g$) be the Haar measure on $G$ (resp. $G/K$) such that $$ dg = \gamma(a_t)\ dk_1 \ da \ dk_2 $$ on $KA^+K$, where $\gamma(a_t) = (e^t-e^{-t})^p (e^{2t}-e^{-2t})^q=2^{p+q}(\sinh t)^p(\sinh 2t)^q$, $da=dt$ and $dk$ is the Haar measure on $K$ normalized so that $K$ has volume 1.


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