volume form in a symmetric space of real rank one - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:57:42Z http://mathoverflow.net/feeds/question/89013 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89013/volume-form-in-a-symmetric-space-of-real-rank-one volume form in a symmetric space of real rank one emiliocba 2012-02-20T13:19:26Z 2012-02-20T14:42:49Z <p>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 $c$?</p> <p>Now, a more detailed exposition. Let $G$ be a connected semisimple Lie group of real rank one and finite center. Let:</p> <ul> <li>$\mathfrak g=\mathfrak k \oplus \mathfrak p$ a Cartan decomposition;</li> <li>$G=NAK$ be an Iwasawa decomposition of $G$;</li> <li>$\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak n$ the corresponding decomposition of $\mathfrak g$;</li> <li>$B$ Killing form on $\mathfrak g$,</li> <li>$\alpha$ the simple root ($G$ has real rank one);</li> <li>$p=\dim \mathfrak n_\alpha$, $q=\dim \mathfrak n_{2\alpha}$;</li> <li>$H_\alpha\in\mathfrak a$ with $\alpha(H_\alpha)=1$; </li> <li>$A^+=\{ a_t:=\exp(t H_\alpha) : t>0 \}$, </li> <li>$M$ the centralizer of $A$ in $K$.</li> </ul> <p>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.</p> <p>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.</p> <p>Thanks.-.</p>