a question about invariant volume forms on homogeneous spaces. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T05:04:24Z http://mathoverflow.net/feeds/question/77459 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77459/a-question-about-invariant-volume-forms-on-homogeneous-spaces a question about invariant volume forms on homogeneous spaces. turtle 2011-10-07T15:21:15Z 2011-10-16T19:33:43Z <p>Here I consider $G$ a connected Lie group, which is assumed to be linear (i.e. embeddable in some $GL_n(\mathbb{R})$, and $X$ a homogeneous space under $G$. Fix a point $x\in X$, one considers the map $m:G\rightarrow X$ sending $g$ to $g(x)$. </p> <p>Does the left (or right) invariant volume form on $G$ passes to an invariant volume form on $X$, under the pushing forward along $m$? Here by pushing forward along $m$, I mean the measure $\mu$ on $X$, such that for a continuous function $f$ of compact support, one has $$\int_X f(x)d\mu(x):=\int_G f(m(g))dg$$, $dg$ being the left (or right) Haar measure on $G$.</p> <p>It seems that one needs to assume that the isotropy subgroup of $x$ in $G$ is compact. Does it matter if $G$ is not unimodular?</p> <p>Many thanks.</p> http://mathoverflow.net/questions/77459/a-question-about-invariant-volume-forms-on-homogeneous-spaces/77471#77471 Answer by Alain Valette for a question about invariant volume forms on homogeneous spaces. Alain Valette 2011-10-07T17:36:11Z 2011-10-07T17:36:11Z <p>For a recent discussion of invariant measures on homogeneous spaces, see e.g. Appendix B in M. Bachir Bekka, Pierre de La Harpe, Alain Valette, Kazhdan's property (T), Cambridge Univ. Press 2008 : <a href="http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf" rel="nofollow">http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf</a></p> <p>The necessary and sufficient condition for the existence of an invariant measure on $X$, is that the restriction of the modular function of $G$ to $G_x$ (= the isotropy subgroup of $x$),coincides with the modular function of $G_x$.</p> http://mathoverflow.net/questions/77459/a-question-about-invariant-volume-forms-on-homogeneous-spaces/78279#78279 Answer by Joseph Wolf for a question about invariant volume forms on homogeneous spaces. Joseph Wolf 2011-10-16T19:33:43Z 2011-10-16T19:33:43Z <p>If $X = G/H$ then it carries a $G$--invariant measure if and only if the quotient $\Delta_G/\Delta_H$ if the modular functions is equal to $1$. So for example if $G$ is unimodular then the condition is that $H$ be unimodular.</p>