a question about invariant volume forms on homogeneous spaces.

Question

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 $g(x)$.

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 $G$.

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?

Many thanks.

Answer 1

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.

Answer 2

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 : http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf

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 $G_x$.