2
$\begingroup$

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.

$\endgroup$

2 Answers 2

5
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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 $G_x$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.