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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 by 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.

1

a question about invariant volume forms on homogeneous spaces.

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 by $m$? It seems that one needs to assume that the isotropy subgroup of $x$ in $G$ is compact.