Suppose $G$ is a $\mathbb{Q}$-algebraic group (I am interested in the semisimple case) acting rationally on a vector space $V_\mathbb{Q}$. Let $x \in V_\mathbb{Q}$ be a non-zero rational vector. Consider the stabilizer group $G_{x}$ of $x$. This is a closed subgroup in $G$.

Consider a compactly supported continuous $f:V_{\mathbb{R}}\rightarrow \mathbb{R}$ and then consider the following integral \begin{equation} \int_{G(\mathbb{R}) / G_{x}(\mathbb{R})} f(gx )dg . \end{equation}

When is this integral well defined for all rational points? That is, when can I perform an integration on the homogeneous space $G(\mathbb{R})/G_x(\mathbb{R})$ for all rational points. In the semisimple case, this is the same as asking if there any general conditions to guarantee that $G_x$ will be unimodular for any rational point $x \in V_\mathbb{Q}$.

Once it is well-defined, when is it finite for any $f$?

For example, some very generous conditions are when $G(\mathbb{R})$ acts transitively on non-zero points, or when $G(\mathbb{R}) x$ forms the non-zero points of a subspace in $V_\mathbb{R}$.

My guess is that such questions must have been considered in representation theory of algebraic groups but I don't really know where to start looking.