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}
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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 $x \in V_\mathbb{Q}$.
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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.
Best Answer
One condition that I came across is that it is sufficient to have $G(\mathbb{C}) \cdot x $ a closed subvariety of $V_\mathbb{C}$. Then $G(\mathbb{C}) \cdot x$ is a closed affine variety. This guarantees in particular that $G_x(\mathbb{C})$ must be reductive from Matsushima's criterion which says if $G$ is a connected reductive $\mathbb{Q}$-group and $H \subseteq G$ is a $\mathbb{Q}$-subgroup then $G/H$ is affine if and only if $H$ is reductive.
This also applies if $G(\mathbb{C}) \cdot x$ is just affine but not closed but I don't know if there is a nice way to check this for some $x \in V_\mathbb{Q}$.