Integrating on orbits of algebraic groups

Question

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}

1. 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 $x \in V_\mathbb{Q}$.

2. 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.

Answer 1

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 $x \in V_\mathbb{Q}$.

Answer 2

In the setup of general topological groups there is a necessary and sufficient condition for the existence of an invariant measure on the homogeneous space $G/H$, where $H$ is a closed subgroup of $G$, which amounts to the coincidence of the modular function of the group $H$ with the restriction of the modular function of $G$ to $H$ (this is Weil's theorem, see Section III.4 of Nachbin's book The Haar integral). If $G$ is semisimple, then it is unimodular, and therefore in this case the necessary and sufficient condition is that $H$ be unimodular as well.

PS Now I am actually puzzled by the interrogation mark at the end of the 3rd paragraph of your question. Are you aware of Weil's theorem and asking for the conditions that would provide unimodularity of a subgroup of $G$?