Following Bernstein-Zelevinski, an $\ell$-space is a Hausdorff, locally compact totally disconnected topological space. For an $\ell$-space $X$, denote $S(X)$ the space of Bruhat-Schwartz functions on $X$, i.e., the space of $\mathbb{C}$-valued locally constant, compact supported functions on $X$. A distribution on $X$ is defined to be an element in $\textrm{Hom}_{\mathbb{C}}(S(X),\mathbb{C})$. Denote by $S^*(X)$ the space of distributions on $X$. Let $G$ be an $\ell$-group which acts on $X$. Then $G$ acts on $S(X)$ as $(g.f)(x)=f(g^{-1}.x),g\in G,f\in S(X), x\in X$. This induces an action of $G$ on the distributions: $(g.T)(f)=T(g^{-1}.f)$. A distribution $T$ is called $G$-invariant if $g.T=T$ for all $g\in G$. Denote by $S^*(X)^G$ the space of $G$-invariant distributions on $X$.

Let $F$ be a $p$-adic field.

$\textbf{Question 1}$: Let $G=F^\times \times F^\times$ and $X=F^3-\{(0,0,0)\}$. Let $U=(F^\times)^3$ which is an open subset of $X$. Let $G$ act on $X$ by $(a,b).(x,y,z)=(ax,by,abz)$. Is there a $G$-invariant distribution on $X$ such that its restriction to $U$ is non-trivial? If such $T$ exists, what can we say about $Supp(T|_U)$?

$\textbf{Question 1'}$: Let $G,X,U$ be as in $\textbf{Question 1}$. Let $\mu=\psi(z/(xy))d^*xd^*yd^*z$, viewed as a distribution on $U$. Here $\psi$ is an additive character of $F$. Is there a $G$-invariant distribution $T$ on $X$ such that $T|_U=\mu$?

Here is one example in my mind that suggests the non-existence of distributions $T$ in Question 1. Consider the multiplicative action of $F^\times$ on $F$, and let $U=F^\times$, which is an open subset of $F$. Then by Tate's thesis, there is no $F^\times$-invariant distribution $T$ on $F$ such that $T|_U$ is non-zero. In fact, the only $F^\times$-invariant distribution on $F$ is the Dirac measure.

But if we consider the action $(a,b).(x,y,z)=(ax,by,a^{-1}b^{-1}z)$ in $\textbf{Question 1}$, there clearly exists $G$-invariant distribution on $X$ (even on $F^3$, say the usual Haar measure $dxdydz$ on $F^3$) such that its restriction to $U$ is non-trivial.

Here are some thoughts on Question 1. Let $Z=X-U$. Then there is an exact sequence $$0\rightarrow S^*(Z)^G\rightarrow S^*(X)^G\rightarrow S^*(U)^G$$ Let's take an element $T_0\in S^*(U)^G$ and asks the question: is it possible to extend $T_0$ to a distribution on $X$? Given a function $\phi\in S(F)$, we can define a distribution $T_0$ on $U$ by $$<T_0,f>=\int_U f(x,y,z)\phi(z/(xy))d^*xd^*yd^*z, f\in S(U).$$ But it seems that there is no obvious way to extend $T_0$ to $X$. For example, if $f\in S(X)$, the integral $ \int_U f(x,y,z)\phi(z/(xy))d^*xd^*yd^*z$ is not well-defined in general, even when $\phi$ has compact support. But I don't know how to characterize $S^*(U)^G$ in general, and I also don't know how to show the above $T_0$ cannot be extended to a $G$-invariant distribution on $X$.

Any thoughts or suggestions? Thanks in advance.

I asked part of the question here https://math.stackexchange.com/questions/2409933/extension-of-certain-invariant-distributions, but did not draw much attention, and thus I decided to move it here. If it is not permitted, I will delete one.