Let $X$ be an $\ell$ space (in the sense of Bernstein-Zelevinski), $H$ be an $\ell$ group which acts on $X$ and $\chi$ be a character of $H$. Denote $C^\infty(X)^{H,\chi}$ the space of locally constant functions $\phi$ on $X$ such that $\phi(h.x)=\chi(h)\phi(x)$ for all $h\in H, x\in X$. Let $C_c^\infty(X)$ be the space of locally constant and compact supported functions on $X$, and let $C_c^\infty(X)(H,\chi)$ be the subspace of $C_c^\infty(X)$ spanned by functions of the form $h.f-\chi(h)f,\forall h\in H,f\in C_c^\infty(X)$. Here $(h.f)(x)=f(h^{-1}.x)$.

Given $f\in C_c^\infty(X)(H,\chi)$ and $\phi\in C^\infty(X)^{H,\chi}$, we have $$\langle f,\phi\rangle:=\int_X f(x)\phi(x)dx=0.$$

My question is: given $f\in C_c^\infty(X)$, if $$\int_X f(x)\phi(x)dx=0, \quad \forall \phi\in C^\infty(X)^{H,\chi},$$ can we conclude that $f\in C_c^\infty(X)(H,\chi)$? If this is not true, what can we say about $f$?

This question can be rephrased in the following way. The integral pair defines an embedding $C^\infty(X)\hookrightarrow C^\infty_c(X)^*$, where $C_c^\infty(X)^*$ denotes the linear dual of $C_c^\infty(X)$. Although $C^\infty(X)$ is not the whole space $C_c^\infty(X)^*$, it separates points in the sense that, given a point $f\in C_c^\infty(X)$, if $\langle f,\phi\rangle=0$ for all $\phi\in C^\infty(X)$, then $f=0$.

Similarly, we have an embedding $C^\infty(X)^{H,\chi}\hookrightarrow (C_c^\infty(X)_{H,\chi})^*=\textrm{Hom}_{H,\chi}(C_c^\infty(X),\textbf{C})$, where $C_c^\infty(X)_{H,\chi}$ is the Jacquet module $C_c^\infty(X)/C_c^\infty(X)(H,\chi)$. The above question then becomes: does $C^\infty(X)^{H,\chi}$ separate points?

Thanks in advance.


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