Let $G \subset \operatorname{GL}_d(\mathbb{R})$ be a non-compact semi-simple Lie group and $K \subset G$ a maximal compact subgroup. Let $\mathfrak{g}$ (resp. $\mathfrak{k}$) be the Lie algebra of $G$ (resp. $K$), consider the Cartan decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ and fix a maximal abelian subspace $\mathfrak{a} \subset \mathfrak{p}$ with complexification $\mathfrak{a}_{\mathbb{C}} = \mathfrak{a} \otimes_{\mathbb{R}} \mathbb{C}$.

We define the Fourier–Harish-Chandra transform of a function $f \in C^{\infty}_c (K \backslash G / K)$ by \begin{equation*} \widehat{f}(\lambda) = \int_G f(g) \varphi_{- \lambda}(g) dg \quad (\lambda \in \mathfrak{a}^*_{\mathbb{C}}) \end{equation*} where $\varphi_{- \lambda}$ is the spherical function of spectral parameter $-\lambda$. One may show that $f \mapsto \widehat{f}$ is bijection from $C^{\infty}_c (K \backslash G / K)$ to the $W$-invariants Paley–Wiener functions on $\mathfrak{a}^*_{\mathbb{C}}$; where $W$ is the Weyl group (see e.g. [Gan, Theorem 3.5]).

Now fix $v \in \mathbb{R}^d$ with stabilizer $H \subset G$ and assume $X = G \cdot v = H \backslash G$ is a closed sub-variety. Given $s \in C^{\infty}_c(X)$, consider the following transform \begin{equation*} \widetilde{s}(\lambda) = \int_G s(g^{-1} \cdot v) \varphi_{- \lambda}(g) dg \quad (\lambda \in \mathfrak{a}^*_{\mathbb{C}}). \end{equation*} My question is: given a compact subset $C \subset \mathfrak{a}^*_{\mathbb{C}}$, is there $s_C \in C^{\infty}_c(X)$ such that $\widetilde{s_C}$ does not vanish on $C$ ? I'm particularly interested in the case $G = O(n,m)$ and $H = O(n)$, for integers $n > m \geq 1$.

In this case, we have $H \subset K := O(n) \times O(m)$ so I guess that a recipe of the form:

- pick a Paley–Wiener function $S$ with support containing $C$;
- invert $S$ to obtain $s \in C_c^{\infty}(K \backslash G / K)$ such that $\widehat{s} = S$;
- view $s$ as function in $C_c^{\infty}(K \backslash G)$, then as a function in $C_c^{\infty}(H \backslash G)$ and finally as a function in $C_c^{\infty}(X)$

should work ? This seems too trivial to me and I'm afraid I'm missing something.

[Gan] R. Gangolli - On the Plancherel formula and the Paley–Wiener theorem for spherical functions on semisimple Lie groups.