Assume $\pi$ is a tempered representation of $GL_n(\mathbb{Q}_p)$. $N_n$ is a maximal unipotent subgroup of $GL_n$, and $\xi$ is a non-degenerate character of $N_n(\mathbb{Q}_p)$. Let $\Pi$ be the Whittaker model of $\pi$ associated to $N_n$ and $\xi$. Assume $W \in \Pi$.

Then the question is whether $W\otimes \overline{W} \in \mathcal{C}^{\omega}(N_n\times N_n \backslash GL_n \times GL_n, \xi \otimes \xi^{-1}),$ where $\mathcal{C}^{\omega}$ is a Harish-Chandra Schwartz space.

Thank you very much.