# A convergence condition on tempered representation

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.