Asymptotics of Schwartz functions: nonarchimedean

Let $G$ be a split, simply connected, simple group, and let $P\le G$ be a maximal parabolic. Braverman and Kazhdan in \cite{BKnormalized} defined a Schwartz space on the affine closure $X_P$ of $X_P^\circ:=P^{\mathrm{der}}\backslash G$. An alternate, more analytically tractable definition was given in \cite{Getz:Hsu:Leslie}, following several earlier works. In the nonarchimedean setting when $G$ is a classical group or $G_2$, we show the two definitions coincide and prove several previously conjectured properties of the Schwartz space that will be useful in applications. In addition, we prove that the quotient of the Schwartz space by the space of compactly supported smooth functions on the open orbit is of finite length and we describe its subquotients. Finally, we use our work to study the set of possible poles of degenerate Eisenstein series under certain assumption at archimedean places.
Comment: Edit throughout. Fix errors and strengthen results in section 5