  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.
