  Let $G$ be a reductive group over a non archimedean local field, and $\psi$ a
non-degenerate character of the unipotent radical $U_0$ of a minimal parabolic
subgroup $P_0=M_0U_0$. For $P=MU\supseteq P_0$, we show that the descent to the
Jacquet module $J_P(\mathcal{W}(G,\psi))$ of Delorme's constant term map from
the space $\mathcal{W}(G,\psi)$ of generalized Whittaker functions on $G$ to
$\mathcal{W}(M,\psi_{|U_0\cap M})$ is the dual map of the inverse of the
isomorphism of Bushnell and Henniart from $J_{P^-}(\mathcal{W}_c(G,\psi^{-1}))$
to $\mathcal{W}_c(M,\psi_{|U_0\cap M}^{-1})$ (in particular the constant term
map is surjective). We give applications of this result. We also provide an
integral version of Lapid and Mao's asymptotic expansion for integral
generalized Whittaker functions in the context of $\ell$-adic representations.
