Generalized Whittaker functions and Jacquet modules

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.