Let $F$ be a $p$-adic field, $n>2$ be an integer, $G=GL(n,F)$, and $Z\subset G$ be the unipotent subgroup of $G$ that consists of matrices of the form $I_n+tE_{1,n},\ t\in F$, where $E_{1,n}$ denotes the matrix with with a 1 at its $(1,n)$-position and 0 elsewhere. Let $\psi:Z\to \mathbb C^\times$ be a smooth additive character of $Z$. I am interested in computing the twisted Jacquet modules $V_{Z,\psi}$ for all irreducible admissible representations $V$ of $GL(n,F)$. Has this computation been done before? Can anyone suggest a method for doing this computation?

Here is some more technical information: Let $P$ be the standard parabolic of $G$ with Levi factor $L=GL(1)\times GL(n-2)\times GL(1)$. Note that the unipotent radical of $P$ is a Heisenberg group. Let $P_1$ be the centralizer of $Z$ in $P$. It is known (and not hard to show) that as a $P_1$-module we have $V_{Z,\psi}\simeq W_V\otimes \rho_\psi$ where $W_V$ is a representation of $GL(n-2)$ and $\rho_\psi$ is a representation of $P_1$ that is obtained by the "Oscilloator extension" of the Schrödinger representation of $H$. What I am interested in is the computation of the $GL(n-2)$-representation $W_V$ for all irreducible admissible $V$. Is this computation doable?