Twisted Jacquet modules associated to maximal parabolic subgroups and cuspidal representations of $GL(n, q)$
Let $\pi$ be a cuspidal representation of $GL(n,F)$ over a finite field $F$. Let $P=MN$ be the Levi decomposition of a maximal parabolic subgroup corresponding to the partition $(k,n-k)$ of $n$. Given a rank $r$ character $\psi_r$ of the unipotent radical $N$, the twisted Jacquet module $\pi_{N, \psi_r}$ is a representation of the subgroup $M_r$ of $M$ which stabilizes $\psi_r$. The problem we solve in this work is to determine the structure of $\pi_{N, \psi_r}$ as a $M_r$-module. This problem was first studied by D. Prasad, who solved it for the case $r=k=n/2$ by calculating the character of $\pi_{N, \psi_r}$ and matching it to a known representation of $M_r$. In this work, we solve the problem for all values of $(r,k,n)$ directly without calculating the character of $\pi_{N, \psi_r}$. Our solution depends on two other key conceptual advances: (i) We generalize the Bernstein-Zelevinsky framework for studying representations of the Mirabolic subgroup of $GL(n,F)$, to maximal parabolic subgroups $P$. In particular, we show that the twisted Jacquet functor which takes a representation of $P$ to its twisted Jacquet modules, gives an equivalence of categories between Rep$(P)$ and the direct sum $\oplus_r \text{Rep}(M_r)$. (ii) Using this, we construct a pair of recursively defined representations $\Pi_{k,n}, \Pi_{n-k,n}^\dagger$ of $P$, which generalizes to $P$, the representation of the Mirabolic subgroup obtained from the trivial representation by recursively applying the Bernstein-Zelevinsky $\Phi^+$ functor. Like the representation $(\Phi^+)^{n-1}(1)$ of the Mirabolic subgroup, the representation $\Pi_{n-k,n}^\dagger$ satisfies a universal property with respect to restrictions to $P$ of cuspidal representations of $GL(n,F)$. Our solution of the main problem is a simple consequence of this universal property.