Twisted Jacquet modules associated to maximal parabolic subgroups and cuspidal representations of $GL(n, q)$

Abstract

Let $π$ be a cuspidal representation $GL(n,F)$ 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 $ψ_r$ of the unipotent radical $N$, the twisted Jacquet module $π_{N, ψ_r}$ is a representation of the subgroup $M_r$ of $M$ which stabilizes $ψ_r$. The main problem we solve in this work is to determine the structure of $π_{N, ψ_r}$ as a $M_r$-module. This problem was first studied by D. Prasad, who solved the problem for the case $r=k=n/2$. This and subsequent works on the problem for special cases of $(r,k,n)$, identify the structure of $π_{N, ψ_r}$ by calculating its character 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 $π_{N, ψ_r}$. Our solution depends on two other key conceptual advances: (i) We show that the twisted Jacquet functor which takes a complex representation of $P$ to its twisted Jacquet modules (one for each rank), gives an equivalence of categories between Rep$(P)$ and the direct sum $\oplus_r \text{Rep}(M_r)$ of the categories Rep$(M_r)$. (ii) We use this equivalence to construct a recursively defined representation $Π_{k,n}$ of $P$, which generalizes to $P$, the representation of the Mirabolic subgroup obtained from the trivial representation by iterating the Bernstein-Zelevinsky $Φ^+$ functor. Like the representation $(Φ^+)^{n-1}(1)$ of the Mirabolic subgroup, the representation $Π_{n-k,n}$ (after composing with the inverse transpose isomorphism) satisfies a universal property with respect to restrictions to $P$ of cuspidal representations of $G_n$. Our solution of the main problem is a simple consequence of this universal property.