A proof of the finite field analogue of Jacquet’s conjecture

In this paper, the proof of the finite-field-analogue of Jacquet's conjecture on local converse theorem for cuspidal representations of general linear groups is given. More precisely, the set of twisted gamma factors of $\pi$, $$ \Big\{\gamma(\pi\times\tau,\psi)\mid \tau \in \scr{G}_t,\ 1\le t \le \Big[{n\over 2}\Big]\Big\}, $$ together with a central character $\omega_\pi$, determine uniquely (up to isomorphism) the irreducible cuspidal representation $\pi$ of ${\rm GL}_n({\Bbb F}_q)$, where ${\cal G}_t$ denotes the set of irreducible generic representations of ${\rm GL}_t({\Bbb F}_q)$, and ${\Bbb F}_q$ denotes a finite field of $q$ elements.