On certain degenerate Whittaker Models for cuspidal representations of GLk⋅n(Fq)

Let π be an irreducible cuspidal representation of GLkn(Fq). Assume that π=πθ, corresponds to a regular character θ of F∗qkn. We consider the twisted Jacquet module of π with respect to a non-degenerate character of the unipotent radical corresponding to the partition (nk) of kn. We show that, as a GLn(Fq)-representation, this Jacquet module is isomorphic to πθ↾F∗n⊗Stk−1, where St is the Steinberg representation of GLn(Fq). This generalizes a theorem of D. Prasad, who considered the case k=2. We prove and rely heavily on a formidable identity involving q-hypergeometric series and linear algebra.