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

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