On Speh representations for level zero supercuspidal representations and Ginzburg-Kaplan gamma factors

We establish a relation between Speh representations of ${\mathrm{G}\mathrm{L}}_n\left({\mathbb{F}}_q\right)$ and Speh representations of ${\mathrm{G}\mathrm{L}}_n\left(F\right)$, where $F$ is a non-archimedean local field. We use irreducible level zero supercuspidal representations to show that these two notions of Speh representations associated to cuspidal representations are related via a commutative diagram, and that their corresponding $(k,c)$ $\psi$-Whittaker models are also related. We use these results to relate the local Ginzburg-Kaplan integrals for level zero supercuspidal representations to their finite field counterparts.