On the Degenerate Whittaker space for some induced representations of ${\mathrm{G}\mathrm{L}}_4({\mathfrak{o}}_2)$

Let ${\mathfrak{o}}_l$ be a finite principal ideal local ring of length $l$. The degenerate Whittaker space associated with a representation of ${\mathrm{G}\mathrm{L}}_{2n}({\mathfrak{o}}_l)$ is a representation of ${\mathrm{G}\mathrm{L}}_n({\mathfrak{o}}_l)$. For strongly cuspidal representations of ${\mathrm{G}\mathrm{L}}_{2n}({\mathfrak{o}}_l)$ the structure of degenerate Whittaker space is described by Prasad’s conjecture, which has been proven for ${\mathrm{G}\mathrm{L}}_4({\mathfrak{o}}_2)$. In this paper, we describe the degenerate Whittaker space for certain induced representations of ${\mathrm{G}\mathrm{L}}_4({\mathfrak{o}}_2)$, specifically those induced from subgroups analogous to the maximal parabolic subgroups of ${\mathrm{G}\mathrm{L}}_4({\mathbb{F}}_q)$.