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

Abstract

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