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

TL;DR

This paper investigates the structure of degenerate Whittaker spaces for certain induced representations of ${ m GL}_4(rak{o}_2)$, extending understanding beyond strongly cuspidal cases and confirming conjectures in this context.

Contribution

It describes the degenerate Whittaker space for specific induced representations of ${ m GL}_4(rak{o}_2)$, expanding the scope of previous results.

Findings

Degenerate Whittaker space characterized for induced representations.

Extension of Prasad's conjecture to new classes of representations.

Provides explicit structure for representations induced from parabolic-like subgroups.

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 ${\rm GL}_4(\mathbb{F}_q)$.