On degenerate Whittaker space for $GL_4(\mathfrak{o}_2)$

TL;DR

This paper explicitly describes the degenerate Whittaker space for irreducible strongly cuspidal representations of $GL_4(rak{o}_2)$, confirming a special case of Prasad's conjecture and showing it is multiplicity free.

Contribution

It provides an explicit description of the degenerate Whittaker space for certain representations of $GL_4(rak{o}_2)$, verifying a conjecture and establishing multiplicity freeness.

Findings

Explicit description of $ ext{Whittaker}$ space for $GL_4(rak{o}_2)$ representations.

Verification of a special case of Prasad's conjecture.

Proof that the Whittaker space is multiplicity free.

Abstract

Let $\mathfrak{o}_2$ be a finite principal ideal local ring of length 2. For a representation $\pi$ of $GL_{4}(\mathfrak{o}_2)$, the degenerate Whittaker space $\pi_{N, \psi}$ is a representation of $GL_2(\mathfrak{o}_2)$. We describe $\pi_{N, \psi}$ explicitly for an irreducible strongly cuspidal representation $\pi$ of $GL_4(\mathfrak{o}_2)$. This description verifies a special case of a conjecture of Prasad. We also prove that $\pi_{N, \psi}$ is a multiplicity free representation.