On degenerate Whittaker space for $G{L}_4({\mathfrak{o}}_2)$

Abstract

Let ${\mathfrak{o}}_2$ be a finite principal ideal local ring of length 2. For a representation $\pi$ of $G{L}_4({\mathfrak{o}}_2)$, the degenerate Whittaker space ${\pi }_{N,\psi }$ is a representation of $G{L}_2({\mathfrak{o}}_2)$. We describe ${\pi }_{N,\psi }$ explicitly for an irreducible strongly cuspidal representation $\pi$ of $G{L}_4({\mathfrak{o}}_2)$. This description verifies a special case of a conjecture of Prasad. We also prove that ${\pi }_{N,\psi }$ is a multiplicity free representation.

26 pages