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

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.