On the degenerate Whittaker space for GL4(o2)

Abstract

Let $o_2$ be a finite principal ideal local ring of length 2. For a representation π of ${\mathrm{GL}}_4\left(o_2\right)$, the degenerate Whittaker space ${\pi }_{N,\psi }$ is a representation of ${\mathrm{GL}}_2\left(o_2\right)$. We describe ${\pi }_{N,\psi }$ explicitly for an irreducible strongly cuspidal representation π of ${\mathrm{GL}}_4\left(o_2\right)$. This description verifies a special case of a conjecture of Prasad. We also prove that ${\pi }_{N,\psi }$ is a multiplicity free representation.