Tempered Symplectic Distinction For Quaternionic $GL_n(D)$: A Conjecture of PRASAD

Source abstract

Let $F$ be a non-Archimedean local field of characteristic zero, let $D/F$ be the quaternion division algebra, and put $G_n = GL_n(D)$ and $H_n = Sp_n(D)$. We prove Prasad's revised conjecture on tempered representations with symplectic period. If $ρ$ is a unitary $H_r$-distinguished supercuspidal representation and $δ_m(ρ)$ denotes the generalized Steinberg representation associated to $ρ$, set $B_0(ρ) = ρ, B_m(ρ) = δ_{m+1}(ρ) \times δ_m(ρ), m \geq 1.$ We prove that an irreducible tempered representation $π$ of $G_n$ is $H_n$-distinguished if and only if $π$ is isomorphic to the product over $i$ of $B_{m_i}(ρ_i),$ where each $ρ_i$ is a unitary distinguished supercuspidal representation.