On Symplectic Periods for Inner forms of ${\mathrm{GL}}_n$

Abstract

In this paper we study the question of determining when an irreducible admissible representation of ${\mathrm{GL}}_n(D)$ admits a symplectic model, that is when such a representation has a linear functional invariant under ${\mathrm{Sp}}_n(D)$, where $D$ is a quaternion division algebra over a non-Archimedian local field $k$ and ${\mathrm{Sp}}_n(D)$ is the unique non-split inner form of the symplectic group ${\mathrm{Sp}}_{2n}(k)$. We show that if a representation has a symplectic model it is necessarily unique. For ${\mathrm{GL}}_2(D)$ we completely classify those representations which have a symplectic model. Globally, we show that if a discrete automorphic representation of ${\mathrm{GL}}_n(D_{\mathbb{A}})$ has a non-zero period for ${\mathrm{Sp}}_n(D_{\mathbb{A}})$, then its Jacquet-Langlands lift also has a non-zero symplectic period. A somewhat striking difference between distinction question for ${\mathrm{GL}}_{2n}(k)$, and ${\mathrm{GL}}_n(D)$(with respect to ${\mathrm{Sp}}_{2n}(k)$ and ${\mathrm{Sp}}_n(D)$ resp.) is that there are supercuspidal representations of ${\mathrm{GL}}_n(D)$ which are distinguished by ${\mathrm{Sp}}_n(D)$. The paper ends by formulating a general question classifying all unitary distinguished representations of ${\mathrm{GL}}_n(D)$, and proving a part of the local conjectures through a global conjecture.