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

Abstract

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