On symplectic periods for inner forms of $\mathrm{GL}_n$

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-archimedean 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 $\mathrm{Sp}_n (DA)$, 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.

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Received 22 January 2016

Accepted 26 August 2016

Published 4 June 2018