In their paper "Uniqueness of linear periods", Compositio Mathematica, $\textbf{102}$ (1996), 65-123, Jacquet and Rallis asked the following question in the middle of page 67.

Let $F$ be a $p$-adic field, and $n$ be a positive integer. Let $H_0=\left\{\begin{pmatrix}a&\\ &a \end{pmatrix},a\in GL_n(F)\right\}\subset GL_{2n}(F)$.

Given an irreducible smooth representation of $GL_{2n}(F)$, if $\textrm{Hom}_{H_0}(\pi,1)\ne 0$, could we conclude that $\pi$ is self-contragredient?

If $n=1$, it is well-known that the answer is yes. How about the general case? What is the current status of this problem?



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