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Let $G$ be a reductive group over a nonarchimedean local field $F$. Let $\pi$ be an irreducible, cuspidal representation of $G$, with contragredient $\tilde{\pi}$. Then $\tilde{\pi}$ is cuspidal.

A character of $G$ is unramified if it is trivial on all compact subgroups of $G$. The group of unramified characters of $G$ acts on the set of isomorphism classes of irreducible, cuspidal $G$-representations. Under this action, is $\tilde{\pi} \sim \pi$?

Note: This question has now been asked at MO.

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