I found that I didn't quite understand how to think about the contragredient of an automorphic representation. I have read this post on Mathoverflow, which is helpful:
https://mathoverflow.net/questions/295897/contragredient-of-a-cuspidal-representation
But maybe my question is: Let $\pi\cong\otimes'\pi_v$ is an automorphic representation of, say, $GL(n,\mathbb{A}_\mathbb{Q})$, with contragredient $\hat{\pi}$. Then is it true that $\hat{\pi}\cong\otimes'\hat{\pi}_v$, where each $\hat{\pi}_v$ is the contragredient of the local component $\pi_v$? I saw on Bump's text:
In proposition 3.3.4 he explained how to realize the space of "contragredient" automorphic forms before we taking their irreducible subquotients. And in the paragraph just before it, he wrote "Let $\hat{\pi}\cong\otimes'\hat{\pi}_v$. This is the contragredient of the representation $(\pi,V)$." Does he mean that each $\hat{\pi}_v$ is the contragredient of the local component $\pi_v$? If this really holds, then how to see this? (I found that from the characerisation of proposition 3.3.4, I could not directly realize this fact.)
By the way, in answers to this post, it seems that self-contragredient automorphic representations are of particular interest. Is there any further reason or background?
Thanks a lot in advance for any explanation!