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Let $F$ be a number field and $\mathbb{A}$ its adele ring. $G$ be a classical group and $ \pi$ be a unitary cuspidal automorphic representation of $G(\mathbb{A})$.

Then I am wondering whether there is a known theorem on the local component of $\pi$. For example, the statement I am expecting is $\pi_v$ is supercuspidal, square-integrable, tempered or generic etc for all places $v$.

Any comments are welcome!

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    $\begingroup$ $\pi_v$ can not be supercuspidal for all $v$, in fact, $\pi_v$ is unramified for almost all $v$. Even for $G=\mathrm{GL}(n)$ whether $\pi_v$ is tempered for almost all $v$ is a part of the Generalized Ramanujan Conjecture. $\endgroup$
    – Subhajit Jana
    Jan 28 at 16:23
  • $\begingroup$ @SubhajitJana, thanks for the comment! $\endgroup$
    – Monty
    Jan 29 at 4:12
  • $\begingroup$ @Echo, thanks for the nice comment! $\endgroup$
    – Monty
    Jan 29 at 4:12

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Let me work in the category of $L^2$-automorphic representations. Assuming your global representation $\pi$ is irreducible, about the only thing you can say about an arbitrary local component $\pi_v$ is that it is an irreducible smooth representation of $G_v$. For almost all $v$, you can also say $\pi_v$ will be spherical.

I don't know exactly which groups you include in classical groups (unitary groups? non-quasi-split forms?) but if you allow compact groups then the trivial representation is cuspidal, and in positive rank it is locally non-generic everywhere.

For (split) SO(5) you have Saito-Kurokawa lifts, which are locally non-tempered. Also, many cuspidal representation of SO(5) are not generic everywhere. See references on Siegel modular forms, SO(5) or GSp(4).

As mentioned in comments, $\pi_v$ can typically only be supercuspidal or discrete series at a finite number of places (e.g., if G is GL($n$) with $n > 1$).

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    $\begingroup$ @paul garrett: that really doesn't sound correct... $\endgroup$
    – Satan's Minion
    Jan 29 at 2:17
  • $\begingroup$ @Kimball, thanks for great answer. I learned much from your answer. By the way, if the condition that $\pi$ is generic os added, then can we say that $\pi_v$’ are generic for all places though we are not sure it is tempered. $\endgroup$
    – Monty
    Jan 29 at 4:09
  • $\begingroup$ @Satan'sMinion, ah, indeed, my comment from yesterday about Siegel modular forms lacking local Whittaker models was inaccurate and dumb. :) I don't know what I was imagining I was remembering. :) Maybe Saito-Kurokawa lifts? Thanks for making me think more carefully. :) $\endgroup$
    – paul garrett
    Jan 29 at 18:26
  • $\begingroup$ @Monty Are you asking if globally generic implies locally generic? The answer to this is yes, by factorization of Whittaker models. $\endgroup$
    – Kimball
    Jan 30 at 3:35
  • $\begingroup$ @Kimball, oh, I see! Thanks for kind reply again! $\endgroup$
    – Monty
    Jan 30 at 3:54

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