5
$\begingroup$

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!

$\endgroup$
3
  • 2
    $\begingroup$ $\pi_v$ 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. $\endgroup$ Jan 28, 2023 at 16:23
  • $\begingroup$ @SubhajitJana, thanks for the comment! $\endgroup$
    – Monty
    Jan 29, 2023 at 4:12
  • $\begingroup$ @Echo, thanks for the nice comment! $\endgroup$
    – Monty
    Jan 29, 2023 at 4:12

1 Answer 1

8
$\begingroup$

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 $n > 1$).

$\endgroup$
5
  • 1
    $\begingroup$ @paul garrett: that really doesn't sound correct... $\endgroup$ Jan 29, 2023 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, 2023 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. :) $\endgroup$ Jan 29, 2023 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, 2023 at 3:35
  • $\begingroup$ @Kimball, oh, I see! Thanks for kind reply again! $\endgroup$
    – Monty
    Jan 30, 2023 at 3:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.