# Local component of cuspidal automorphic representation

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 $$v$$.

• $\pi_v$ 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. Jan 28 at 16:23
• @SubhajitJana, thanks for the comment! Jan 29 at 4:12
• @Echo, thanks for the nice comment! Jan 29 at 4:12

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 $$n > 1$$).

• @paul garrett: that really doesn't sound correct... Jan 29 at 2:17
• @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. Jan 29 at 4:09
• @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. :) Jan 29 at 18:26
• @Monty Are you asking if globally generic implies locally generic? The answer to this is yes, by factorization of Whittaker models. Jan 30 at 3:35
• @Kimball, oh, I see! Thanks for kind reply again! Jan 30 at 3:54