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!
Best Answer
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 $n > 1$).