Question

Assume $v$ is a place of a number field $k$, finite or not. Let $\pi_v$ be an irreducible admissible generic representation of $GL_n(k_v)$. Is it always true that we can find some irreducible generic automorphic representation $\Pi$ of $GL_n(\mathbb{A}_k)$ with $v$-component exactly isomorphic to $\pi_v$?

A form of the famous generalized Ramanujan conjecture says that if $\Pi$ is cuspidal, then every component is tempered. So the above question is kind of converse to Ramanujan conjecture.

It is known that if $v$ is a finite place, and $\pi_v$ is supercuspidal, then $\Pi$ always exists, and in fact we can take $\Pi$ to be a cuspidal representation.

Many thanks for any answer or references related to this question.

Answer 1

I don't know if you're still interested in the question, but Arthur proves something like this in his upcoming book on representations of classical groups (http://www.claymath.org/cw/arthur/pdf/Book.pdf):

Lemma 6.2.2 says that for $G=SO(n)$ or $Sp(2n)$, a local field $F\neq\Bbb C$, and a square-integrable irreducible representation $\pi$ of $G(F)$, there is a global field $K$, a place $v$, and an automorphic representation $\Pi$ in the discrete spectrum that has $\pi$ as the $v$-component and is spherical at all other finite places. (He needs the lemma to use the trace formula over the constructed global field $K$.)

And I think I've seen something like this also for other groups somewhere else, at least for $GL(n)$ (with weaker conditions on the remaining places).