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.

5more comments