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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.

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What do you mean by generic, in this context? –  Alain Valette Aug 8 '11 at 15:03
    
Here it means there exists some automorphic form $\phi$ in $\Pi$, such that the associated Whittaker function $W_{\phi}$ is not identically zero. –  user1832 Aug 8 '11 at 15:30
    
Aren't there uncountably many local representations, but only countably many automorphic representations? –  doug Aug 8 '11 at 19:28
    
@AD: Yes, if you restrict to cuspidal GL(n) aut. reps. –  David Hansen Aug 8 '11 at 21:20
    
@David: What exactly do you mean by automorphic representation? I used to understand a representation which occurs discretely in $L^2(G({\mathbb Q})\backslash G({\mathbb A})$. If you fix a level, ie a compact open at the finite places, you get countably many automorphic representations by spectral geometry (cuspidals plus residues of Eisenstein series). As there are essentially countably many levels, you're done. What am I missing? –  doug Aug 9 '11 at 7:20
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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).

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Thank you very much! –  user1832 Nov 21 '11 at 21:50
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