Local to Global principle for reductive groups

Question

Let $G$ be a reductive group over an algebraic number field $k$. Denote with $k_v$ a local field and with $A$ the ring of its adeles, let $G_k$, $G_{k_v}$ resp. $G_A$ be the group of its $k$- resp. $A$- points. What are necessary and sufficient conditions for a local representation $\pi_v$ of $G_{k_v}$ to appear as $\otimes_v$ $\pi_v$ in the right regular representation of $G_k \backslash G_A$? What is a good reference to study this local to global process?

Answer 1

If $\pi_v$ is supercuspidal, then (after making a twist if necessary) it should be possible to find an automorphic $\pi$ with $\pi_v$ as the local factor at $v$. This kind of result is usually proved (although their are sometimes other possibilities) by an application of the simple trace formula. This method will also give good (although perhaps not complete) control of the ramification at other places. If you look at the paper of Deligne, Kazhdan, and Vigneras on Jacquet--Langlands for $GL_n$, I think you will find an expose of the technique.

If $\pi_v$ is not supercuspidal (or if one is not willing to make a twist), then this is not generally possible, just for cardinality reasons. The link that David Loeffler provides to the discussion of the $GL_2$ case is somewhat indicative of the situation.

Answer 2

There is also a theorem of I think Hakim (my apologies if this is not Jeff's theorem), generalized by Prasad and Schulze-Pillot, that allows you to globalize representations distinguished with respect to a subgroup. This one uses a simple relative trace formula.

Answer 3

Also, if $\pi_v$ is square-integrable and $G=Sp$ or $SO$, Arthur proves this in his upcoming book. See my reply to the question embedding of local representation into automorphic representation