MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
The special case of this for $\mathrm{GL}_2 / \mathbf{Q}$ was asked a little while ago, here:… – David Loeffler Oct 28 '10 at 11:08
up vote 8 down vote accepted

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.

share|cite|improve this answer
I think you mean Deligne-Kazhdan-Vigneras. Rogawski also came up with an independent proof, and personally I find Rogawski's paper easier to read. – Kimball Oct 28 '10 at 16:34
Dear Kimball, Thanks! – Emerton Oct 28 '10 at 23:57

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.

share|cite|improve this answer

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.