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Hi, this is my first question (of many). I am blogging for the Fields Medal Symposium and would like to get into the mathematics involved with our program.

In an attempt to sort through the articles and through all the conversations that I've been having in and around the Fields Institute, I'm having trouble seeing how all these concepts relate to each other.

In particular, the Langlands Program began with an aim to discover quadratic reciprocity laws, for which methods of representation theory of automorphic forms were applied. How and when do orbital integrals come into the picture and why is stability important?

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Richard, I gather that you are an aspiring mathematics journalist, which is laudable: the public understanding of what mathematicians do is poor, and we mathematicians could probably use some help from enthusiastic people who want to spread the word. But (even though my area is not very close to the Langlands Program), your question does not seem to have the requisite level or focus for this site. This is a site essentially for sharply focused research questions and answers between professionals, and the software is not really adapted to the type of discussion you seem to be after. Sorry. –  Todd Trimble Sep 27 '12 at 1:31
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While it's true that the question is not of the type desired for this site, I wonder where he COULD ask such a question. The people with the right level of understanding are the users of MO, not, say, StackExchange, I think. I don't know the answer.. Tom –  Tom Dickens Sep 27 '12 at 3:48
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Dear Richard, You can find a very basic introduction to the Langlands Programme in my Splitting Primes, available on the arXiv (arxiv.org/abs/1007.4426). On p.13, there are references to other expository articles, written by professionals such as Arthur, Gelbart, Knapp and Taylor. Best wishes, –  Chandan Singh Dalawat Sep 27 '12 at 5:23
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Perhaps the word "quadratic" should be removed from "quadratic reciprocity laws". –  S. Carnahan Sep 27 '12 at 6:19
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One potential place for such an open-ended question / discussion would be in comments to an appropriate post on Richard Cerezo's blog itself (linked to in his profile); it might be a good approach to write to well-known math bloggers and have them advertise this blog so that interested parties can contribute. –  j.c. Sep 27 '12 at 16:28
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1 Answer

One of the main problems in the Langlands program is functoriality, which is roughly speaking: A "map", say $G \mapsto G'$, between groups should "transfer" certain representations. I will remain pretty vague what "map" and "transfer" are, simply because I only understand special cases of this. The representations are (cuspidal) automorphic representations. Essentially you want to show that automorphic forms are a category with the transforms being the morphisms.

I will describe the Jacquet-Langlands correspondence. A classical way to proof such transfers is via the trace formula. Roughly speaking, it is an identity between the representations and its conjugacy classes. It suffices for proving the transfer that the terms involving the conjugacy classes of $G$ and $G'$ coincide in some sense. Now, these terms are (weighted) orbital integrals associated to each conjugacy class.

Another part of Langlands program is the correspondence with Galois representations, which is probably what you mean with quadratic reciprocity in this context. Here, the strategy is similar only that given a Galois representation, you have to let it act on some geometry which is comparable with the conjugacy classes of $G$. I think that was the key ingredient in Laurent Lafforgue's proof for $GL(n)$ over a global function field.

So the trace formula is the standard strategy for both correspondence and functoriality. It is useful to have a good understanding of orbital integrals therefore.

Now let me address stability: Quote from Harris http://www.institut.math.jussieu.fr/projets/fa/bpFiles/Intro_Harris.pdf, Section 3, which is by the way an excellent survey.

Semi-simple conjugacy classes in inner forms of $GL(n)$ are completely characterized by their characteristic polynomials. Similar characterizations are possible for other groups, but only over an algebraically closed field.

Now, global fields are never algebraically closed and to overcome the issues involved in that matter is precisely the problem of stability, I think.

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