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Robert Langlands is famous in number theory for making famous and deep conjectures about very abstract things called automorphic forms, somewhere in the 60s.

There's a very interesting article by Langlands called Where stands functoriality today which describes the development of the subject from Langlands' point of view as of 1990s.

But if somebody was to write an overview of the current state of Langlands functoriality, what would it say?

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Again, hard to select one answer as the accepted one. –  Ilya Nikokoshev Oct 31 '09 at 13:08
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Heh, the year in the title may need to be updated to 2010. –  Victor Protsak Aug 12 '10 at 3:28
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@Victor: only if someone posts an answer in 2010 (other than the spam answer which just bumped this thread to the top of the list, whereupon the answer was deleted by a moderator). Strictly speaking I guess anyone who answers now should stick with what was known in 2009 ;-) –  Kevin Buzzard Aug 12 '10 at 10:44
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5 Answers

The work of Ngo should allow for the treatment of all endoscopic cases of functoriality; this is a kind of technical condition, but includes transfer from classical groups to GL(n) and base change for unitary groups. Previously, cyclic base change for GL(n) was known (by Arthur and Clozel), as well as some very interesting isolated cases like the symmetric square, third, and fourth power of GL2 (due to Gelbart-Jacquet, Kim-Shahidi, and Kim respectively) and the tensor products of GL2 x GL2 (Ramakrishnan) and GL2 x GL3 (Kim-Shahidi). These results are all fundamental tools in modern number theory.

The general Langlands conjectures are still wide open. I have heard one prominent expert on the trace formula remark that "the symmetric nth powers of GL2 should be as hard as the general conjecture", which you may interpret however you may. Langlands has an idea which he refers to as "Beyond endoscopy", however it seems very difficult to get started and the only cases where it has been worked out are endoscopic and were already known.

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Btw, the functoriality of the symmetric $n$-th power lift for $\mathrm{GL}(n)$ implies the general Petersson-Ramanujan conjecture. –  plusepsilon.de Dec 8 '10 at 13:15
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Here are few remarks which might be relevant, although I understand almost nothing of the global Langlands program.

Lafforgue is currently working on problems relating to functoriality. There are a number of recent preprints and notes on his webpage, see for example "Quelques remarques sur le principe de fonctorialité". If you don't read French, maybe the lectures of Lafforgue in Cambridge a few months ago would be useful. They are available in various video formats at the Newton Institute webpage. To find them, see this list, and scroll down to May - there is a total of 5 talks by Lafforgue, the first one on May 5th.

My impression of Lafforgue's work is that he aims for a proof of functoriality in a fairly general setting, and (amazingly!) he hopes that the method would work also in the number field case and not only for function fields (although I might have misunderstood this). The method has at least some vague similarity with Tate's thesis, I think.

For more general background on functoriality and related things, see maybe Knapp's survey on the Langlands program, the Clay Summer School Proceedings from 2003 (here is the Google Books page), and this short note of Rapoport on Lafforgue's earlier work.

Edit: Thanks to "unknown" and David for pointing out the work of Ngo! I should have added that Laumon also gave a talk in May at the Newton Institute, on Ngo's proof, this is available here (both video and slides). See also the discussion at SBS. On the functoriality principle in general, there is also this 15-page expository presentation of Arthur.

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I'm somewhat surprised that noone has mentioned James Cogdell in their answers. In any case, Cogdell is an expert on functoriality, and has written many papers and given many talks on the subject, and on "where it stands today".

Here is a series of lectures he gave within the past year :

http://www.math.osu.edu/~cogdell/cimpa-www.pdf

Here is a paper of his (with Piatetski-Shapiro, Shahidi) on functoriality for quasisplit classical groups :

http://www.math.osu.edu/~cogdell/lift3-www.pdf

And here are some notes from a colloquium talk on L-functions and functoriality :

http://www.math.osu.edu/~cogdell/lff-www.ps

There is also a nice exposition about functoriality (written by Cogdell) in the book "An Introduction to the Langlands program", which is a book published in 2003.

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Perhaps the best answer to the question "Where stands functoriality today?" is given by Langlands himself in his informal write-up of two recent lectures (March 2011) given at the IAS, available here:

http://publications.ias.edu/sites/default/files/functoriality.pdf

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Ngo Bao Chau has recently proved the so called "Fundamental Lemma" (he is very likely to get the Fields medal for his work!) and this will have many dramatic applications to the Langlands program, though the full functoriality conjecture of Langlands is still far from being proved (but I am not an expert). James Arthur gave a talk at UBC on applications of the FL to the Langlands program but I have not been able to find a writeup of the talk.

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@Ilya Nokokoshev, apropos Arthur's UBC talk, he just delivered a rather dramatic summing up of the current understanding of the subject in Banff. He's writing a monograph, “The endoscopic classification of representations: orthogonal and symplectic groups”, a partial draft of which is available at claymath.org/cw/arthur/pdf/Book.pdf. –  L Spice Jul 6 '11 at 6:38
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