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My question, roughly speaking is, what happened to the function fields Langlands conjecture? I understand around 2000 (or slightly earlier perhaps), Lafforgue proved the function fields Langlands correspondence for $GL(n)$ in full generality (proving all aspects of the conjectures). Since then, what's happened to the function fields Langlands conjectures, and what work have people been doing in this direction? (Or has this field died out after Lafforgue's monumental achievement?)

I have been trying but haven't found a good reference for this, but since the function fields Langlands conjectures can be defined for all reductive groups (though I understand in some sense, it is not as "strong" as it is for $GL$ in the general case) - what is the status of these conjectures? Have partial results been obtained?

I understand that geometric Langlands is very intensely researched today, but I consider geometric Langlands as being slightly different (even though it is an analogue of the function fields Langlands correspondence - from reading Frenkel's article on geometric Langlands, my impression was not that geometric Langlands encapsulates all of the information in function fields Langlands conjectures), and I'm asking what work has been done since specifically on function fields Langlands. Or am I misunderstanding things and do the geometric Langlands conjectures actually encapsulate all the information from function fields Langlands as well?

I understand that the Fundamental Lemma has been proven recently, and that Lafforgue is doing some things relating to Langlands functoriality currently. Here's a related thread about Langlands functoriality: Where stands functoriality in 2009?..

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  • $\begingroup$ I'm pretty sure that this kind of question is not appropriate for MO and belongs on a blog or discussion forum. $\endgroup$
    – Harry Gindi
    Commented Jan 3, 2010 at 11:10
  • $\begingroup$ I refer everyone to the meta discussion where the op asked if this kind of question was okay, and got the answer "no." tea.mathoverflow.net/discussion/121/… $\endgroup$
    – Harry Gindi
    Commented Jan 3, 2010 at 11:50
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    $\begingroup$ This may be a stupid question. But is there any other online math discussion forum where this type of question will be answered? If so, please give me the URL and I will be most grateful. I had been looking for one and was most unsuccessful. In a blog only the people who blog can post, and the online math discussions forums I could find were all not advanced enough. $\endgroup$
    – Anweshi
    Commented Jan 3, 2010 at 15:11
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    $\begingroup$ Harry- I would say that this not the sort of thread being discussed in that meta question. Specifically, this is a question with a clear answer (which I'll admit I don't know and am mildly curious about). I think this is fine as an individual question, but I would recommend questioners to not make too much of a habit of questions like this. $\endgroup$
    – Ben Webster ♦
    Commented Jan 3, 2010 at 15:27
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    $\begingroup$ @Anweshi: "is there any other online math discussion forum..." -- this is not a discussion form, this is a Q&A forum. $\endgroup$
    – David Zureick-Brown
    Commented Jan 3, 2010 at 19:30

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(1) Regarding the relationship between geometric Langlands and function field Langlands: typically research in geometric Langlands takes place in the context of rather restricted ramification (everywhere unramified, or perhaps Iwahori level structure at a finite number of points). There are investigations in some circumstances involving wild ramification (which is roughly the same thing as higher than Iwahori level), but I believe that there is not a definitive program in this direction at this stage.

Also, Lafforgue's result was about constructing Galois reps. attached to automorphic forms. Given this, the other direction (from Galois reps. to automorphic forms), follows immediatly, via converse theorems, the theory of local constants, and Grothendieck's theory of $L$-functions in the function field setting.

On the other hand, much work in the geometric Langlands setting is about going from local systems (the geometric incarnation of an everywhere unramified Galois rep.) to automorphic sheaves (the geometric incarnation of an automorphic Hecke eigenform) --- e.g. the work of Gaitsgory, Mirkovic, and Vilonen in the $GL_n$ setting does this. I don't know how much is known in the geometric setting about going backwards, from automorphic sheaves to local systems.

(2) Regarding the status of function field Langlands in general: it is important, and open, other than in the $GL_n$ case of Lafforgue, and various other special cases. (As in the number field setting, there are many special cases known, but these are far from the general problem of functoriality. Langlands writes in the notes on his collected works that "I do not believe that much has yet been done beyond the group $GL(n)$''.) Langlands has initiated a program called ``Beyond endoscopy'' to approach the general question of functoriality. In the number field case, it seems to rely on unknown (and seemingly out of reach) problems of analytic number theory, but in the function field case there is some chance to approach these questions geometrically instead. This is a subject of ongoing research.

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  • $\begingroup$ A comment: It is mentioned in pages about Drinfel'd that he proved the functional field Langlands. Lafforgue is perhaps more general? $\endgroup$
    – Anweshi
    Commented Jan 3, 2010 at 18:10
  • $\begingroup$ Drinfeld proved functional Langlands for $GL(n)$ when $n=2$, which is clearly less general than Lafforgues $GL(n)$ for any $n$. $\endgroup$
    – t3suji
    Commented Jan 3, 2010 at 18:52
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    $\begingroup$ Of course Matthew is right in general, but I wanted to add that there are general conjectures in geometric Langlands with arbitrary ramification - starting in general terms in the work of Frenkel-Gaitsgory and developed further in the work in progress of Gaitsgory-Lurie. The extended topological field theory picture of geometric Langlands includes (and one could say requires) arbitrary ramification. But other than formal structure again Matthew is right, most of the understanding is in the tamely ramified case. $\endgroup$
    – David Ben-Zvi
    Commented Jan 3, 2010 at 19:15
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    $\begingroup$ My understanding is that away from GL_n (where packets have size 1) it's still in some sense an open question to even formulate a local Langlands conjecture rigorously. Matt/David: feel free to correct me if I'm wrong! I thought that the conjecture's current form was "there is a canonical bijection between (packets of representation-theoretic objects) and (certain Galois-theoretic objects)" but if one doesn't have a true definition of "canonical" (and do we?) then what next? For GL_n we have epsilon factors of pairs but we have nothing like this for general G, right? $\endgroup$
    – Kevin Buzzard
    Commented Jan 4, 2010 at 9:17
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    $\begingroup$ In the geometric (ie categorified function field) setting there is a general formulation of local Langlands for any group (rough version due to Frenkel-Gaitsgory, refined by Gaitsgory-Lurie in a preprint). It is a conjectural equivalence between [homotopical versions of 2-categories of] categories acted on by the loop group (categorified analogs of admissible reps of p-adic groups) and categories sheafifed over the stack of dual-group local systems over the punctured disc (analog of Galois representations). It is fixed more canonically by a kind of compatibility with Satake parameters.. $\endgroup$
    – David Ben-Zvi
    Commented Jan 4, 2010 at 14:49

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