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?.](https://mathoverflow.net/questions/1252/where-stands-functoriality-in-2009).



What is the current status of the function fields Langlands conjectures?
(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](http://publications.ias.edu/rpl/series.php?series=51) that "I do not believe that much has yet been done beyond the group $GL(n)$''.) Langlands has initiated a program called [``Beyond endoscopy''](http://publications.ias.edu/rpl/comments.php?paper=263) 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.


