# What is the relation between L. Lafforgue and Frenkel-Gaitsgory-Vilonen results on Langlands correspondence ?

What is the relation between Lafforgue's result on Langlands and Frenkel-Gaitsgory-Vilonen ? ( http://arxiv.org/abs/math/0012255 , http://arxiv.org/abs/math/0204081 ) Does one imply other ? If not why ?

More technical: Do FGV work only with unramified Galois irreps (Seems Yes) ? If Yes, is it difficult to cover ramified case ? If yes, what is the problem ?

Is there clear relation between irreps of GL(Adels) and Hecke-eigen-sheaves on BunGL ?

How to see in FGV setup that Hecke eigenvalues should correspond to Frobenius eigs ?

Background

$GL_n$ Langlands correspondence is bijective correspondence between (1) and (2), where

(1) n-dimensional Irreps of (almost) Galois group

Main requirement that Frobenius eigenvalues should be equal to Hecke eigenvalues for each point "p".

Consider the case of "function fields" i.e. Galois group is taken for some curve over finite field and adels over this curve.

Lafforgue's proved the correspondence above for the curves over finite-fields. His proof follows strategy proposed and worked out by Drinfeld in GL_2 case. He considers moduli spaces of "schtukas" where both groups acts. And proves that "functions" on it can be decomposed as $\sum V\otimes V^t \otimes W$

V - irrep of GL(Adels), W - of Galois. As far as I understand main difficulties are of "technical" nature - one should find correct compactifications and introduce "negligible" pieces which are not interesting...

It is completely different from the strategy of FGV, proposed by Drinfeld(?) and Laumon. In this setup starts from the Galois irrep (=local system on curve) and constructs certain sheaf on BunGL which is Hecke-eigensheaf (with "eigenvalue" given exactly by the local system from which one starts).

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Check out sections 1 and 2 here arxiv.org/abs/0906.2747 – YangMills Jan 29 '12 at 0:00
@YangMills Thank you for the link, may be I did not look carefully, but I did not see... – Alexander Chervov Jan 29 '12 at 14:37
You might wish to check out this beautiful article by Frenkel : math.berkeley.edu/~frenkel/review.pdf – anādimadhyānta Jan 29 '12 at 16:41

Let me try to answer. [FGV] is only about unramified representations of the Galois group but they prove a stronger fact in this case (existence of certain "automorphic sheaf"). Lafforgue's result doesn't follow from there for several reasons:

a) Formally [FGV] use Lafforgue, but this was actually taken care of by a later paper of Gaitsgory ("On the vanishing conjecture..."). So that is really not a problem now.

b) Extending [FGV] to the ramified case is not trivial. I actually suspect that it can be done using the thesis of Jochen Heinloth but this has never been done (even the formulation is not completely clear in the ramified case)

c) In the unramified case what follows immediately from [FGV] is that you can attach a cuspidal automorphic form to a Galois representation. It is not obvious to me that the converse statement follows (Lafforgue's argument actually goes in the opppsite direction: he proves that a cuspidal automorphic form corresponds to a Galois representation and then the converse statement follows immediately from the converse theorem of Piatetski-Shapiro et. al. and from the fact that you know everything about Galois L-functions in the functional field case).

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For b: certainly you mean in the tamely ramified case, yes? – Moosbrugger Feb 1 '12 at 20:52
@Alexander Thank You ! – Alexander Chervov Feb 1 '12 at 20:54
Well, actually it is known that the tamely ramified case implies the general case. In fact, it is even enough to deal with the tamely ramified case with unipotent monodtromy - this will again imply everything (although for non-trivial reasons - you need to use the existence of global cycling lifting for $GL(n)$, which is known). – Alexander Braverman Feb 1 '12 at 20:55
More precisely: if you only want to prove the "classical" (i.e. not geometric) Langlands conjecture for $GL(n)$ and if you are only interested to show that you can attach an automorphic form to a Galois representation, then it is enough to look at the tamely ramified case with unipotent monodromy. – Alexander Braverman Feb 1 '12 at 20:58
My comment was about the geometric side of things. One might hope for a geometric story in wild ramification as well (as conjecturally there is one in the characteristic zero/D-modules case). – Moosbrugger Feb 1 '12 at 21:21