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
(2) Certain Irreps of GL_n(Adels).
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).
Best Answer
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).