# 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).

• 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