What is the relation between L. Lafforgue and Frenkel-Gaitsgory-Vilonen results on Langlands correspondence ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:00:30Z http://mathoverflow.net/feeds/question/86853 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86853/what-is-the-relation-between-l-lafforgue-and-frenkel-gaitsgory-vilonen-results-o What is the relation between L. Lafforgue and Frenkel-Gaitsgory-Vilonen results on Langlands correspondence ? Alexander Chervov 2012-01-27T19:16:16Z 2012-02-01T20:32:40Z <p>What is the relation between Lafforgue's result on Langlands and Frenkel-Gaitsgory-Vilonen ? ( <a href="http://arxiv.org/abs/math/0012255" rel="nofollow">http://arxiv.org/abs/math/0012255</a> , <a href="http://arxiv.org/abs/math/0204081" rel="nofollow">http://arxiv.org/abs/math/0204081</a> ) Does one imply other ? If not why ? </p> <p>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 ? </p> <p>Is there clear relation between irreps of GL(Adels) and Hecke-eigen-sheaves on BunGL ?</p> <p>How to see in FGV setup that Hecke eigenvalues should correspond to Frobenius eigs ? </p> <hr> <p>Background</p> <p>$GL_n$ Langlands correspondence is bijective correspondence between (1) and (2), where</p> <p>(1) n-dimensional Irreps of (almost) Galois group </p> <p>(2) Certain Irreps of GL_n(Adels).</p> <p>Main requirement that Frobenius eigenvalues should be equal to Hecke eigenvalues for each point "p". </p> <p>Consider the case of "function fields" i.e. Galois group is taken for some curve over finite field and adels over this curve.</p> <hr> <p>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 $\sum V\otimes V^t \otimes W$</p> <p>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...</p> <p>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). </p> http://mathoverflow.net/questions/86853/what-is-the-relation-between-l-lafforgue-and-frenkel-gaitsgory-vilonen-results-o/87273#87273 Answer by Alexander Braverman for What is the relation between L. Lafforgue and Frenkel-Gaitsgory-Vilonen results on Langlands correspondence ? Alexander Braverman 2012-02-01T20:32:40Z 2012-02-01T20:32:40Z <p>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:</p> <p>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.</p> <p>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)</p> <p>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).</p>