International Congress of Mathematicians

572 D. GaitsgoryUnfortunately, the geometric Langlands conjecture, i.e., the conjecture predictingthe existence of 3>, is not well formulated, because it is not known (at leastto the author) what "sufficiently generic" is for an arbitrary group G.The exception is the case when G = GL n , and "sufficiently generic" is understoodas "irreducible". In this situation, the geometric Langlands conjecture in theabove form was formulated by G. Laumon in [Lai], following the pioneering workof V. Drinfeld, [Dr], where the corresponding statement was proved for GL 2 .0.3. Although it is still not clear under what circumstances 3> should exist, at leastthree different constructions have been suggested by A. Beilinson and V. Drinfeld,in addition to the original Drinfeld's construction (the latter is, however, specificto the case G = GL n ). To the best of my knowledge, there are no theorems thatestablish the equivalence between any two of the four constructions described below.The first construction works in the D-module context (in particular, we have toassume that char(fc) = 0), under an additional assumption on a: one assumes thata is an oper, cf. [BD]. In this case, the corresponding 3> is a D-module (and notjust an object of the derived category). Moreover, 3> is holonomic. (Unfortunately,it is not clear from the construction whether 3> has regular singularities.)The second construction, via the so-called chiral Hecke algebra, also takes placein the D-module context. Here a can be arbitrary, but it is not clear under whatassumptions on a the object 3> thus constructed is non-zero, or when it is a singleD-module. It is not clear either whether the corresponding complex always hasholonomic (or even finitely generated) cohomologies.In the above two constructions, the fact that we work with D-modules is usedvery essentially, as the corresponding 3> is constructed by generators and relations.The other two constructions are more geometric in the sense that 3> is producedstarting from a, viewed as a sheaf on X, using the 6 functors.The third construction, uses the "spectral projector", and makes sense overany fc and for any a. It is again not clear under what assumption on a, the resulting3> is non-zero or when it lies in the bounded derived category.0.4. Finally, the fourth approach which, as was mentioned above, works for GL nonly and goes back to the original work of Drinfeld, is the subject of the presentpaper. In this case it can actually be shown that for an irreducible representation a,which can be thought of as an n-dimensional local system E on X, the correspondingautomorphic sheaf 3> (or JE) indeed exists and has all the desired properties.Let us add a few words about the history of this approach. After [Dr], wherethe case of n = 2 was solved, the (conjectural) generalization of the construction ofJE was suggested by G. Laumon in [Lai] and [La2]. Laumon's approach was furtherdeveloped by E. Frenkel, D. Kazhdan, K. Vilonen and the author (cf. [FGKV] and[FGV1]). The present paper can be regarded as a summary of these works. Finally,a certain vanishing result that was missing in order to complete the proof of theconjecture has been established in [Ga].Let us now briefly explain how the paper is organized. In Section 1 we reviewthe classical (i.e., function-theoretic) Langlands correspondence for GL n . In Section