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International Congress of Mathematicians

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572 D. GaitsgoryUnfortunately, the geometric Langlands conjecture, i.e., the conjecture predictingthe existence <strong>of</strong> 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 work<strong>of</strong> 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 <strong>of</strong> my knowledge, there are no theorems thatestablish the equivalence between any two <strong>of</strong> 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 <strong>of</strong> 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 <strong>of</strong> Drinfeld, is the subject <strong>of</strong> the presentpaper. In this case it can actually be shown that for an irreducible representation a,which can be thought <strong>of</strong> 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 <strong>of</strong> this approach. After [Dr], wherethe case <strong>of</strong> n = 2 was solved, the (conjectural) generalization <strong>of</strong> the construction <strong>of</strong>JE 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 <strong>of</strong> these works. Finally,a certain vanishing result that was missing in order to complete the pro<strong>of</strong> <strong>of</strong> 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

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