International Congress of Mathematicians

576 D. Gaitsgoryrespects the S^-equivariant structure for any d. 1Few remarks are in order:Remark 1. In addition to the "basic" Hecke functor H, one can introduce thefunctors H* : D(Bun„) —t D(X x Bun„) for i = 1, ...,n (classically, they correspondto the generators TJ of the Hecke algebra of E(GL n (X x ),GL n (O x jj). We haveH ~ H 1 , and H" is the pull-back functor with respect to the multiplication map(x, M) —¥ M(x). One can show (cf. [FGV1], Sect. 2) that if JE is a Hecke eigensheafwith respect to E, then ïV(7 E ) ~ A i (E) M J E -Remark 2. Note, that formally in the definition of eigen-sheaves, we did not usethe fact that the local system E was n-dimensional. However, one can show (usingRemark 1 above) that if JE £ D(Bun„) is a Hecke eigen-sheaf with respect to Eand the rank of E is different from n, then JE = 0.2.5. The following is the statement of the (unramified) geometric Langlands correspondencefor GL n , conjectured by G. Laumon in [Lai] and proved in [FGV1] and[Ga]:2.6. Theorem. Suppose that the local system E is irreducible. Then there existsa Hecke eigen-sheaf JE £ D(Bun„) with respect to E, which is, moreover, anirreducible perverse sheaf over every connected component of Bun„, and cuspidal. 2Remark 3. Of course, if JE is a Hecke eigen-sheaf, then so is JE ® K, whereK is any complex of vector spaces. An additional conjecture, which has not beenfully established yet, is that any Hecke eigen-sheaf with respect to E has this form,where JE is the eigen-sheaf constructed in [FGV1].3. Geometric Whittaker modelsFrom now on our goal will be to sketch the steps involved in the construction3.1. Let BunJ, be the stack classifying the data of (M,K), where M is a rank nvector bundle, and K is a non-zero map Q" _1 —t M. 3 Yet n denote the naturalprojection Bun^ —t Bun„.We will construct an object J' E £ D(Bun^), and then show that it descendsto the sought-for perverse sheaf JE on Bun„.For us, the category D(Bun^) is the geometric analog of the space of functionson the quotient P(K)\GL n (K)/GL n (