International Congress of Mathematicians

Geometrie Langlands Correspondence for GL n 581latter. Unfortunately, it is not true that the "elementary" functor Av„^ is exact.However, it will be exact when regarded as a functor acting on a certain quotientcategory.5.3. We introduce the category D(Bun„) as the quotient of D(Bun„) by a triangulatedsubcategory Dd egen (Bun n ). (An object J £ D(Bun„) belongs to Drf ese „(Bun„)essentially when it is degenerate, i.e., when it vanishes in the Whittaker model.)The quotient D(Bun„) possesses the following three crucial properties:1) The perverse i-structure on D(Bun„) induces a well-defined i-structure on D(Bun^2) The Hecke functors E, x : D(Bun„) —t D(Bun„) gives rise to well-defined functorsD(Bun„) —t D(Bun„), and the latter functors are exact in the sense of the t-structure on D(Bun„).3) The subcategory Drf ese „(Bun„) is orthogonal to the subcategory of cuspidalsheaves. I.e., if 3^1,3^2 £ D(Bun„) are such that 7i is cuspidal and the imageof 3^ in D(Bun„) vanishes, then Hom D ( B un„)(3 r i ) 3^) — Hom D ( B un„)(3 r 2 ) 3 r i) = 0.5.4. The main step in the proof that the functor Av n E is the following:5.5. Theorem. The functor AY„ : E : D(Bun„) —t D(Bun„) descends to a welldefinedfunctor D(Bun„) —t D(Bun„), and the latter functor is exact.The main idea behind the proof of Theorem 5.5. is the same phenomenon asthe one that forbids the existence of Hecke eigen-sheaves on Bun„ with respect tolocal systems of a wrong rank. Namely, if J is a perverse sheaf on Bun„ such thatthe corresponding object of D(Bun„) violates Theorem 5.5., then by looking at thebehavior of H d (J) £ Y)(X d x Bun„) around the various diagonals in X d , we arriveto a contradiction.Using Theorem 5.5. the proof of the exactness of Av^ E proceeds as follows:From (5.1) (and using the fact that taking invariants is an exact functor), we obtainthat the functor Av^ E is well-defined and exact on the quotient category D(Bun„).Moreover, by induction on n we can assume that for any J £ D(Bun„), Av n E (J)is cuspidal.Hence, if J is a perverse sheaf on Bun„, in the cohomological truncationsarrowsA°(Av^(3-)) and r