International Congress of Mathematicians

580 D. Gaitsgory4.4. The next step is to show that perversity and irreducibility of J' E on Bun^* 'implies that J' E descends to a perverse sheaf on Bun„, cf. [FGV1], Section 5.First, one shows that ^Ißun»" descends to a perverse sheaf on Bun^*. Thisis done using a trick with Euler characteristics. It turns out, that since we alreadyknow that ^Ißun»* ' is perverse and irreducible, it is sufficient to show that theEuler-Poincaré characteristics of the stalks of J' E are constant along the fibers ofthe map n : Bun^ —t Bun„. Secondly, one observes that the above Euler-Poincarécharacteristics are actually independent of the local system E (and depend only onits rank).Thus, it is sufficient to find just one local system of a given rank n, for whichthe above constancy of Euler-Poincaré characteristics takes place, and one easilyfinds one like that.Finally, using the Hecke eigen-property of J' E , one shows that there exists aperverse sheaf JE defined on the entire Bun„, whose (cohomologically normalized)pull-back to Bun^ is isomorphic to J' E - The fact that the resulting sheaf JE iscuspidal follows from the construction.5. The vanishing resultIn this section we will indicate the main ideas involved in the proof of Theorem4.3., following [Ga]. We will change the notation slightly, and replace n' by nand E' by E.5.1. First, let us rewrite the definition of the functor Av n E . Consider first thefunctor A\- n>E : D(Bun„) —t D(Bun„) given byJ^p,(q*(E l )®H(Jj),where p,q are the two projections from X x Bun„ to Bun„ and X, respectively.From Proposition 2.3. it follows that the d-fold iteration ItAv^ E := Av„^ 0...0k\' n} E' : D(Bun„) —^ D(Bun„) maps to the equivariant derived category D Sd (Bun„),where the E^-action on the base Bun„ is, of course, trivial.Moreover, it follows from the definition of Laumon's sheaf H E , that there is afunctorial isomorphismAVIE(J) CZ (ltAvl E (Jjffi (5.1)where the superscript E^ designates the functor of Ed-invariants.5.2. The first step in the proof of Theorem 4.3. is the observation that instead ofproving that the functor AY„ : E vanishes, it is in fact enough to show that it is exactin the sense of the perverse i-structure.The fact that the seemingly weaker exactness assertion is equivalent to vanishingis proved using the Euler characteristics trick, similar to what we did in theprevious section.Since, as we have seen above, the functor Av^ E can be expressed throughmore elementary functors Av n: E, we will analyze the exactness properties of the