International Congress of Mathematicians

578 D. Gaitsgoryelementary steps (i.e. nk+i,k0symmetric power of X. This map sends the data of (At,Ki,...,K n ) to the divisor ofzeroes of the map K„, which, we recall, is a map between line bundles.The second map is denoted by ev : Q„ : „ —t G a , and it is defined as follows. Bythe definition of Q n , n , a point of this stack defines a complete flag of vector bundles0 = Mo C Mi C ... C M n = M, and identifications Mj/Mj_i ~ ft n-i for i < n, anda map G —¥ M n /M n _i for i = n. Therefore, we have n — 1 short exact sequences0 —¥ iV +1 —¥ M J —¥ iV —t 0, where M J is the corresponding rank 2 subquotient ofM, and each such extension defines a class in 77 1 (X, Q) ~ G a . The value of ev onthe above point of Q„ : „ is the sum of the above extension n — 1 classes in G a .oWe define the perverse sheaf J E on Q n^n as a tensor product of r* (U E^ ) (heredE^ denotes the symmetric power of E which lives on X^) and ev*(A-Sch), whereA-Sch is the Artin-Schreier sheaf on G a . We apply an appropriate cohomologicalshift to the above tensor product to make it a perverse sheaf.oFinally, we define J E £ D(Q„ : j) a s Goresky-MacPherson extension of J l E , i.e.o o rJ E := j\*(J x E). The fact that J E belongs to the Whittaker category D (Q„,„)follows from the construction.3.6. In the next section we will explain how the irreducibility assumption on Eimplies that the complex J' E on Bun^, constructed in this way, descends to Bun„.Here we will comment on the action of the Hecke functors.In [Ga], Section 6, it was shown that one can lift the Hecke functors E, x andH from Bun„ on the stacks Q n ,k-More precisely, for every fc we have the functors H^"'* : D(Q ni fc) —^ D(Q ni fc),which preserve the subcategory D H ' (Q n ,k)- Moreover, for fc = 2, ...,n, the functorsWk-i,k>^k,k-iW^k,k-i* intertwine in the natural sense the functors E n x - k andjj^n.fc-i_ p or n — i ; t Vj e pull-back functor (with an appropriate cohomological shift)n* intertwines H^"' 1 acting on D(Bun^) and E x acting on D(Bun„).Analogusly, we have the global Hecke functors H n - k : D(Q ni fc) —^ D(X x Q n ,k),where the assertions similar to the above ones hold.The basic fact about the perverse sheaf J E is its eigen-property, cf. [FGV1],Appendix:3.7. Proposition. There is a canonical isomorphism E n ' n (J E ) ~ E[l] M J E ,compatible with iterations (cf. Section 2).