We work on either the split or the nonsplit form. We begin with a lisse sheaf *F* on a dense open set *j* : *U ⊂ G* which is geometrically irreducible, pure of weight zero, and not geometrically isomorphic to (the restriction to *U* of) any Kummer sheaf *L _{χ}*. Wedenoteby

**Theorem 18.1.** *Suppose that N is not geometrically isomorphic to any nontrivial multiplicative translate of itself, and that N is symplectically self-dual. Suppose further that for either of the two possible geometric isomorphisms G/k* ≅ G* _{m}/k, both* F(0)

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