The orthogonal case is more difficult than the symplectic one because of the need to distinguish between *SO*(*n*) and *O*(*n*), which we do not in general know how to do. 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}*. We denote by

**Theorem 19.1.** *Suppose that N is not geometrically isomorphic to any nontrivial ...*

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