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# Orthogonal Examples, Especially SO(n) Examples

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 : UG which is geometrically irreducible, pure of weight zero, and not geometrically isomorphic to (the restriction to U of) any Kummer sheaf Lχ. We denote by G := j? F its middle extension to G. Then the object N := G(1/2)[1] Parith is pure of weight zero and geometrically irreducible. The following result is the orthogonal version of Theorem 18.1.

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

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