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# GL(n)Examples

Here 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 Lχ. We denote by G := j? F its middle extension to G. Then the object N := G(1/2) Parith is pure of weight zero and geometrically irreducible.

Theorem 17.1. Suppose that N is not geometrically isomorphic to any nontrivial multiplicative translate of itself. Suppose further that for one of the two possible geometric isomorphisms G/kGm/k, F(0)unip is a single Jordan block Unip(e) for some e ≥ 1, and F(∞)unip = 0. For n := dim(!(N)) we have

Ggeom,N = Garith,N = GL

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