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Convolution and Equidistribution by Nicholas M. Katz

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CHAPTER 11

A Second Construction of Autodual Objects

In this construction, we work on the split form Gm/k, Spec(k[x, 1/x]). We begin with a geometrically irreducible lisse sheaf F on an open dense set UGm which is ι-pure of weight zero and which is self-dual: FF.

Denote by d the rank of F. We view F |U as a d-dimensional representation ρ of image(U), toward either the orthogonal group O(d)/Q, if the autoduality is orthogonal, or toward the symplectic group Sp(d)/Q if the autoduality is symplectic (which forces d to be even). We denote by Ggeom,F the Zariski closure of the image ρ((U)) of the geometric fundamental group.1

We have a finite morphism ...

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