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Convolution and Equidistribution by Nicholas M. Katz

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CHAPTER 20

GL(n) × GL(n) × … × GL(n) Examples

In this chapter, we investigate the following question. Suppose we have a geometrically irreducible middle extension sheaf G on Gm/k which is pure of weight zero, such that the object N := G(1/2)[1] Parith has “dimension” n and has Ggeom,N = Garith,N = GL(n). Suppose in addition we are given s ≥ 2 distinct characters χi of k×. We want criteria which insure that for the objects

Ni := NLχi,

the direct sum ⊕iNi has Ggeom,⊕iNi = Garith,⊕iNi = ∏i GL(n). Because we have a priori inclusions Ggeom,⊕iNiGarith,⊕iNi ⊂ ∏i GL(n), it suffices to prove that Ggeom,⊕iNi = ∏i GL(n). To show this, it suffices to show both of the following two statements.

(1) The determinants in the Tannakian sense “det”(

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