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Convolution and Equidistribution by Nicholas M. Katz

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CHAPTER 6

Group-Theoretic Facts about Ggeom and Garith

Theorem 6.1. Suppose N in Parith is geometrically semisimple. Then Ggeom,N is a normal subgroup of Garith,N.

Proof. Because N is geometrically semisimple, the group Ggeom,N is reductive, so it is the fixer of its invariants in all finite dimensional representations of the ambient Garith,N. By noetherianity, there is a finite list of representations of Garith,N such that Ggeom,N is the fixer of its invariants in these representations. Taking the direct sum of these representations, we get a single representation of Garith,N such that Ggeom,N is the fixer of its invariants in that single representation. This representation corresponds to an object M in <N>arith, and a Ggeom,N- invariant in ...

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