The main purpose of this paper is to prove a group-theoretic generalization of a theorem of Katz on isocrystals. Along the way we reprove the group-theoretic generalization of Mazurs inequality for isocrystals due to Rapoport-Richartz, and generalize from split groups to unramified groups a result of Kottwitz-Rapoport which determines when an affine Deligne-Lusztig subset of the affine Grassmannian is non-empty.
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