2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67971The 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 Mazur's 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.Representation Theory11S25 (Primary); 14L05, 14F30 (Secondary)text