2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/228812We prove a conjecture of Kottwitz and Rapoport which implies a converse to Mazur's Inequality for all split and quasi-split (connected) reductive groups. These results are related to the non-emptiness of certain affine Deligne-Lusztig varieties.Added quasi-split case; simplified proof of split caseRepresentation TheoryNumber Theorytext