| Résume | Kottwitz and Rapoport conjectured a root-system type statement that implies the converse to Mazur's Inequality for all (split) groups, and that can be used to establish a criterion for the non-emptiness of certain affine Deligne-Lusztig varieties. We prove their conjecture. In addition, we show how this result gives the vanishing of higher cohomology groups for certain line bundles on toric varieties associated with root systems. |
