Vanishing Results for Toric Varieties Associated To GLn and G2

Toric varieties associated with root systems appeared very naturally in the theory of group compactifications. Here they are considered in a very different context. We prove the vanishing of higher cohomology groups for certain line bundles on toric varieties associated to GL n and G 2 . This can be considered of general interest and it improves the previously known results for these varieties. We also show how these results give a simple proof of a converse to Mazur’s inequality for GL n and G 2 respectively. It is known that the latter imply the nonemptiness of some affine Deligne–Lusztig varieties.