Vanishing Results for Toric Varieties Associated To GL n 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 G₂. 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 G₂ respectively. It is known that the latter imply the nonemptiness of some affine Deligne–Lusztig varieties. Dedicated to Scarlett MccGwire and Dr. Christian Duhamel