Affine Deligne-Lusztig varieties associated to additive affine Weyl group elements

Affine Deligne-Lusztig varieties can be thought of as affine analogs of classical Deligne-Lusztig varieties, or Frobenius-twisted analogs of Schubert varieties. We provide a method for proving a non-emptiness statement for affine Deligne-Lusztig varieties inside the affine flag variety associated to affine Weyl group elements satisfying a certain length additivity hypothesis. In particular, we prove that non-emptiness holds whenever it is conjectured to do so for alcoves in the shrunken dominant Weyl chamber, providing a partial converse to the emptiness results of Goertz, Haines, Kottwitz, and Reuman. Our technique involves the work of Geck and Pfeiffer on cuspidal conjugacy classes, in addition to an analysis of the combinatorics of certain fully commutative elements in the finite Weyl group.