  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.
