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

摘要

Affine Deligne-Lusztig varieties can be thought of as affine analogs ofclassical Deligne-Lusztig varieties, or Frobenius-twisted analogs of Schubertvarieties. We provide a method for proving a non-emptiness statement for affineDeligne-Lusztig varieties inside the affine flag variety associated to affineWeyl group elements satisfying a certain length additivity hypothesis. Inparticular, we prove that non-emptiness holds whenever it is conjectured to doso for alcoves in the shrunken dominant Weyl chamber, providing a partialconverse to the emptiness results of Goertz, Haines, Kottwitz, and Reuman. Ourtechnique involves the work of Geck and Pfeiffer on cuspidal conjugacy classes,in addition to an analysis of the combinatorics of certain fully commutativeelements in the finite Weyl group.