  Affine Lusztig varieties encode the orbital integrals of Iwahori-Hecke
functions and serve as building blocks for the (conjectural) theory of affine
character sheaves. In this paper, we establish a close relationship between
affine Lusztig varieties and affine Deligne-Lusztig varieties. Consequently, we
give an explicit nonemptiness pattern and dimension formula for affine Lusztig
varieties in most cases.
