Affine Deligne-Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport-Zink spaces; their irreducible components give rise to an interesting class of cycles on the special fiber of Shimura varieties. In this talk we give a description of the top-dimensional irreducible components of ADLV’s modulo the action of a natural symmetry group, thus verifying a conjecture of Miaofen Chen and Xinwen Zhu. We obtain as a corollary, a description of the irreducible components of the basic locus of certain Shimura varieties in terms of a class set for an inner form of the structure group, generalizing classical results of Deuring and Serre. A key input for our approach is an analysis of certain twisted orbital integrals using techniques from local harmonic analysis. This is based on joint work in progress with X. He and Y. Zhu.