Irreducible components of affine Deligne-Lusztig varieties and orbital integrals

Affine Deligne-Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport-Zink spaces; their irreducible components give rise to interesting algebraic cycles on the special fiber of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which relates the number of irreducible components of ADLV's to a certain weight multiplicity for a representation of the Langlands dual group. Our approach is to use techniques from local harmonic analysis to compute the asymptotics of a certain twisted orbital integral which counts the number of F_q-points on the ADLV as q goes to infinity. This is joint work with Yihang Zhu.