Yihang Zhu (Columbia)
Irreducible components of affine Deligne-Lusztig varieties, and orbital integrals.

Abstract: Affine Deligne-Lusztig varieties (ADLV) naturally arise from the study of Shimura varieties and Rapoport-Zink spaces. Their irreducible components provide interesting algebraic cycles on special fibers of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which equates the number of irreducible components of an ADLV (modulo a symmetry group) with a weight multiplicity of the Langlands dual group. Our method is to count the number of F_q points on the ADLV, and study the growth rate of the counting as q grows. This allows us to apply tools from local harmonic analysis, including twisted orbital integrals and the Base Change Fundamental Lemma. We reduce the problem to a problem about q-analogues of Kostant's partition functions. After estimating these partition functions, we reduce the conjecture to the previously known cases proved by Hamacher-Viehmann and Nie. Along the way we also use an independent-of-p argument, which allows us to "set p=0". This is joint work with Rong Zhou.