Affine Deligne--Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport--Zink spaces. Their irreducible components provide an interesting class of 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 count the number of F_q points as q goes to infinity; this boils down to computing a certain twisted orbital integral. After applying techniques from local harmonic analysis, we reduce to computing a particular coefficient of the matrix for the inverse Satake transform. Using an interpretation of this coefficient in terms of a q-analogue of Kostant's partition function, we are able to reduce the problem to the previously known special cases of the conjecture proved by Hamacher--Viehmann and Nie. This is joint work with Yihang Zhu.

An etale p-adic local system on a rigid analytic variety can be regarded as a family of p-adic Galois representations parametrized by the variety, and p-adic Hodge theory has brought many results and applications on such objects, including a p-adic Riemann-Hilbert correspondence by Diao, Lan, Liu and Zhu. I will discuss constancy of a key invariant (generalized Hodge-Tate weights) of general p-adic local systems

We prove the modularity of some two-dimensional residually reducible p-adic Galois representations over Q when p is at least 5. In the first talk, I will present a generalization of Emerton's local-global compatibility result. In the second talk, I will use this compatibility result to make a patching argument for completed homology in this setting.