The Langlands-Kottwitz method seeks to express the trace of Frobenius-Hecke operators on the cohomology of Shimura varieties in a formula resembling the geometric side of the Arthur-Selberg trace formula. This formula involves, among other things, orbital and twisted orbital integrals, which are intriguing objects in local harmonic analysis.

 

In the second and third lectures, we will explore a local analogue of the global formula. This is ongoing joint work with Rong Zhou. The local formula relates the cohomology of some local Shimura varieties (which are spaces closely related to Rapoport-Zink spaces and p-adic period domains) with twisted orbital integrals. The discrete summation over Kottwitz triples in the global formula is replaced by a "continuous" integral over conjugacy classes. 

 

We expect that potential applications of the local formula are two-fold: One can use techniques from local harmonic analysis and trace formula to try and understand the cohomology of local Shimura varieties, and conversely use knowledge about the latter to prove things about local harmonic analysis.  As an illustration of the  second direction, we explain a new approach towards Rapoport's conjecture on the vanishing of certain twisted orbital integrals. This  conjecture is a natural statement in local harmonic analysis, and is also a key ingredient in the global Langlands-Kottwitz method for a non-quasi-split prime. In this approach, we reduce the problem to a conjecture about local Shimura varieties, and the latter is susceptible to a local proof using Xinwen Zhu’s recent work on categorical local Langlands. To this end, we will explain what we understand of Xinwen Zhu's work. 

 

Organizer

Chia-Fu Yu(Academia Sinica)