The test function conjecture for parahoric local models
Timo Richarz (Jussieu)
Parahoric local models describe the singularities in terms of linear algebra which arise in the reduction of Shimura varieties (or Shtuka spaces) with a parahoric level structure. For arithmetic applications, one is interested in determining the semisimple trace of Frobenius function on their sheaf of nearby cycles. A conjecture of Kottwitz, proven by Pappas and Zhu in mixed, resp. Gaitsgory in equal characteristic, asserts that the function is naturally a central function in the corresponding parahoric Hecke algebra. In joint work with T. Haines, we characterize this central function for general reductive groups uniquely by its action on weakly unramified principal series representations in terms of traces of their Satake parameters. The new ingredient in the proof is the study of fiberwise $\mathbb G_m$-actions on local models, and an application of the geometric Satake equivalence.