Parahoric motivic Hecke categories in equal and mixed characteristic

Abstract

We study constructible $\mathbb{Z}[\tfrac{1}{p}]$-linear \'etale motives on parahoric Hecke stacks associated with quasi-split tamely ramified reductive groups. We prove a canonical equivalence between their equal-characteristic and Witt-vector incarnations that is monoidal with respect to convolution. This extends Bando's comparison, which concerns split reductive groups and constructible \'etale sheaves. The proof combines the study of a family of affine Grassmannians for Pappas--Zhu group schemes with the use of the motivic nearby cycles functor.