The integral motivic Satake equivalence for ramified groups

TL;DR

This paper develops a geometric Satake equivalence for ramified quasi-split reductive groups over nonarchimedean local fields using étale motives, extending previous work and generalizing key isomorphisms.

Contribution

It introduces a new geometric Satake framework for ramified groups using étale motives and extends the connection between LS galleries and MV cycles to residually split groups.

Findings

Constructed the geometric Satake equivalence for ramified groups.

Extended Gaussent--Littelmann's work to residually split groups.

Generalized Zhu's integral Satake isomorphism to ramified groups.

Abstract

We construct the geometric Satake equivalence for quasi-split reductive groups over nonarchimedean local fields, using \'etale Artin-Tate motives with $\mathbb{Z}[\frac{1}{p}]$-coefficients. We consider local fields of both equal and mixed characteristic. Along the way, we extend the work of Gaussent--Littelmann on the connection between LS galleries and MV cycles to the case of residually split reductive groups. As an application, we generalize Zhu's integral Satake isomorphism for spherical Hecke algebras to ramified groups. Moreover, for residually split groups, we define generic spherical Hecke algebras, and construct generic Satake and Bernstein isomorphisms.