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May 15, 2026

The integral motivic Satake equivalence for ramified groups

Key Points

This research aims to construct the geometric Satake equivalence for quasi-split reductive groups over nonarchimedean local fields.
Utilized étale Artin–Tate motives with Z[1/p]-coefficients.
Extended Gaussent–Littelmann's work on the connection between LS galleries and MV cycles.
Generalized Zhu’s integral Satake isomorphism for spherical Hecke algebras.
Developed a geometric Satake equivalence for ramified groups, confirming its applicability to both equal and mixed characteristic local fields.
Defined generic spherical Hecke algebras for residually split groups.
Constructed generic Satake and Bernstein isomorphisms related to the established equivalence.

Abstract

Abstract We construct the geometric Satake equivalence for quasi-split reductive groups over nonarchimedean local fields, using étale Artin–Tate motives with Z ⁢ 1 p Z1p -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.

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Cite This Study

Thibaud Van Den Hove (Wed,) studied this question.

synapsesocial.com/papers/6a06b8f8e7dec685947ab71b https://doi.org/https://doi.org/10.1515/crelle-2026-0031

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