The base change fundamental lemma for central elements in parahoric Hecke algebras

TL;DR

This paper proves a fundamental lemma for the centers of parahoric Hecke algebras, extending previous results to a broader class of groups and aiding the study of Shimura varieties with parahoric level structures.

Contribution

It establishes the base change fundamental lemma for centers of parahoric Hecke algebras, generalizing prior work on spherical Hecke algebras for unramified groups.

Findings

Proved the fundamental lemma for centers of parahoric Hecke algebras.

Extended the base change fundamental lemma to a new class of Hecke algebras.

Provided a key ingredient for studying Shimura varieties with parahoric level structures.

Abstract

Clozel and Labesse proved the base change fundamental lemma for spherical Hecke algebras attached to an unramified group over a p-adic field. This paper proves an analogous fundamental lemma for centers of parahoric Hecke algebras attached to the same class of groups. This provides an ingredient needed for the author's program to study Shimura varieties with parahoric level structure at p.