Relative Fundamental Lemmas for Spherical Hecke Algebras and Multiplicative Hitchin Fibrations: the Jacquet--Rallis Case

TL;DR

This paper proves the Jacquet--Rallis fundamental lemma for spherical Hecke algebras over local function fields by employing multiplicative Hitchin fibrations, extending previous Lie algebra case methods.

Contribution

It introduces a novel proof of the fundamental lemma using multiplicative Hitchin fibrations in the context of spherical Hecke algebras.

Findings

Proof of the Jacquet--Rallis fundamental lemma in the function field setting

Extension of multiplicative Hitchin fibration techniques to new algebraic structures

Connection between Hitchin fibrations and fundamental lemmas in representation theory

Abstract

We prove the Jacquet--Rallis fundamental lemma for spherical Hecke algebras over local function fields using multiplicative Hitchin fibrations. Our work is inspired by the proof of [Yun11] in the Lie algebra case and builds upon the general framework of multiplicative Hitchin fibrations in [Wang26].