Multiplicative Hitchin Fibrations and the Fundamental Lemma

TL;DR

This paper proves the fundamental lemma for the spherical Hecke algebra of a reductive group over a finite field using multiplicative Hitchin fibrations, under certain characteristic conditions.

Contribution

It introduces a novel approach employing multiplicative Hitchin fibrations to establish the fundamental lemma in this setting.

Findings

Proves the fundamental lemma for the spherical Hecke algebra of G.

Uses multiplicative Hitchin fibrations as a key tool.

Requires characteristic of k to be larger than twice the Coxeter number.

Abstract

Let $k$ be a finite field and let $G$ be a reductive group over $k[[\pi]]$. Suppose $\mathrm{char}(k)$ is larger than twice the Coxeter number of $G$, we prove the standard endoscopic fundamental lemma for the spherical Hecke algebra of $G$ using multiplicative Hitchin fibrations.