Close fields, affine Springer fibers and fundamental lemmas

TL;DR

This paper proves a local constancy theorem for affine Springer fibers over close local fields, enabling the transfer of fundamental lemmas from characteristic zero to positive characteristic.

Contribution

It establishes geometric local constancy for affine Springer fibers and extends fundamental lemmas to positive characteristic fields.

Findings

Stable orbital integrals are locally constant in families of close local fields.

Fundamental lemmas transfer from characteristic zero to positive characteristic.

Supports broader applications in the Langlands program.

Abstract

We prove a geometric local constancy theorem for affine Springer fibers in families of close local fields. Consequently, stable orbital integrals are locally constant in these families, and both the base change fundamental lemma and the standard endoscopic fundamental lemma transfer from characteristic zero to arbitrary positive characteristic.