Remarks on affine Springer fibres

TL;DR

This paper studies the structure of affine Springer fibers associated with a regular semisimple element in a complex simple Lie algebra, focusing on the action of a free abelian group and describing a fundamental domain.

Contribution

It introduces a description of a fundamental domain for the action of a free abelian group on a variety related to affine Springer fibers.

Findings

Identifies a free action of a free abelian group on the variety.

Describes a fundamental domain for this group action.

Provides structural insights into affine Springer fibers.

Abstract

Let h be a regular semisimple element in a complex simple Lie algebra g. Let t be an indeterminate. We consider the "variety" of Iwahori subalgebras of g tensored with power series in t which contain t times h. This variety admits a free action of a free abelian group of rank equal to the rank of g. We describe a fundamental domain for this action.