Low elements in dominant Shi regions

TL;DR

This paper explains why minimal elements of dominant Shi regions are low using ad-nilpotent ideals and surveys the bijections related to dominant Shi regions in affine Weyl groups.

Contribution

It provides a new explanation for low elements in dominant Shi regions and surveys the bijections involved in their study.

Findings

Minimal elements of dominant Shi regions are low, explained via ad-nilpotent ideals.

Provides a comprehensive survey of bijections in the study of dominant Shi regions.

Connects algebraic ideals with geometric regions in affine Coxeter groups.

Abstract

This note is a complement of a recent paper about low elements in affine Coxeter groups. We explain in terms of ad-nilpotent ideals of a Borel subalgebra why the minimal elements of dominant Shi regions are low. We also give a survey of the bijections involved in the study of dominant Shi regions in affine Weyl groups.