Affine Springer Fibers and the Affine Matrix Ball Construction for Rectangular Type Nilpotents

TL;DR

This paper investigates affine Springer fibers for rectangular type nilpotent elements in type A, using the affine matrix ball construction to relate component positions to Kazhdan-Lusztig structures, extending previous work.

Contribution

It introduces a new application of the affine matrix ball construction to relate component positions with Kazhdan-Lusztig cells in affine Springer fibers for rectangular nilpotents.

Findings

The relative position map aligns with Kazhdan-Lusztig cell structure.

Generalizes Steinberg and van Leeuwen's work.

Provides explicit calculations for type A affine Springer fibers.

Abstract

In this paper, we study the affine Springer fiber $\mathcal{F} l_N$ in type $A$ for rectangular type semisimple nil-element $N$ and calculate the relative position between irreducible components. In particular, we use the affine matrix ball construction to show the relative position map is compatible with the Kazhdan-Lusztig cell structure, generalizing the work of Steinberg and van Leeuwen.