Geometric Structure of Affine Deligne-Lusztig Varieties for $GL_3$

TL;DR

This paper investigates the geometric structure of affine Deligne-Lusztig varieties for $GL_3$, identifying their irreducible components and conditions under which these components are classical Deligne-Lusztig varieties times affine spaces.

Contribution

It provides a complete classification of irreducible components of affine Deligne-Lusztig varieties for $GL_3$ with basic $b$, including when they are disjoint and have classical structures.

Findings

Classified all irreducible components for $GL_3$ affine Deligne-Lusztig varieties.

Identified conditions where components are classical Deligne-Lusztig varieties times affine spaces.

Proved that in certain cases, irreducible components are pairwise disjoint.

Abstract

In this paper we study the geometric structure of affine Deligne-Lusztig varieties for $GL_3$ and $b$ basic. We completely determine the irreducible components of the affine Deligne-Lusztig variety. In particular, we classify the cases where all of the irreducible components are classical Deligne-Lusztig varieties times finite-dimensional affine spaces. If this is the case, then the irreducible components are pairwise disjoint.