On some simple geometric structure of affine Deligne-Lusztig varieties for $GL_n$

TL;DR

This paper investigates the geometric structure of affine Deligne-Lusztig varieties for GL_n with basic b, revealing conditions under which these varieties decompose into classical Deligne-Lusztig varieties times affine spaces.

Contribution

It introduces a new condition on , leading to a geometric decomposition of affine Deligne-Lusztig varieties for GL_n.

Findings

Affine Deligne-Lusztig varieties decompose into classical Deligne-Lusztig varieties times affine spaces under certain conditions.

The paper provides a geometric criterion for the structure of these varieties.

This advances understanding of their geometric and combinatorial properties.

Abstract

In this paper we study the geometric structure of affine Deligne-Lusztig varieties for $GL_n$ and $b$ basic. We introduce a new condition on $\lambda$. If this is satisfied, then the corresponding affine Deligne-Lusztig variety is the disjoint union of classical Deligne-Lusztig varieties times finite-dimensional affine spaces.