Dimension formula of the affine Deligne-Lusztig variety $X(\mu, b)$

TL;DR

This paper provides an explicit formula for the dimension of affine Deligne-Lusztig varieties $X(, b)$ for large dominant coweights, advancing understanding of their geometric structure in relation to Shimura varieties.

Contribution

It introduces a new explicit dimension formula for affine Deligne-Lusztig varieties $X(, b)$ when the coweight is sufficiently large, linking geometric and arithmetic aspects.

Findings

Explicit dimension formula for $X(, b)$

Applicable to large dominant coweights

Enhances understanding of affine Deligne-Lusztig varieties

Abstract

The study of certain union $X(\mu, b)$ of affine Deligne-Lusztig varieties in the affine flag varieties arose from the study of Shimura varieties with Iwahori level structure. In this paper, we give an explicit formula of $\dim X(\mu, b)$ for sufficiently large dominant coweight $\mu$.