Closure of Steinberg fibers and affine Deligne-Lusztig varieties

TL;DR

This paper explores the relationship between Steinberg fiber closures in the wonderful compactification and affine Deligne-Lusztig varieties, providing new insights and proofs regarding their structure and nonemptiness patterns.

Contribution

It establishes connections between Steinberg fiber closures and affine Deligne-Lusztig varieties, offering a new proof of the explicit description of Steinberg fiber closures.

Findings

Describes the emptiness/nonemptiness pattern of affine Deligne-Lusztig varieties with quasi-regular translation.

Provides a new proof of the explicit description of Steinberg fiber closures.

Connects geometric structures in algebraic groups with affine Deligne-Lusztig varieties.

Abstract

We discuss some connections between the closure $\bar F$ of a Steinberg fiber in the wonderful compactification of an adjoint group and the affine Deligne-Lusztig varieties $X_w(1)$ in the affine flag variety. Among other things, we describe the emptiness/nonemptiness pattern of $X_w(1)$ if the translation part of $w$ is quasi-regular. As a by-product, we give a new proof of the explicit description of $\bar F$, first obtained in \cite{H1}.