Closure of Steinberg fibers and 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}.

References in corpus (4)

• A guide to the reduction modulo p of Shimura varieties
• Introduction to Shimura varieties with bad reduction of parahoric type
• Minimal length elements in conjugacy classes of extended affine Weyl group
• $G$-stable pieces and partial flag varieties