Closure of Steinberg Fibers and Affine Deligne–Lusztig Varieties

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 Xw(1) in the affine flag variety. Among other things, we describe the emptiness/nonemptiness pattern of Xw(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 He [“Unipotent variety in the group compactification.” Advances in Mathematics 203 (2006): 109–31].