  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}.
