  We study the $J_b(F)$-action on the set of top-dimensional irreducible
components of affine Deligne--Lusztig varieties in the affine Grassmannian. We
show that the stabilizer of any such component is a parahoric subgroup of
$J_b(F)$ of maximal volume, verifying a conjecture of X.~Zhu. As an
application, we give a description of the set of top-dimensional irreducible
components in the basic locus of Shimura varieties.
