  Affine Deligne-Lusztig varieties with various level structures show up in the
study of Shimura varieties and moduli spaces of shtukas. Among is the Iwahori
level structure which is the most refined one. We study the nonemptiness
problem of single affine Deligne-Lusztig varieties at Iwahori level in the
basic case. Under a genericity condition (the ``shrunken Weyl chambers''
condition), an explicit criterion is known. However, no explicit criterion has
been available without the condition even conjecturally. We conjecture a new
criterion in full generality, and prove it except for finitely many cases. As
an application, the nonemptiness problem for special cases and a new
conjectural dimension formula are discussed.
