  Rapoport and Kottwitz defined the affine Deligne-Lusztig varieties
$X_{\tilde{w}}^P(b\sigma)$ of a quasisplit connected reductive group $G$ over
$F = \mathbb{F}_q((t))$ for a parahoric subgroup $P$. They asked which pairs
$(b, \tilde{w})$ give non-empty varieties, and in these cases what dimensions
do these varieties have. This paper answers these questions for $P=I$ an
Iwahori subgroup, in the cases $b=1$, $G=SL_2$, $SL_3$, $Sp_4$. This
information is used to get a formula for the dimensions of the
$X_{\tilde{w}}^K(\sigma)$ (all shown to be non-empty by Rapoport and Kottwitz)
for the above $G$ that supports a general conjecture of Rapoport. Here $K$ is a
special maximal compact subgroup.
