Formulas for the dimensions of some affine Deligne-Lusztig Varieties

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 $b=1$, $G=S{L}_2$, $S{L}_3$, $S{p}_4$. 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.