  This paper concerns the dimensions of certain affine Deligne-Lusztig
varieties, both in the affine Grassmannian and in the affine flag manifold.
Rapoport conjectured a formula for the dimensions of the varieties X_mu(b) in
the affine Grassmannian. We prove his conjecture for b in the split torus; we
find that these varieties are equidimensional; and we reduce the general
conjecture to the case of superbasic b. In the affine flag manifold, we prove a
formula that reduces the dimension question for X_x(b) with b in the split
torus to computations of dimensions of intersections of Iwahori orbits with
orbits of the unipotent radical. Calculations using this formula allow us to
verify a conjecture of Reuman in many new cases, and to make progress toward a
generalization of his conjecture.
