Dimensions of some affine Deligne-Lusztig varieties

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.