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Dimensions of some affine Deligne-Lusztig varieties
Authors:Ulrich Görtz
Institution:Mathematisches Institut, Universität Bonn, Beringstr. 1, 53115 Bonn, Germany; Mathematics Department, University of Maryland, College Park, MD 20742-4015, USA; Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637, USA; Laboratory of Populations, Rockefeller University, 1230 York Ave., New York, NY 10021, USA
Abstract: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 Xμ(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 Xx(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.
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