Alcoves associated to special fibers of local models

Abstract

The special fiber of the local model of a PEL Shimura variety with Iwahori-type level structure admits a cellular decomposition. The set of strata is in a natural way a finite subset of the affine Weyl group determined by the Shimura data. In this paper, we generalize and give a new proof of a theorem of Kottwitz and Rapoport concerning the description of this set for the case of linear and symplectic groups. For higher rank groups not of type A_n, we produce counterexamples showing the situation is not as nice in general. Also, we recover a theorem of Deodhar characterizing the Bruhat order on S_n.

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