Abstract
Let k be an algebraically closed field of positive characteristic p. We first classify the D -truncations mod p of Shimura F -crystals over k and then we study stratifications defined by inner isomorphism classes of these D -truncations. This generalizes previous works of Kraft, Ekedahl, Oort, Moonen, and Wedhorn. As a main tool we introduce and study Bruhat F -decompositions; they generalize the combined form of Steinberg theorem and of classical Bruhat decompositions for reductive groups over k (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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