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The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties. (English) Zbl 1157.14013

The paper under review deals with algebraic loop groups and perverse shaves on certain homogeneous spaces under these groups. These homogeneous spaces arise from local models for Shimura varieties, and the perverse sheaves from vanishing cycles. As usual, the key is to decompose them into simple objects. For this one needs that they are mixed which is shown (in great generality) using de Jong’s alterations. The main tools involved are traces of Frobenius which define elements in a Hecke-Iwahori algebra.

MSC:

14G35 Modular and Shimura varieties
14C25 Algebraic cycles
14M15 Grassmannians, Schubert varieties, flag manifolds
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