Weight spectral sequence and Hecke correspondence on Shimura varieties
The purpose of this paper is twofold. First, we elaborate the construction of T. Saito on the action of algebraic correspondence on the l-adic cohomology of varieties over local fields with semistable reduction and derive a recursive formula to compute the actions on the weight spectral sequence. Secondly, applying this to a certain kind of unitary Shimura varieties, we elaborate and complement the work of R. Taylor and the author on the compatibility of local and global Langlands correspondences. More specifically, we explicitly compute the action of (affine) Iwahori-Hecke algebra of GLn over p-adic field on the weight spectral sequence for unitary Shimura varieties with Iwahori level structure. This enables a full representation-theoretic elaboration of the results in the previous work, in particular the key vanishing result on the cohomology of Igusa varieties, and reproves the local-global compatibility somewhat more directly. Also, the local nature of the computation of the Hecke correspondence is interesting in its own right---it is an analogue of the classical Eichler-Shimura congruence relation, which amounts to finding some part of the cohomology of the Shimura variety by directly computing the action of Hecke correspondences on vanishing cycle sheaves.