The Jordan-Holder series for nearby cycles on some Shimura varieties and affine flag varieties

We study the Jordan-Hoder series for nearby cycles on certain Shimura varieties and Rapoport-Zink local models, and on finite-dimensional pieces of Beilinson's deformation of the affine Grassmannian to the affine flag variety (and their p-adic analogues). We give a formula for the multiplicities of irreducible constituents in terms of certain cohomology groups, and we also provide an algorithm to compute multiplicities, in terms of the affine Hecke algebra.