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

TL;DR

This paper investigates the structure of nearby cycles on Shimura varieties and related geometric objects, providing formulas and algorithms to compute the multiplicities of their irreducible components using cohomology and affine Hecke algebra techniques.

Contribution

It introduces a new formula and an algorithm for calculating the Jordan-Hoelder series of nearby cycles on Shimura varieties and affine flag varieties, linking geometric and algebraic methods.

Findings

Derived a formula for multiplicities of irreducible constituents.

Developed an algorithm to compute multiplicities via affine Hecke algebra.

Applied results to Rapoport-Zink models and deformation of affine Grassmannian.

Abstract

We study the Jordan-Hoelder 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.