Convex Polytopes for the Central Degeneration of the Affine Grassmannian

TL;DR

This paper explores the algebraic geometry and combinatorics of the central degeneration in type A affine Grassmannian, focusing on semi-infinite orbits, MV cycles, and related polytopes, revealing new geometric and combinatorial insights.

Contribution

It provides a detailed analysis of the central degeneration of semi-infinite orbits, MV cycles, and their polytopes, connecting these to Demazure modules and affine Deligne-Lusztig varieties.

Findings

Descriptions of the central degeneration of semi-infinite orbits.

Transformations of Mirkovi$ ext{c}$-Vilonen polytopes under degeneration.

Insights into the geometry of Iwahori MV cycles and generalized MV cycles.

Abstract

We study the algebraic geometry and combinatorics of the central degeneration (the degeneration that shows up in local models of Shimura varieties and Gaitsgory's central sheaves) in type A. More specifically, we elucidate the central degeneration of semi-infinite orbits and explain its relations with Levi restriction. Also, we discuss the central degeneration of Mirkovi$\acute{\text{c}}$-Vilonen cycles in the affine Grassmannian, and the corresponding transformations of Mirkovi$\acute{\text{c}}$-Vilonen polytopes. In addition, we shed some light on the geometry of Iwahori MV cycles in the affine Grassmannian and generalized MV cycles in the affine flag variety, which are closely related to Demazure modules and affine Deligne-Lusztig varieties respectively.