Relation between the two geometric Satake equivalence via nearby cycle

TL;DR

This paper explores the connection between two versions of the geometric Satake equivalence, one over the Fargues--Fontaine curve and the other using Witt vector affine Grassmannians, through the concept of nearby cycles.

Contribution

It establishes a relationship between two different geometric Satake equivalences by analyzing their connection via nearby cycle techniques.

Findings

Demonstrates the relation between the two geometric Satake equivalences.

Provides a framework to compare different geometric realizations of the Satake equivalence.

Enhances understanding of the geometric structures underlying the Satake correspondence.

Abstract

Fargues and Scholze proved the geometric Satake equivalence over the Fargues--Fontaine curve. On the other hand, Zhu proved the geometric Satake equivalence using a Witt vector affine Grassmannian. In this paper, we explain the relation between the two version of the geometric Satake equivalence via nearby cycle.