Two monoidal structures on Satake category in mixed characteristic

TL;DR

This paper explains the coincidence of two geometric Satake equivalences in mixed characteristic, highlighting the equivalence of their symmetric monoidal structures on the Satake category, connecting recent advances by Fargues-Scholze and Zhu.

Contribution

It establishes the equivalence of two monoidal structures on the Satake category in mixed characteristic, unifying different approaches by Fargues-Scholze and Zhu.

Findings

Coincidence of two geometric Satake equivalences

Equivalence of symmetric monoidal structures on the Satake category

Unification of approaches in mixed characteristic

Abstract

Fargues and Scholze proved the geometric Satake equivalence over the Fargues-Fontaine curve. This can be transferred to the geometric Satake equivalence concerning a Witt vector affine Grassmannian via nearby cycle. On the other hand, Zhu proved the geometric Satake equivalence concerning a Witt vector affine Grassmannian. In this paper, we explain the coincidence of these two geometric Satake equivalences, including the coincidence of the two symmetric monoidal structures on the Satake category.