Derived geometric Satake equivalence on the Beilinson-Drinfeld Grassmannian with one leg in mixed characteristic

TL;DR

This paper extends the geometric Satake equivalence to a derived setting on the Beilinson-Drinfeld Grassmannian with one leg in mixed characteristic, advancing the geometric Langlands program.

Contribution

It proves the derived geometric Satake equivalence with one leg in mixed characteristic, building on Fargues-Scholze's framework for the geometric Langlands program.

Findings

Established derived category equivalence for etale sheaves on the local Hecke stack

Connected the category of sheaves to L-group-equivariant perfect complexes

Extended geometric Satake to mixed characteristic setting

Abstract

Fargues-Scholze developed a framework for the geometric Langlands program on the Fargues-Fontaine curve. In particular, they proved the geometric Satake equivalence on the moduli space of closed Cartier divisors on the curve. We prove the derived version of this equivalence with one leg. Namely, we show that the derived category of etale sheaves on the local Hecke stack is equivalent to the category of L-group-equivariant perfect complexes over the symmetric algebra of the shifted and (-1)-Tate-twisted Lie algebra.