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

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.