# The integral geometric Satake equivalence in mixed characteristic

## TL;DR

This paper establishes an equivalence between categories of equivariant perverse sheaves on affine Grassmannians and representations of the Langlands dual group in mixed characteristic, extending geometric Satake to this setting.

## Contribution

It proves the integral geometric Satake equivalence in mixed characteristic, connecting equivariant perverse sheaves with Langlands dual group representations over various coefficient rings.

## Key findings

- Equivalence of categories established for $ ext{Z}_ ext{ell}$ and $ ext{F}_ ext{ell}$ coefficients.
- Extension of geometric Satake to mixed characteristic case.
- Supports Langlands program through categorical correspondence.

## Abstract

Let $k$ be an algebraically closed field of characteristic $p$. Denote by $W(k)$ the ring of Witt vectors of $k$. Let $F$ denote a totally ramified finite extension of $W(k)[1/p]$ and $\mathcal{O}$ the its ring of integers. For a connected reductive group scheme $G$ over $\mathcal{O}$, we study the category $P_{L^+G}(Gr_G,\Lambda)$ of $L^+G$-equivariant perverse sheaves in $\Lambda$-coefficient on the affine Grassmannian $Gr_G$ where $\Lambda=\mathbb{Z}_{\ell}$ and $\mathbb{F}_{\ell}$ and prove it is equivalent as a tensor category to the category of finitely generated $\Lambda$-representations of the Langlands dual group of $G$.