# An example of the derived geometrical Satake correspondence over integers

## TL;DR

This paper explores a specific instance of the derived geometric Satake correspondence over integers, linking sheaf morphisms on the affine Grassmannian to coherent sheaf morphisms on the Lie algebra.

## Contribution

It provides an explicit description of morphisms in the derived Satake correspondence over integers, connecting equivariant sheaves on the affine Grassmannian with coherent sheaves on the Lie algebra.

## Key findings

- Describes morphisms of equivariant complexes on the affine Grassmannian.
- Relates sheaf morphisms to coherent sheaf morphisms on the Lie algebra.
- Provides an example of the derived Satake correspondence over integers.

## Abstract

Let G^\vee be a complex simple algebraic group. We describe certain morphisms of G^\vee(\calO)-equivariant complexes of sheaves on the affine Grassmannian \Gr of G^\vee in terms of certain morphisms of G-equivariant coherent sheaves on \frakg, where G is the Langlands dual group of G^\vee and \frakg is its Lie algebra. This can be regarded as an example of the derived Satake correspondence.

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Source: https://tomesphere.com/paper/0910.5702