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.

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.