Geometric Satake equivalence in mixed characteristic and Springer correspondence

TL;DR

This paper establishes a connection between the geometric Satake equivalence and Springer correspondence in mixed characteristic settings, constructing monoidal structures and isomorphisms under ramification assumptions.

Contribution

It extends the geometric Satake equivalence to mixed characteristic affine Grassmannians and constructs related monoidal structures and isomorphisms.

Findings

Proved the Satake-Springer relation for étale sheaves in mixed characteristic.

Constructed a monoidal structure on the restriction functor of Satake categories.

Established a canonical isomorphism between mixed and equal characteristic affine Grassmannians.

Abstract

The geometric Satake equivalence and the Springer correspondence are closely related when restricting to small representations of the Langlands dual group. We prove this result for \'etale sheaves, including the case of the mixed characteristic affine Grassmannian, assuming a sufficient ramification. In this process, we construct a monoidal structure on the restriction functor of Satake categories. We construct also a canonical isomorphism between a mixed characteristic affine Grassmannian under a sufficient ramification and an equal characteristic one.