Geometric Langlands duality and representations of algebraic groups over commutative rings

TL;DR

This paper presents a geometric reformulation of the Satake isomorphism, establishing an equivalence between dual group representations and perverse sheaves on the affine Grassmannian over arbitrary commutative rings.

Contribution

It introduces a geometric version of the Satake isomorphism applicable over any commutative ring, extending classical representation theory to a broader algebraic context.

Findings

Equivalence between dual group representations and perverse sheaves on affine Grassmannian

Extension of Satake isomorphism to arbitrary commutative rings

Recovery of dual group representation theory over commutative rings

Abstract

In this paper we give a geometric version of the Satake isomorphism. Given a connected complex reductive algebraic group, we show that the category of representations of its Langlands dual is naturally equivalent to a certain category of perverse sheaves on the complex affine Grassmannian. We can work with perverse sheaves with coefficients in an arbitrary commutative ring and then we recover the representation theory of the split form of the dual group over the commutative ring.