Geometric Langlands duality and forms of reductive groups

TL;DR

This paper explores geometric and Tannakian methods to construct and understand various forms of the Langlands dual group, including split, quasi-split, and inner forms, via perverse sheaves and gerbes.

Contribution

It provides a new geometric and Tannakian framework for constructing and analyzing different forms of the Langlands dual group over various fields.

Findings

Constructs split form of the dual group via perverse sheaves.

Provides Tannakian construction of quasi-split forms.

Describes gerbes associated with inner forms of the dual group.

Abstract

The category of perverse sheaves on the affine Grassmannian of a complex reductive group $G$ gives a canonical geometric construction of the split form of the Langlands dual group $\check G_\bZ$ over the integers. Given a field $k$, we give a Tannakian construction of the quasi-split forms of $\check G_k$, as well as a construction of the gerbe associated to an inner form of $\check G_k$.