Geometric Langlands duality and forms of reductive groups

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 ${\check{G}}_k$.