Geometric Langlands duality and forms of reductive groups

The Satake category is the category of perverse sheaves on the affine Grassmannian of a complex reductive group G. The global cohomology functor induces a tensor equivalence between the Satake category and the category of finite-dimensional representations of the split form of the Langlands dual group of G. We give several variants of this result in the non-split case. The representations of quasi-split groups arise as sheaves that are invariant with respect to the semi-linear action of a finite Galois group combined with the natural action of the group of outer automorphisms of G. Moreover, we show that representations of an inner form are given by perverse sheaves with coefficients in a locally constant sheaf of division algebras. However, in this case the fibre functors are only given implicitly. We construct the fibre functors on the Satake category to produce the inner forms of adjoint groups and inner forms of type A in characteristic zero.