Galois and Cartan Cohomology of Real Groups

Abstract

Real forms of a complex reductive group are classified by Galois cohomology H^1(Gamma,G_ad) where G_ad is the adjoint group. Cartan's classification of real forms in terms of maximal compact subgroups can be stated in terms of H^(Z/2Z,G_ad) where the action is by a (holomorphic) Cartan involution. The main result is that for any complex reductive group, possibly disconnected, there is a canonical isomorphism between H^1(Gamma,G) and H^1(Z/2Z,G). As applications we give short proofs of some well known results, including the Sekiguchi correspondence, Matsuki duality, results on Cartan subgroups, the rational Weyl group, and strong real forms. We also compute H^1(Gamma,G) for all simple, simply connected real groups.