Galois and Cartan Cohomology of Real Groups

TL;DR

This paper establishes a canonical isomorphism between Galois and Cartan cohomology for complex reductive groups, providing new proofs and computations related to real forms, Cartan subgroups, and dualities.

Contribution

It introduces a canonical isomorphism between Galois and Cartan cohomology for complex reductive groups, unifying various classification results and simplifying proofs.

Findings

Proves a canonical isomorphism between H^1(Gamma,G) and H^1(Z/2Z,G).

Provides explicit computations of H^1(Gamma,G) for all simple, simply connected real groups.

Offers new proofs for the Sekiguchi correspondence, Matsuki duality, and related results.

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.