Galois Cohomology of Real Groups

Abstract

Real forms of a complex reductive group are classified in terms of Galois cohomology $H^1(\Gamma,G_{ad})$ where $G_{ad}$ is the adjoint group. Alternatively, the theory of the Cartan involution gives a description in terms of cohomology with respect to a holomorphic involution: $H^1(\mathbb Z/2\mathbb Z,G_{ad})$ where the non trivial element acts by a holomorphic involution $\theta$. The main theorem is that in general, if $\theta$ is the Cartan involution of a real form $\sigma$, there is a canonical isomorphism $H^1(\Gamma,G)\simeq H^1(\mathbb Z/2\mathbb Z,G)$. This has applications to the structure and representation theory of real groups. We give two such applications. The first is a simple proof of Matsuki's result on conjugacy classes of tori in real groups. The second is a computation of $H^1(\Gamma,G)$ in general. The answer is expressed in terms of the notion of strong real forms. We include tables for all simply connected simple groups.