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.

revision 1: highlighted definition of real forms; divided Proposition 8.2 into Prop. 8.2/Corollary 8.3; fixed several typos and 2 references revision 2: added discussion of rational Weyl group and cohomology of spin groups

Cited by in corpus (8)

• A `relative' local Langlands correspondence
• Galois cohomology of reductive algebraic groups over the field of real numbers
• On the Stability and Gelfand Property of Symmetric Pairs
• Galois cohomology of real semisimple groups
• Real homogenous spaces, Galois cohomology, and Reeder puzzles
• Pseudo-real principal $G$-bundles over a real curve
• Solutions of Reeder's Puzzle
• Regular subalgebras and nilpotent orbits of real graded Lie algebras