Jeffrey Adams, Olivier Taïbi

Duke Math. J. 167 (6), 1057-1097, (15 April 2018) DOI: 10.1215/00127094-2017-0052
KEYWORDS: Galois cohomology, Lie groups, 11E72, 20G10

Suppose that $G$ is a complex, reductive algebraic group. A real form of $G$ is an antiholomorphic involutive automorphism $\sigma $, so $G\left(\mathbb{R}\right)=G(\mathbb{C}{)}^{\sigma}$ is a real Lie group. Write ${H}^{1}(\sigma ,G)$ for the Galois cohomology (pointed) set ${H}^{1}(Gal(\mathbb{C}/\mathbb{R}),G)$. A Cartan involution for $\sigma $ is an involutive holomorphic automorphism $\theta $ of $G$, commuting with $\sigma $, so that $\theta \sigma $ is a compact real form of $G$. Let ${H}^{1}(\theta ,G)$ be the set ${H}^{1}({\mathbb{Z}}_{2},G)$, where the action of the nontrivial element of ${\mathbb{Z}}_{2}$ is by $\theta $. By analogy with the Galois group, we refer to ${H}^{1}(\theta ,G)$ as the Cartan cohomology of $G$ with respect to $\theta $. Cartan’s classification of real forms of a connected group, in terms of their maximal compact subgroups, amounts to an isomorphism ${H}^{1}(\sigma ,{G}_{\mathrm{ad}})\simeq {H}^{1}(\theta ,{G}_{\mathrm{ad}})$, where ${G}_{\mathrm{ad}}$ is the adjoint group. Our main result is a generalization of this: there is a canonical isomorphism ${H}^{1}(\sigma ,G)\simeq {H}^{1}(\theta ,G)$.

We apply this result to give simple proofs of some well-known structural results: the Kostant–Sekiguchi correspondence of nilpotent orbits; Matsuki duality of orbits on the flag variety; conjugacy classes of Cartan subgroups; and structure of the Weyl group. We also use it to compute ${H}^{1}(\sigma ,G)$ for all simple, simply connected groups and to give a cohomological interpretation of strong real forms. For the applications it is important that we do not assume that $G$ is connected.