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Galois and Cartan cohomology of real groups. (English) Zbl 1410.11036
Let $$G$$ be a complex reductive Lie group. A real form of $$G$$ is an anti-holomorphic involutive group automorphism $$\sigma:G\to G$$; this datum is equivalent to specifying a real algebraic group whose $$\mathbb{C}$$-points are $$G$$ and such that $$\sigma$$ is induced by complex conjugation. The fixed points, $$G^\sigma$$, are then the real points of the given real algebraic group. Call $$\sigma$$ a compact real form if $$G^\sigma$$ is compact. A Cartan involution for a real form $$\sigma$$ is a holomorphic involution $$\theta:G\to G$$ such that $$\sigma\theta=\theta\sigma$$ and $$\theta\sigma$$ is a compact real form. It always exists and is unique up to conjugation by an element of $$(G^\sigma)^0$$; this is known when $$G$$ is connected, and a proof for arbitrary $$G$$ is given in the paper.
Let $$\sigma$$ be a real form of $$G$$ with Cartan involution $$\theta$$. The involutions $$\sigma$$ and $$\theta$$ induce two $$\mathbb{Z}/2\mathbb{Z}$$-actions on $$G$$, giving rise to corresponding $$\mathbb{Z}/2\mathbb{Z}$$-cohomology pointed sets denoted $$\mathrm{H}^1(\sigma,G)$$ and $$\mathrm{H}^1(\theta,G)$$, respectively. The former is the Galois cohomology of the real algebraic group $$G^\sigma$$; the authors call the latter set the Cartan cohomology of $$G$$ (relative to $$\theta$$).
The authors show that there is a canonical isomorphism of pointed sets $\mathrm{H}^1(\sigma,G)\cong \mathrm{H}^1(\theta,G) .$ If $$X$$ is a homogeneous $$G$$-space with commuting involutions $$\sigma_X$$, $$\theta_X$$ compatible with $$\sigma$$ and $$\theta$$, then the authors also show that there is a canonical bijection $$X^{\sigma_X}/G^\sigma\cong X^{\theta_X}/G^\theta$$, provided $$Gx\cap X^{\sigma_X\theta_X}\neq \emptyset$$ for all $$x\in X^{\sigma_X}\cup X^{\theta_X}$$.
The authors use these results to give elegant proofs to the Kostant-Sekiguchi correspondence, the Matsuki duality and a number of facts concerning Cartan subgroups and the Weyl group of $$G$$.
When $$G$$ is connected, the authors proceed with associating with every strong real form of $$G$$ a central invariant taking values in a group which is canonically isomorphic to $$Z^\sigma/\{z^\sigma z\,|\,z\in Z\}$$, with $$Z$$ being the center of $$G$$. They establish a bijection between $$\mathrm{H}^1(\sigma,G)$$ and conjugacy classes of strong real forms of $$G$$ with the same central invariant as $$\sigma$$, and use this bijection together with the previous results to compute $$\mathrm{H}^1(\sigma,G)$$ for $$G$$ simple and simply connected.

MSC:
 11E72 Galois cohomology of linear algebraic groups 22E46 Semisimple Lie groups and their representations 20G20 Linear algebraic groups over the reals, the complexes, the quaternions
Software:
Atlas of Lie Groups
Full Text:
References:
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