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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.

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
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