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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
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