Suppose that is a complex, reductive algebraic group. A real form of is an antiholomorphic involutive automorphism , so is a real Lie group. Write for the Galois cohomology (pointed) set . A Cartan involution for is an involutive holomorphic automorphism of , commuting with , so that is a compact real form of . Let be the set , where the action of the nontrivial element of is by . By analogy with the Galois group, we refer to as the Cartan cohomology of with respect to . Cartan’s classification of real forms of a connected group, in terms of their maximal compact subgroups, amounts to an isomorphism , where is the adjoint group. Our main result is a generalization of this: there is a canonical isomorphism .
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 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 is connected.
"Galois and Cartan cohomology of real groups." Duke Math. J. 167 (6) 1057 - 1097, 15 April 2018. https://doi.org/10.1215/00127094-2017-0052