Duke Mathematical Journal
- Duke Math. J.
- Volume 167, Number 6 (2018), 1057-1097.
Galois and Cartan cohomology of real groups
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.
Duke Math. J., Volume 167, Number 6 (2018), 1057-1097.
Received: 23 November 2016
Revised: 5 October 2017
First available in Project Euclid: 13 March 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11E72: Galois cohomology of linear algebraic groups [See also 20G10]
Secondary: 20G10: Cohomology theory
Adams, Jeffrey; Taïbi, Olivier. Galois and Cartan cohomology of real groups. Duke Math. J. 167 (2018), no. 6, 1057--1097. doi:10.1215/00127094-2017-0052. https://projecteuclid.org/euclid.dmj/1520928011