15 April 2018 Galois and Cartan cohomology of real groups
Jeffrey Adams, Olivier Taïbi
Duke Math. J. 167(6): 1057-1097 (15 April 2018). DOI: 10.1215/00127094-2017-0052


Suppose that G is a complex, reductive algebraic group. A real form of G is an antiholomorphic involutive automorphism σ, so G(R)=G(C)σ is a real Lie group. Write H1(σ,G) for the Galois cohomology (pointed) set H1(Gal(C/R),G). A Cartan involution for σ is an involutive holomorphic automorphism θ of G, commuting with σ, so that θσ is a compact real form of G. Let H1(θ,G) be the set H1(Z2,G), where the action of the nontrivial element of Z2 is by θ. By analogy with the Galois group, we refer to H1(θ,G) as the Cartan cohomology of G with respect to θ. Cartan’s classification of real forms of a connected group, in terms of their maximal compact subgroups, amounts to an isomorphism H1(σ,Gad)H1(θ,Gad), where Gad is the adjoint group. Our main result is a generalization of this: there is a canonical isomorphism H1(σ,G)H1(θ,G).

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 H1(σ,G) 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 G is connected.


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Jeffrey Adams. Olivier Taïbi. "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


Received: 23 November 2016; Revised: 5 October 2017; Published: 15 April 2018
First available in Project Euclid: 13 March 2018

zbMATH: 06870401
MathSciNet: MR3786301
Digital Object Identifier: 10.1215/00127094-2017-0052

Primary: 11E72
Secondary: 20G10

Keywords: Galois cohomology , Lie groups

Rights: Copyright © 2018 Duke University Press


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Vol.167 • No. 6 • 15 April 2018
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