Duke Mathematical Journal

Galois and Cartan cohomology of real groups

Jeffrey Adams and Olivier Taïbi

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Abstract

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.

Article information

Source
Duke Math. J., Volume 167, Number 6 (2018), 1057-1097.

Dates
Received: 23 November 2016
Revised: 5 October 2017
First available in Project Euclid: 13 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1520928011

Digital Object Identifier
doi:10.1215/00127094-2017-0052

Mathematical Reviews number (MathSciNet)
MR3786301

Zentralblatt MATH identifier
06870401

Subjects
Primary: 11E72: Galois cohomology of linear algebraic groups [See also 20G10]
Secondary: 20G10: Cohomology theory

Keywords
Galois cohomology Lie groups

Citation

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


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