Duke Mathematical Journal

Galois and Cartan cohomology of real groups

Jeffrey Adams and Olivier Taïbi

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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

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

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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

Galois cohomology Lie groups


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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  • [1] J. Adams, “Guide to the Atlas software: Computational representation theory of real reductive groups” in Representation Theory of Real Reductive Lie Groups, Contemp. Math. 472, Amer. Math. Soc., Providence, 2008, 1–37.
  • [2] J. Adams, D. Barbasch, and D. A. Vogan, Jr., The Langlands Classification and Irreducible Characters for Real Reductive Groups, Progr. Math. 104, Birkhäuser, Boston, 1992.
  • [3] J. Adams and F. du Cloux, Algorithms for representation theory of real reductive groups, J. Inst. Math. Jussieu 8 (2009), 209–259.
  • [4] J. Adams and D. A. Vogan, Jr., $L$-groups, projective representations, and the Langlands classification, Amer. J. Math. 114 (1992), 45–138.
  • [5] A. Borel, “Automorphic L-functions” in Automorphic Forms, Representations and $L$-Functions (Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math. 33, Amer. Math. Soc., Providence, 1979, 27–61.
  • [6] A. Borel, Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991.
  • [7] A. Borel and J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111–164.
  • [8] M. Borovoi, Galois cohomology of real reductive groups and real forms of simple Lie algebras (in Russian), Funktsional. Anal. i Prilozhen. 22, no. 2 (1988), 63–63; English translation in Funct. Anal. Appl. 22 (1988), 135–136.
  • [9] M. Borovoi, Galois cohomology of reductive algebraic groups over the field of real numbers, preprint, arXiv:1401.5913v1 [math.GR].
  • [10] M. Borovoi and D. A. Timashev, Galois cohomology of real semisimple groups, preprint, arXiv:1506.06252v1 [math.GR].
  • [11] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin, 1990.
  • [12] N. Bourbaki, Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 7–9, Elem. Math. (Berlin), Springer, Berlin, 2005.
  • [13] G. Hochschild, The Structure of Lie Groups, Holden-Day, San Francisco, 1965.
  • [14] T. Kaletha, Rigid inner forms of real and $p$-adic groups, Ann. of Math. (2) 184 (2016), 559–632.
  • [15] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The Book of Involutions, Amer. Math. Soc. Colloq. Publ. 44, Amer. Math. Soc., Providence, 1998.
  • [16] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809.
  • [17] R. E. Kottwitz, Stable trace formula: Cuspidal tempered terms, Duke Math. J. 51 (1984), 611–650.
  • [18] R. E. Kottwitz, Stable trace formula: Elliptic singular terms, Math. Ann. 275 (1986), 365–399.
  • [19] T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), 331–357.
  • [20] G. D. Mostow, Self-adjoint groups, Ann. of Math. (2) 62 (1955), 44–55.
  • [21] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure Appl. Math. 139, Academic Press, Boston, 1994.
  • [22] J. Sekiguchi, Remarks on real nilpotent orbits of a symmetric pair, J. Math. Soc. Japan 39 (1987), 127–138.
  • [23] J.-P. Serre, Galois Cohomology, corrected reprint of 1997 English edition, Springer Monogr. Math., Springer, Berlin, 2002.
  • [24] T. A. Springer, Linear Algebraic Groups, 2nd ed., Progr. Math. 9, Birkhäuser, Boston, 1998.
  • [25] R. Steinberg, Endomorphisms of Linear Algebraic Groups, Mem. Amer. Math. Soc. 80, Amer. Math. Soc., Providence, 1968.
  • [26] D. A. Vogan, Jr., Representations of Real Reductive Lie Groups, Progr. Math. 15, Birkhäuser, Boston, 1981.
  • [27] D. A. Vogan, Jr., Irreducible characters of semisimple Lie groups, IV: Character-multiplicity duality, Duke Math. J. 49 (1982), 943–1073.
  • [28] D. A. Vogan, Jr., “The local Langlands conjecture” in Representation Theory of Groups and Algebras, Contemp. Math. 145, Amer. Math. Soc., Providence, 1993, 305–379.
  • [29] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups, II, Grundlehren Math. Wiss. 189, Springer, New York, 1972.