## Duke Mathematical Journal

### Galois and Cartan cohomology of real groups

#### Abstract

Suppose that $G$ is a complex, reductive algebraic group. A real form of $G$ is an antiholomorphic involutive automorphism $\sigma$, so $G(\mathbb{R})=G(\mathbb{C})^{\sigma}$ is a real Lie group. Write $H^{1}(\sigma,G)$ for the Galois cohomology (pointed) set $H^{1}(\operatorname{Gal}(\mathbb{C}/\mathbb{R}),G)$. A Cartan involution for $\sigma$ is an involutive holomorphic automorphism $\theta$ of $G$, commuting with $\sigma$, so that $\theta\sigma$ is a compact real form of $G$. Let $H^{1}(\theta,G)$ be the set $H^{1}(\mathbb{Z}_{2},G)$, where the action of the nontrivial element of $\mathbb{Z}_{2}$ is by $\theta$. By analogy with the Galois group, we refer to $H^{1}(\theta,G)$ as the Cartan cohomology of $G$ with respect to $\theta$. Cartan’s classification of real forms of a connected group, in terms of their maximal compact subgroups, amounts to an isomorphism $H^{1}(\sigma,G_{\mathrm{ad}})\simeq H^{1}(\theta,G_{\mathrm{ad}})$, where $G_{\mathrm{ad}}$ is the adjoint group. Our main result is a generalization of this: there is a canonical isomorphism $H^{1}(\sigma,G)\simeq H^{1}(\theta,G)$.

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 $H^{1}(\sigma,G)$ 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
Revised: 5 October 2017
First available in Project Euclid: 13 March 2018

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
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

#### References

• [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.