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Galois and Cartan cohomology of real groups. (English) Zbl 1410.11036
Let $$G$$ be a complex reductive Lie group. A real form of $$G$$ is an anti-holomorphic involutive group automorphism $$\sigma:G\to G$$; this datum is equivalent to specifying a real algebraic group whose $$\mathbb{C}$$-points are $$G$$ and such that $$\sigma$$ is induced by complex conjugation. The fixed points, $$G^\sigma$$, are then the real points of the given real algebraic group. Call $$\sigma$$ a compact real form if $$G^\sigma$$ is compact. A Cartan involution for a real form $$\sigma$$ is a holomorphic involution $$\theta:G\to G$$ such that $$\sigma\theta=\theta\sigma$$ and $$\theta\sigma$$ is a compact real form. It always exists and is unique up to conjugation by an element of $$(G^\sigma)^0$$; this is known when $$G$$ is connected, and a proof for arbitrary $$G$$ is given in the paper.
Let $$\sigma$$ be a real form of $$G$$ with Cartan involution $$\theta$$. The involutions $$\sigma$$ and $$\theta$$ induce two $$\mathbb{Z}/2\mathbb{Z}$$-actions on $$G$$, giving rise to corresponding $$\mathbb{Z}/2\mathbb{Z}$$-cohomology pointed sets denoted $$\mathrm{H}^1(\sigma,G)$$ and $$\mathrm{H}^1(\theta,G)$$, respectively. The former is the Galois cohomology of the real algebraic group $$G^\sigma$$; the authors call the latter set the Cartan cohomology of $$G$$ (relative to $$\theta$$).
The authors show that there is a canonical isomorphism of pointed sets $\mathrm{H}^1(\sigma,G)\cong \mathrm{H}^1(\theta,G) .$ If $$X$$ is a homogeneous $$G$$-space with commuting involutions $$\sigma_X$$, $$\theta_X$$ compatible with $$\sigma$$ and $$\theta$$, then the authors also show that there is a canonical bijection $$X^{\sigma_X}/G^\sigma\cong X^{\theta_X}/G^\theta$$, provided $$Gx\cap X^{\sigma_X\theta_X}\neq \emptyset$$ for all $$x\in X^{\sigma_X}\cup X^{\theta_X}$$.
The authors use these results to give elegant proofs to the Kostant-Sekiguchi correspondence, the Matsuki duality and a number of facts concerning Cartan subgroups and the Weyl group of $$G$$.
When $$G$$ is connected, the authors proceed with associating with every strong real form of $$G$$ a central invariant taking values in a group which is canonically isomorphic to $$Z^\sigma/\{z^\sigma z\,|\,z\in Z\}$$, with $$Z$$ being the center of $$G$$. They establish a bijection between $$\mathrm{H}^1(\sigma,G)$$ and conjugacy classes of strong real forms of $$G$$ with the same central invariant as $$\sigma$$, and use this bijection together with the previous results to compute $$\mathrm{H}^1(\sigma,G)$$ for $$G$$ simple and simply connected.

##### MSC:
 11E72 Galois cohomology of linear algebraic groups 22E46 Semisimple Lie groups and their representations 20G20 Linear algebraic groups over the reals, the complexes, the quaternions
##### Software:
Atlas of Lie Groups
Full Text:
##### 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. · Zbl 1175.22001 [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. · Zbl 0756.22004 [3] J. Adams and F. du Cloux, Algorithms for representation theory of real reductive groups, J. Inst. Math. Jussieu 8 (2009), 209-259. · Zbl 1221.22017 [4] J. Adams and D. A. Vogan, Jr., $$L$$-groups, projective representations, and the Langlands classification, Amer. J. Math. 114 (1992), 45-138. · Zbl 0760.22013 [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. · Zbl 0726.20030 [7] A. Borel and J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111-164. · Zbl 0143.05901 [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. · Zbl 1139.17002 [13] G. Hochschild, The Structure of Lie Groups, Holden-Day, San Francisco, 1965. · Zbl 0131.02702 [14] T. Kaletha, Rigid inner forms of real and $$p$$-adic groups, Ann. of Math. (2) 184 (2016), 559-632. · Zbl 1393.22009 [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. · Zbl 0955.16001 [16] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753-809. · Zbl 0224.22013 [17] R. E. Kottwitz, Stable trace formula: Cuspidal tempered terms, Duke Math. J. 51 (1984), 611-650. · Zbl 0576.22020 [18] R. E. Kottwitz, Stable trace formula: Elliptic singular terms, Math. Ann. 275 (1986), 365-399. · Zbl 0577.10028 [19] T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), 331-357. · Zbl 0396.53025 [20] G. D. Mostow, Self-adjoint groups, Ann. of Math. (2) 62 (1955), 44-55. · Zbl 0065.01404 [21] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure Appl. Math. 139, Academic Press, Boston, 1994. · Zbl 0841.20046 [22] J. Sekiguchi, Remarks on real nilpotent orbits of a symmetric pair, J. Math. Soc. Japan 39 (1987), 127-138. · Zbl 0627.22008 [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. · Zbl 0927.20024 [25] R. Steinberg, Endomorphisms of Linear Algebraic Groups, Mem. Amer. Math. Soc. 80, Amer. Math. Soc., Providence, 1968. · Zbl 0164.02902 [26] D. A. Vogan, Jr., Representations of Real Reductive Lie Groups, Progr. Math. 15, Birkhäuser, Boston, 1981. · Zbl 0469.22012 [27] D. A. Vogan, Jr., Irreducible characters of semisimple Lie groups, IV: Character-multiplicity duality, Duke Math. J. 49 (1982), 943-1073. · Zbl 0536.22022 [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. · Zbl 0802.22005 [29] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups, II, Grundlehren Math. Wiss. 189, Springer, New York, 1972. · Zbl 0265.22021
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