Galois and Cartan cohomology of real groups.

*(English)*Zbl 1410.11036Let \(G\) 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.

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.

Reviewer: Uriya A. First (Vancouver)

##### 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 |

##### Keywords:

Galois cohomology; Lie groups; Cartan involution; strong real form; reductive algebraic group##### Software:

Atlas of Lie Groups
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\textit{J. Adams} and \textit{O. Taïbi}, Duke Math. J. 167, No. 6, 1057--1097 (2018; Zbl 1410.11036)

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##### 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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