Orbits of real forms in complex flag manifolds

Altomani, Andrea; Medori, Costantino; Nacinovich, Mauro

Altomani, Andrea; Medori, Costantino; Nacinovich, Mauro

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 1, p. 69-109

Abstract

We investigate the CR geometry of the orbits $M$ of a real form $𝐆_{0}$ of a complex semisimple Lie group $𝐆$ in a complex flag manifold $X = 𝐆 / 𝐐$ . We are mainly concerned with finite type and holomorphic nondegeneracy conditions, canonical $𝐆_{0}$ -equivariant and Mostow fibrations, and topological properties of the orbits.

MR 2668874 | Zbl 1198.53051

Classification: 53C30, 14M15, 17B20, 32V05, 32V35, 32V40, 57T20

@article{ASNSP_2010_5_9_1_69_0,
     author = {Altomani, Andrea and Medori, Costantino and Nacinovich, Mauro},
     title = {Orbits of real forms in complex flag manifolds},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {1},
     year = {2010},
     pages = {69-109},
     zbl = {1198.53051},
     mrnumber = {2668874},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2010_5_9_1_69_0}
}

Altomani, Andrea; Medori, Costantino; Nacinovich, Mauro. Orbits of real forms in complex flag manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 1, pp. 69-109. http://www.numdam.org/item/ASNSP_2010_5_9_1_69_0/

References

[1] J. Adams and F. Du Cloux, Algorithims for representation theory of real groups, J. Inst. Math. Jussieu 8 (2009), 209–256. | MR 2485793

[2] A. Altomani, C. Medori and M. Nacinovich, The CR structure of minimal orbits in complex flag manifolds, J. Lie Theory 16 (2006), 483–530. | MR 2248142 | Zbl 1120.32023

[3] A. Altomani, C. Medori and M. Nacinovich, On the topology of minimal orbits in complex flag manifolds, Tohoku Math. J. 60 (2008), 403–422. | MR 2453731 | Zbl 1160.57033

[4] S. Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962), 1–34. | MR 153782

[5] M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, “Real Submanifolds in Complex Space and their Mappings”, Vol. 47, Princeton University Press, Princeton, NJ, 1999. | MR 1668103

[6] J. Berndt, S. Console and A. Fino, On index number and topology of flag manifolds, Differential Geom. Appl. 15 (2001), 81–90. | MR 1845178 | Zbl 1067.53041

[7] T. Bloom and I. Graham, A geometric characterization of points of type $m$ on real submanifolds of $𝐂^{n}$ , J. Differential Geom. 12 (1977), 171–182. | MR 492369 | Zbl 0363.32013

[8] N. Bourbaki, “Éléments de mathématique”, Hermann, Paris, 1975, Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364. | MR 453824 | Zbl 0329.17002

[9] R. Bremigan and J. Lorch, Orbit duality for flag manifolds, Manuscripta Math. 109 (2002), 233–261. | MR 1935032 | Zbl 1019.22013

[10] L. Casian and R. J. Stanton, Schubert cells and representation theory, Invent. Math. 137 (1999), 461–539. | MR 1709874 | Zbl 0954.22011

[11] G. Fels, Locally homogeneous finitely nondegenerate CR-manifolds, Math. Res. Lett. 14 (2007), 893–922. | MR 2357464 | Zbl 1155.32027

[12] G. Fels, A. T. Huckleberry and J. A. Wolf, “Cycle Spaces of Flag Domains. A Complex Geometric Viewpoint”, Progress in Mathematics, Vol. 245, Birkhäuser Boston Inc., Boston, MA, 2006. | MR 2188135 | Zbl 1084.22011

[13] A. Fino, Systems of roots and topology of complex flag manifolds, Geom. Dedicata 71 (1998), 299–308. | MR 1631691 | Zbl 0966.53037

[14] S. Helgason, “Differential Geometry, Lie Groups, and Symmetric Spaces”, Pure and Applied Mathematics, Vol. 80, Academic Press, New York, 1978. | MR 514561 | Zbl 0451.53038

[15] J. E. Humphreys, “Linear Algebraic Groups”, n. 21, Springer-Verlag, New York Graduate Texts in Mathematics, 1975. | MR 396773

[16] A. W. Knapp, “Lie Groups Beyond an Introduction”, second ed., Progress in Mathematics, Vol. 140, Birkhäuser Boston Inc., Boston, MA, 2002. | MR 1920389 | Zbl 1075.22501

[17] T. Matsuki, Closure relations for orbits on affine symmetric spaces under the action of parabolic subgroups. Intersections of associated orbits, Hiroshima Math. J. 18 (1988), 59–67. | MR 935882 | Zbl 0652.53035

[18] C. Medori and M. Nacinovich, Levi-Tanaka algebras and homogeneous CR manifolds, Compositio Math. 109 (1997), 195–250. | MR 1478818 | Zbl 0955.32029

[19] C. Medori and M. Nacinovich, Classification of semisimple Levi-Tanaka algebras, Ann. Mat. Pura Appl. 174 (1998), 285–349. | MR 1746933 | Zbl 0999.17039

[20] C. Medori and M. Nacinovich, Complete nondegenerate locally standard CR manifolds, Math. Ann. 317 (2000), 509–526. | MR 1776115 | Zbl 1037.32031

[21] C. Medori and M. Nacinovich, Algebras of infinitesimal CR automorphisms, J. Algebra 287 (2005), 234–274. | MR 2134266 | Zbl 1132.32013

[22] G. D. Mostow, On covariant fiberings of Klein spaces, Amer. J. Math. 77 (1955), 247–278. | MR 67901 | Zbl 0067.16003

[23] G. D. Mostow, Covariant fiberings of Klein spaces. II, Amer. J. Math. 84 (1962), 466–474. | MR 142688 | Zbl 0123.16303

[24] M. Takeuchi, Two-number of symmetric $R$ -spaces, Nagoya Math. J. 115 (1989), 43–46. | MR 1018081 | Zbl 0662.53042

[25] M. Takeuchi and S. Kobayashi, Minimal imbeddings of $R$ -spaces, J. Differential Geom. 2 (1968), 203–215. | MR 239007 | Zbl 0165.24901

[26] D. A. Vogan Jr. , Irreducible characters of semisimple Lie groups. IV. Character-multiplicity duality, Duke Math. J. 49 (1982), 943–1073. | MR 683010 | Zbl 0536.22022

[27] M. Wiggerman, The fundamental group of a real flag manifold, Indag. Mat. 9 (1998), 141–153. | MR 1618211 | Zbl 0921.22011

[28] J. A. Wolf, The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121–1237. | MR 251246 | Zbl 0183.50901