Abstract
A complex flag manifold \(\textsf {F}{=}{{\textbf {G}}}/{{\textbf {Q}}}\) decomposes into finitely many real orbits under the action of a real form \({{\textbf {G}}}^\upsigma \) of \({{\textbf {G}}}\). Their embedding into \(\textsf {F}\) defines on them CR manifold structures. We characterize and list all the closed real orbits which are finitely nondegenerate.
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Notes
if there is not such an integer k.This means that its complexification \({\mathfrak {g}}\) is simple.
if there is not such an integer k.References
Altomani, A., Medori, C.: On homogeneous CR manifolds and their CR algebras. Int. J. Geom. Methods Mod. Phys. 3(5–6), 1199–1214 (2006)
Altomani, A., Medori, C., Nacinovich, M.: The CR structure of minimal orbits in complex flag manifolds. J. Lie Theory 16(3), 483–530 (2006)
Altomani, A., Medori, C., Nacinovich, M.: On the topology of minimal orbits in complex flag manifolds. Tohoku Math. J. (2) 60(3), 403–422 (2008)
Altomani, A., Medori, C., Nacinovich, M.: Orbits of real forms in complex flag manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9(1), 69–109 (2010)
Altomani, A., Medori, C., Nacinovich, M.: Reductive compact homogeneous CR manifolds. Transform. Groups 18(2), 289–328 (2013)
Andreotti, A., Fredricks, G.A.: Embeddability of real analytic Cauchy–Riemann manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6(2), 285–304 (1979)
Araki, S.: On root systems and an infinitesimal classification of irreducible symmetric spaces. J. Math. Osaka City Univ. 13, 1–34 (1962)
Bourbaki, N.: Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics (Berlin). Springer-Verlag, Berlin, Translated from the 1975 and 1982 French originals by Andrew Pressley (2005)
Doubrov, B., Merker, J., The, D.: Classification of simply-transitive Levi non-degenerate hypersurfaces in \({\mathbb{C} }^3\). Int. Math. Res. Not. 2022(19), 15421–15473 (2021)
Fels, G.: Locally homogeneous finitely nondegenerate CR-manifolds. Math. Res. Lett. 14(6), 893–922 (2007)
Freeman, M.: Local biholomorphic straightening of real submanifolds. Ann. Math. (2) 106(2), 319–352 (1977)
Gregorovič, J.: On equivalence problem for 2-nondegenerate CR geometries with simple models. Adv. Math. 384, 107718, 35 (2021)
Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, vol. 80. Academic Press, New York (1978)
Kaup, W., Zaitsev, D.: On local CR-transformation of Levi-degenerate group orbits in compact Hermitian symmetric spaces. J. Eur. Math. Soc. (JEMS) 8(3), 465–490 (2006)
Marini, S., Medori, C., Nacinovich, M.: Higher order Levi forms on homogeneous CR manifolds. Math. Z. 299(1–2), 563–589 (2021)
Marini, S., Medori, C., Nacinovich, M., Spiro, A.: On transitive contact and \(CR\) algebras. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 20(2), 771–795 (2020)
Marini, S., Nacinovich, M.: Orbits of real forms, Matsuki duality and \(CR\)-cohomology, Complex and symplectic geometry. Springer INdAM Ser., vol. 21, Springer, Cham, pp. 149–162 (2017)
Medori, C., Nacinovich, M.: Algebras of infinitesimal CR automorphisms. J. Algebra 287(1), 234–274 (2005)
Medori, C., Spiro, A.: The equivalence problem for five-dimensional Levi degenerate CR manifolds. Int. Math. Res. Not. IMRN 20, 5602–5647 (2014)
Montgomery, D.: Simply connected homogeneous spaces. Proc. Am. Math. Soc. 1, 467–469 (1950)
Mostow, G.D.: On maximal subgroups of real Lie groups. Ann. Math. 74(2), 503–517 (1961)
Onishchik, A.L. (Ed.), Lie Groups and Lie Algebras. I. Encyclopaedia of Mathematical Sciences, vol. 20, Springer-Verlag, Berlin (1993)
Santi, A.: Homogeneous models for Levi degenerate CR manifolds. Kyoto J. Math. 60(1), 291–334 (2020)
Sykes, D., Zelenko, I.: Maximal dimension of groups of symmetries of homogeneous 2-nondegenerate cr structures of hypersurface type with a 1-dimensional levi kernel. Transformation Groups (2022)
Wolf, J.A.: The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components. Bull. Am. Math. Soc. 75, 1121–1237 (1969)
Wolf, J.A.: Real groups transitive on complex flag manifolds. Proc. Am. Math. Soc. 129(8), 2483–2487 (2001)
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This work has been financially supported by the Program “FIL-Quota Incentivante” of University of Parma, co-sponsored by Fondazione Cariparma, by the PRIN project “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics”- CUP D94I19000800001, and by the group G.N.S.A.G.A. of I.N.d.A.M.
Appendix A. Elementary conjugation diagrams
Appendix A. Elementary conjugation diagrams
We conclude presenting a table, taken from explicit computations contained in [7, pp.18–21], that summarize all roots conjugation rules in all different subgraph types.
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Marini, S., Medori, C. & Nacinovich, M. On finitely nondegenerate closed homogeneous CR manifolds. Annali di Matematica 202, 2715–2747 (2023). https://doi.org/10.1007/s10231-023-01337-8
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DOI: https://doi.org/10.1007/s10231-023-01337-8