Skip to main content
SpringerLink
Log in

CLASSIFICATION OF REDUCTIVE REAL SPHERICAL PAIRS I. THE SIMPLE CASE

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

This paper gives a classification of all pairs \( \left(\mathfrak{g},\mathfrak{h}\right) \) with \( \mathfrak{g} \) a simple real Lie \( \mathfrak{h}\subset \mathfrak{g} \) algebra and a reductive subalgebra for which there exists a minimal parabolic subalgebra \( \mathfrak{p}\subset \mathfrak{g} \) such that \( \mathfrak{g}=\mathfrak{h}+\mathfrak{p} \) as vector sum.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Akhiezer, Real group orbits in flag manifolds, in: Lie groups: Structure, Actions, and Representations, Progress in Mathematics, Vol. 306, Birkhäuser, Basel, 2013, pp. 1–24.

  2. Е. М. Андреев, Э. Б. Винберг, А. Г. Элашвили, Орбиты наибольшей размерности полупростых линейных групп Ли, Функ. и его прил. 1. (1967), вьш. 4, 3–7. Engl. transl.: E. M. Andreev, É. B. Vinberg, A. G. Élashvili, Orbits of greatest dimension in semi-simple linear Lie groups, Functional Anal. Appl. 1 (1967), 257–261.

  3. E. P. van den Ban, H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space. I–II, Invent. Math. 161 (2005), 453–566 and 567–628.

  4. M. Berger, Les espaces symétriques noncompacts, Ann. Sci. École Norm. Sup. (3) 74 (1957), 85–177.

  5. A. Borel and J. de Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200–221.

    Article  MathSciNet  MATH  Google Scholar 

  6. N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968.

  7. P. Bravi, G. Pezzini, The spherical systems of the wonderful reductive subgroups, J. Lie Theory 25 (2015), 105–123.

    MathSciNet  MATH  Google Scholar 

  8. M. Brion, Classification des espaces homogènes sphériques, Compositio Math. 63 (1987), no. 2, 189–208.

    MathSciNet  MATH  Google Scholar 

  9. S. Chen, On subgroups of the noncompact real exceptional Lie group \( {F}_4^{\ast } \), Math. Ann. 204 (1973), 271–284.

    Article  MathSciNet  Google Scholar 

  10. P. Delorme, Formule de Plancherel pour les espaces symétriques réductifs, Ann. of Math. 147 (1998) 417–452.

    Article  MathSciNet  MATH  Google Scholar 

  11. Е. Б. Дынкин, Максимальные подгруппы классических групп, Труды ММО 1 (1952), 39–166. Engl. transl.: E. B. Dynkin, The maximal subgroups of the classical groups, in: Selected Papers by E. B. Dynkin with Commentary, Amer. Math Soc. and Internat. Press of Boston, 2000, pp. 37–174.

  12. Е. Б. Дынкин, Полупростые подалгебры полупростых алгебр Ли, Матем, сб. 30(72) (1952), no. 2, 349–462. Engl. transl.: E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, in: Selected Papers by E. B. Dynkin with Commentary, Amer. Math Soc. and Internat. Press of Boston, 2000, pp. 175–312.

  13. Harish-Chandra, Collected Papers, IV, 1970–1983, Springer, Heidelberg, 1984.

  14. F. R. Harvey, Spinors and Calibrations, Perspectives in Mathematics, Vol. 9, Academic Press, Boston, MA, 1990.

  15. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, New York, 1978.

  16. B. Kimelfeld, Homogeneous domains on flag manifolds, J. Math. Anal. Appl. 121 (1987), 506–588.

    Article  MathSciNet  MATH  Google Scholar 

  17. L. Knauss, C. Miebach, Classification of spherical algebraic subalgebras of real simple Lie algebras of rank 1, J. Lie Theory 28 (2018), 265–307.

    MathSciNet  MATH  Google Scholar 

  18. F. Knop, Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc. 9 (1996), 153–174.

    Article  MathSciNet  MATH  Google Scholar 

  19. F. Knop, B. Krötz, Reductive group actions, arXiv:1604.01005 (2016).

  20. F. Knop, B. Krötz, E. Sayag, H. Schlichtkrull, Volume growth, temperedness and integrability of matrix coefficients on a real spherical space, J. Funct. Anal. 271 (2016), 12–36.

    Article  MathSciNet  MATH  Google Scholar 

  21. F. Knop, B. Krötz, H. Schlichtkrull, The local structure theorem for real spherical spaces, Compositio Math. 151 (2015), 2145–2159.

    Article  MathSciNet  MATH  Google Scholar 

  22. F. Knop, B. Krötz, H. Schlichtkrull, The tempered spectrum of a real spherical space, Acta Math. 218 (2017), 319–383.

    Article  MathSciNet  MATH  Google Scholar 

  23. F. Knop, B. Krötz, H. Schlichtkrull, Classification of reductive real spherical pairs II: The semisimple case, arXiv:1703.08048 (2017).

  24. Б. П. Комраков, Максимальные подалгебры вещественных алгебр Ли и проблема Софуса Ли, ДАН СССР 311 (1990), no. 3, 528–532. Engl. transl.: B. P. Komrakov, Maximal subalgebras of real Lie algebras and a problem of Sophus Lie, Soviet Math. Dokl. 41 (1990), no. 2, 269–273.

  25. B. P. Komrakov, Primitive actions and the Sophus Lie problem, in: The Sophus Lie Memorial Conference, Scand. Univ. Press, Oslo, 1994, pp. 187–269.

    Google Scholar 

  26. M. Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math. 38 (1979), no. 2, 129–153.

    MathSciNet  MATH  Google Scholar 

  27. B. Krötz, H. Schlichtkrull, Finite orbit decomposition of real flag manifolds, J. Eur. Math. Soc. 18 (2016), 1391–1403.

    Article  MathSciNet  MATH  Google Scholar 

  28. B. Krötz, H. Schlichtkrull, Multiplicity bounds and the subrepresentation theorem for real spherical spaces, Trans. AMS 368 (2016), no. 4, 2749–2762.

    Article  MathSciNet  MATH  Google Scholar 

  29. B. Krötz, H. Schlichtkrull, Harmonic analysis for real spherical spaces, Acta Math. Sinica, doi/10.1007/s10114-017-6557-9.

  30. LiE, A Computer algebra package for Lie group computations, available at http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/.

  31. И. В. Микюк, Об интегрируемости инвариантных гамильтоновых систем с однородными конфигурационными пространствами, Матем. сб. 129(171) (1986), no. 4, 514–534. Engl. transl.: I. V. Mikityuk, On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Math. USSR-Sbornik 57 (1987), no. 2, 527–546.

  32. А. Л. Онищик, отношения включения между транзитивными компактными группами преобразований, Труды. ММО 11 (1962), 199–242. Engl. transl.: A. L. Onishchik, Inclusion relations among transitive compact transformation groups, Amer. Math. Soc. Transl. (2) 50 (1966), 5–58.

  33. А. Л. Онищик, Разложения редуктивных групп Ли, Матем. сб. 80(122) (1969), no. 4(12), 553–599. Engl. transl.: A. L. Onishchik, Decompositions of reductive Lie groups, Math. USSR-Sbornik 9 (1969), 515–554.

  34. M. Sato, T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977) 1–155.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Wolf, Harmonic Analysis on Commutative Spaces, Mathematical Surveys and Monographs, Vol. 142, American Mathematical Society, Providence, RI, 2007.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department Mathematik, FAU Erlangen-Nürnberg, Cauerstraβe 11, 91058, Erlangen, Germany

    FRIEDRICH KNOP

  2. Institut für Mathematik, Universität Paderborn, Warburger Straβe 100, D-33098, Paderborn, Germany

    BERNHARD KRÖTZ & TOBIAS PECHER

  3. Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100, Copenhagen Ø, Denmark

    HENRIK SCHLICHTKRULL

Authors
  1. FRIEDRICH KNOP
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. BERNHARD KRÖTZ
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. TOBIAS PECHER
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. HENRIK SCHLICHTKRULL
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to HENRIK SCHLICHTKRULL.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

KNOP, F., KRÖTZ, B., PECHER, T. et al. CLASSIFICATION OF REDUCTIVE REAL SPHERICAL PAIRS I. THE SIMPLE CASE. Transformation Groups 24, 67–114 (2019). https://doi.org/10.1007/s00031-017-9470-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-017-9470-5

Search

Navigation