Advertisement

Transformation Groups

, Volume 24, Issue 1, pp 67–114 | Cite as

CLASSIFICATION OF REDUCTIVE REAL SPHERICAL PAIRS I. THE SIMPLE CASE

  • FRIEDRICH KNOP
  • BERNHARD KRÖTZ
  • TOBIAS PECHER
  • HENRIK SCHLICHTKRULLEmail author
Article

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [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.Google Scholar
  2. [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.Google Scholar
  3. [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.Google Scholar
  4. [4]
    M. Berger, Les espaces symétriques noncompacts, Ann. Sci. École Norm. Sup. (3) 74 (1957), 85–177.Google Scholar
  5. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [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.Google Scholar
  7. [7]
    P. Bravi, G. Pezzini, The spherical systems of the wonderful reductive subgroups, J. Lie Theory 25 (2015), 105–123.MathSciNetzbMATHGoogle Scholar
  8. [8]
    M. Brion, Classification des espaces homogènes sphériques, Compositio Math. 63 (1987), no. 2, 189–208.MathSciNetzbMATHGoogle Scholar
  9. [9]
    S. Chen, On subgroups of the noncompact real exceptional Lie group \( {F}_4^{\ast } \), Math. Ann. 204 (1973), 271–284.MathSciNetCrossRefGoogle Scholar
  10. [10]
    P. Delorme, Formule de Plancherel pour les espaces symétriques réductifs, Ann. of Math. 147 (1998) 417–452.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [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.Google Scholar
  12. [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.Google Scholar
  13. [13]
    Harish-Chandra, Collected Papers, IV, 1970–1983, Springer, Heidelberg, 1984.Google Scholar
  14. [14]
    F. R. Harvey, Spinors and Calibrations, Perspectives in Mathematics, Vol. 9, Academic Press, Boston, MA, 1990.Google Scholar
  15. [15]
    S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, New York, 1978.Google Scholar
  16. [16]
    B. Kimelfeld, Homogeneous domains on flag manifolds, J. Math. Anal. Appl. 121 (1987), 506–588.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [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.MathSciNetzbMATHGoogle Scholar
  18. [18]
    F. Knop, Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc. 9 (1996), 153–174.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    F. Knop, B. Krötz, Reductive group actions, arXiv:1604.01005 (2016).Google Scholar
  20. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    F. Knop, B. Krötz, H. Schlichtkrull, The local structure theorem for real spherical spaces, Compositio Math. 151 (2015), 2145–2159.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    F. Knop, B. Krötz, H. Schlichtkrull, The tempered spectrum of a real spherical space, Acta Math. 218 (2017), 319–383.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    F. Knop, B. Krötz, H. Schlichtkrull, Classification of reductive real spherical pairs II: The semisimple case, arXiv:1703.08048 (2017).Google Scholar
  24. [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.Google Scholar
  25. [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. [26]
    M. Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math. 38 (1979), no. 2, 129–153.MathSciNetzbMATHGoogle Scholar
  27. [27]
    B. Krötz, H. Schlichtkrull, Finite orbit decomposition of real flag manifolds, J. Eur. Math. Soc. 18 (2016), 1391–1403.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    B. Krötz, H. Schlichtkrull, Multiplicity bounds and the subrepresentation theorem for real spherical spaces, Trans. AMS 368 (2016), no. 4, 2749–2762.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    B. Krötz, H. Schlichtkrull, Harmonic analysis for real spherical spaces, Acta Math. Sinica, doi/10.1007/s10114-017-6557-9.Google Scholar
  30. [30]
    LiE, A Computer algebra package for Lie group computations, available at http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/.
  31. [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.Google Scholar
  32. [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.Google Scholar
  33. [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.Google Scholar
  34. [34]
    M. Sato, T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977) 1–155.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    J. Wolf, Harmonic Analysis on Commutative Spaces, Mathematical Surveys and Monographs, Vol. 142, American Mathematical Society, Providence, RI, 2007.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  • FRIEDRICH KNOP
    • 1
  • BERNHARD KRÖTZ
    • 2
  • TOBIAS PECHER
    • 2
  • HENRIK SCHLICHTKRULL
    • 3
    Email author
  1. 1.Department MathematikFAU Erlangen-NürnbergErlangenGermany
  2. 2.Institut für MathematikUniversität PaderbornPaderbornGermany
  3. 3.Department of MathematicsUniversity of CopenhagenCopenhagen ØDenmark

Personalised recommendations