Transformation Groups

, Volume 18, Issue 2, pp 507–537 | Cite as

Commuting involutions and degenerations of isotropy representations

  • Dmitri I. Panyushev

Let σ 1 and σ 2 be commuting involutions of a semisimple algebraic group G. This yields a \( {{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2} \)-grading of \( \mathfrak{g} \) = Lie(G), \( \mathfrak{g}={\oplus_{i,j=0,1 }}{{\mathfrak{g}}_{ij }} \), and we study invariant-theoretic aspects of this decomposition. Let \( \mathfrak{g}\left\langle {{\sigma_1}} \right\rangle \) be the \( {{\mathbb{Z}}_2} \)-contraction of \( \mathfrak{g} \) determined by σ 1. Then both σ 2 and σ 3 := σ 1 σ 2 remain involutions of the non-reductive Lie algebra \( \mathfrak{g}\left\langle {{\sigma_1}} \right\rangle \). The isotropy representations related to \( \left( {\mathfrak{g}\left\langle {{\sigma_1}} \right\rangle, {\sigma_2}} \right) \) and \( \left( {\mathfrak{g}\left\langle {{\sigma_1}} \right\rangle, {\sigma_3}} \right) \) are degenerations of the isotropy representations related to \( \left( {\mathfrak{g},{\sigma_2}} \right) \) and \( \left( {\mathfrak{g},{\sigma_3}} \right) \), respectively. These degenerated isotropy representations retain many good properties. For instance, they always have a generic stabiliser and their algebras of invariants are often polynomial. We also develop some theory on Cartan subspaces for various \( {{\mathbb{Z}}_2} \)-gradings associated with the \( {{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2} \)-grading of \( \mathfrak{g} \) and study the special case in which σ 1 and σ 2 are conjugate.


Symmetric Space Conjugacy Class Algebraic Group Maximal Rank Cartan Subalgebra 
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  1. [1]
    Л. B. Антонян, O классификацию однородных элементы \( {{\mathbb{Z}}_2} \) -градуированных полупростых алгебр Ли, Вестник Моск. Ун-та, Cер. 1 Матем Мех. (1982), no. 2, 29–34. Engl. transl.: L. V. Antonyan, On the classification of homogeneous elements of \( {{\mathbb{Z}}_2} \)-graded semisimple Lie algebras, Moscow Univ. Math. Bull. 37 (1982), no. 2, 36–43.Google Scholar
  2. [2]
    A. Bialynicki-Birula, On homogeneous affine spaces of linear algebraic groups, Amer. J. Math. 85 (1963), 577–582.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    M. Bulois, Sheets of symmetric Lie algebras and Slodowy slices, J. Lie Theory 21 (2011), 1–54.MathSciNetMATHGoogle Scholar
  4. [4]
    M. Duo, Opérateurs différentiels invariants sur un espace symétrique, C. R. Acad. Sci., Paris, Sér. A−B 289 (1979), no. 2, A135−A137.Google Scholar
  5. [5]
    F. Gonzalez, S. Helgason, Invariant differential operators on Grassmann manifolds, Adv. Math. 60 (1986), 81–91.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.MATHGoogle Scholar
  7. [7]
    A. Helminck, G. Schwarz, Orbits and invariants associated with a pair of commuting involutions, Duke Math. J. 106 (2001), 237–279.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    A. Helminck, G. Schwarz, Smoothness of quotients associated with a pair of commuting involutions, Canad. J. Math. 56 (2004), 945–962.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    A. Kollross, Exceptional \( {{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2} \)-symmetric spaces, Pacific J. Math. 242 (2009), no. 1, 113–130.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    D. Luna, Sur les orbites fermées des groupes algébriques réductifs, Invent. Math. 16 (1972), 1–5.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    T. Matsuki, Double coset decompositions of reductive Lie groups arising from two involutions, J. Algebra 197 (1997), 49–91.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    A. L. Onishchik, E. B. Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag, Berlin, 1990.MATHCrossRefGoogle Scholar
  15. [15]
    D. I. Panyushev, On spherical nilpotent orbits and beyond, Ann. Inst. Fourier 49 (1999), 1453–1476.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    D. I. Panyushev, On invariant theory of θ-groups, J. Algebra 283 (2005), 655–670.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    D. I. Panyushev, Semi-direct products of Lie algebras and their invariants, Publ. R.I.M.S. 43 (2007), no. 4, 1199–1257.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    D. I. Panyushev, On the coadjoint representation of \( {{\mathbb{Z}}_2} \)-contractions of reductive Lie algebras, Adv. Math. 213 (2007), 380–404.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Э. Б. Винберг, В. Л. Попов, Теория инвариантов, Итоги науки и техн., Соврем. пробл. матем., Фундам. направл., т. 55, Алгебраическaя геометрия−4, ВИНИТИ, М., 1989, cтр. 137–314. Engl. transl.: V. L. Popov, E. B. Vinberg, Invariant Theory, in: Algebraic Geometry, IV, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer-Verlag, Berlin, 1994, pp. 123–284.Google Scholar
  20. [20]
    R. W. Richardson, Orbits, invariants, and representations associated to involutions of reductive groups, Invent. Math. 66 (1982), no. 2, 287–312.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    R. W. Richardson, Normality of G-stable subvarieties of a semisimple Lie algebra, in: Algebraic Groups’, Lecture Notes Mathematics, Vol. 1271, Springer, Berlin, 1987, pp. 243–264.Google Scholar
  22. [22]
    R. W. Richardson, Derivatives of invariant polynomials on a semisimple Lie algebra, in: Miniconference on Harmonic Analysis and Operator Algebras (Canberra, 1987), Proc. Centre Math. Anal. Austral. Nat. Univ., Vol. 15, Austral. Nat. Univ., Canberra, 1987, pp. 228–241.Google Scholar
  23. [23]
    M. Rosenlicht, On quotient varieties and the affine embedding of certain homogeneous spaces, Trans. Amer. Math. Soc. 101 (1961), 211–223.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    G. Schwarz, C. Zhu, Invariant differential operators on symplectic Grassmann manifolds, Manuscr. Math. 82 (1994), no. 2, 191–206.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    W. Smoke, Commutativity of the invariant differential operators on a symmetric space, Proc. Amer. Math. Soc. 19 (1968), 222–224.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    T. A. Springer, Regular elements of finite reection groups, Invent. Math. 25 (1974), 159–198.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    M. Vergne, Instantons et correspondance de Kostant−Sekiguchi, C. R. Acad. Sci., Paris, Sér. I 320 (1995), no. 8, 901–906.MathSciNetMATHGoogle Scholar
  28. [28]
    E. B. Vinberg, Short SO3-structures on simple Lie algebras and associated quasiel-liptic planes, in: E. Vinberg ed., Lie Groups and Invariant Theory, American Mathematical Society Translations, Series 2, Vol. 213, 2005, American Mathematical Society, Providence, RI, pp. 243–270.Google Scholar
  29. [29]
    O. S. Yakimova, One-parameter contractions of Lie-Poisson brackets, J. Europ. Math. Soc. (2013), to appear, arXiv:1202.3009.Google Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsRASMoscowRussia

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