Transformation Groups

, Volume 24, Issue 4, pp 951–986 | Cite as


  • M. BOOS
  • M. BULOISEmail author


We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of the general linear group on the variety of nilpotent matrices in its Lie algebra. Lie-theoretically, it is natural to wonder about the number of orbits of this action. We translate the setup to a representation-theoretic one and obtain a finiteness criterion which classifies all actions with only a finite number of orbits over an arbitrary infinite field. These results are applied to commuting varieties and nested punctual Hilbert schemes.


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  1. [1]
    A. Aitken, H. W. Turnbull, An Introduction to the Theory of Canonical Matrices, Dover, New York, 1961.zbMATHGoogle Scholar
  2. [2]
    I. Assem, D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras, Vol. 1, Techniques of Representation Theory, London Mathematical Society Student, Vol. 65, Cambridge University Press, Cambridge, 2006.Google Scholar
  3. [3]
    K. Bongartz, P. Gabriel, Covering spaces in representation-theory, Invent. Math. 65 (1981/82), no. 3, 331–378.MathSciNetCrossRefGoogle Scholar
  4. [4]
    M. Boos, Finite parabolic conjugation on varieties of nilpotent matrices, Algebr. Represent. Theory 17 (2014), no. 6, 1657–1682.MathSciNetCrossRefGoogle Scholar
  5. [5]
    M. Boos, Staircase algebras and graded nilpotent pairs, J. Pure Appl. Algebra 221 (2017), no. 8, 2032–2052.MathSciNetCrossRefGoogle Scholar
  6. [6]
    T. Brüstle, L. Hille, Finite, tame, and wild actions of parabolic subgroups in GL(V) on certain unipotent subgroups. J. Algebra 226 (2000), 347–380.MathSciNetCrossRefGoogle Scholar
  7. [7]
    M. Bulois, L. Evain, Nested punctual hilbert schemes and commuting varieties of parabolic subalgebras, J. Lie Theory 26 (2016), no. 2, 497–533.MathSciNetzbMATHGoogle Scholar
  8. [8]
    J. Cheah, Cellular decompositions for nested Hilbert schemes of points, Pacific J. Math. 183 (1998), no. 1, 39–90.MathSciNetCrossRefGoogle Scholar
  9. [9]
    V. Dlab, C. M. Ringel, The module theoretical approach to quasi-hereditary algebras. in: Representations of Algebras and Related Topics, London Math. Soc. Lecture Note Ser., Vol. 168, Cambridge University Press, Cambridge, 1992, pp. 200–224.Google Scholar
  10. [10]
    P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71–103, correction, ibid., 309.MathSciNetCrossRefGoogle Scholar
  11. [11]
    P. Gabriel, The universal cover of a representation-finite algebra, in: Representations of Algebras (Puebla, 1980), Lecture Notes in Math., Vol. 903, Springer, Berlin, 1981, pp. 68–105.Google Scholar
  12. [12]
    C. Geiss, B. Leclerc, J. Schröer, Quivers with relations for symmetrizable cartan matrices i: Foundations, Invent. Math. 209 (2017), no. 1, 61–158.MathSciNetCrossRefGoogle Scholar
  13. [13]
    R. Goddard, S. M. Goodwin, On commuting varieties of parabolic subalgebras, J. Pure Appl. Algebra 222 (2018), no. 3, 481–507.MathSciNetCrossRefGoogle Scholar
  14. [14]
    S. M. Goodwin, G. Röhrle, On commuting varieties of nilradicals of borel subalgebras of reductive lie algebras, Proc. Edinburgh Math. Soc. (2) 58 (2015), 169–181.MathSciNetCrossRefGoogle Scholar
  15. [15]
    D. Happel, D. Vossieck, Minimal algebras of infinite representation type with pre-projective component, Manuscripta Math. 42 (1983), no. 2-3, 221–243.MathSciNetCrossRefGoogle Scholar
  16. [16]
    L. Hille, G. Röhrle, A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical, Transform. Groups 4 (1999), no. 1, 35–52.MathSciNetCrossRefGoogle Scholar
  17. [17]
    M. E. C. Jordan, Traité des Substitutions et des Équations Algébriques, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1989.Google Scholar
  18. [18]
    A. G. Keeton, Commuting Varieties Associated with Symmetric Pairs, PhD thesis, University of California, San Diego, 1996.Google Scholar
  19. [19]
    S. H. Murray, Conjugacy classes in maximal parabolic subgroups of general linear groups, J. Algebra 233 (2000), 135–155.MathSciNetCrossRefGoogle Scholar
  20. [20]
    H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, University Lecture Series, Vol. 18, American Mathematical Society, 1999.Google Scholar
  21. [21]
    A. Premet, Nilpotent commuting varieties of reductive lie algebras, Invent. Math. 154 (2003), no. 3, 653–683.MathSciNetCrossRefGoogle Scholar
  22. [22]
    R. W. Richardson, Commuting varieties of semisimple lie algebras and algebraic groups, Compositio Math. 38 (1979), no. 3, 311–327.MathSciNetzbMATHGoogle Scholar
  23. [23]
    C. M. Ringel, Iyama’s finiteness theorem via strongly quasi-hereditary algebras, J. Pure Appl. Algebra 214 (2010), no. 9, 1687–1692.MathSciNetCrossRefGoogle Scholar
  24. [24]
    G. Röhrle, On the modality of parabolic subgroups of linear algebraic groups, Manuscripta Math. 98 (1999), no. 1, 9–20.MathSciNetCrossRefGoogle Scholar
  25. [25]
    J-P. Serre, Espaces fibrés algébriques, Séminaire Claude Chevalley 3 (1958), 1–37.Google Scholar

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Authors and Affiliations

  1. 1.Ruhr-University Bochum, Faculty of Mathematics, Chair of AlgebraBochumGermany
  2. 2.Univ Lyon, Université Jean Monnet, Saint-Étienne, CNRS UMR 5208, Institut Camille JordanSaint-EtienneFrance

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