Skip to main content

PARABOLIC CONJUGATION AND COMMUTING VARIETIES

Abstract

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.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    A. Aitken, H. W. Turnbull, An Introduction to the Theory of Canonical Matrices, Dover, New York, 1961.

    MATH  Google 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.

  3. [3]

    K. Bongartz, P. Gabriel, Covering spaces in representation-theory, Invent. Math. 65 (1981/82), no. 3, 331–378.

    MathSciNet  Article  Google Scholar 

  4. [4]

    M. Boos, Finite parabolic conjugation on varieties of nilpotent matrices, Algebr. Represent. Theory 17 (2014), no. 6, 1657–1682.

    MathSciNet  Article  Google Scholar 

  5. [5]

    M. Boos, Staircase algebras and graded nilpotent pairs, J. Pure Appl. Algebra 221 (2017), no. 8, 2032–2052.

    MathSciNet  Article  Google 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.

    MathSciNet  Article  Google 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.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    J. Cheah, Cellular decompositions for nested Hilbert schemes of points, Pacific J. Math. 183 (1998), no. 1, 39–90.

    MathSciNet  Article  Google 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.

  10. [10]

    P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71–103, correction, ibid., 309.

    MathSciNet  Article  Google 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.

  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.

    MathSciNet  Article  Google Scholar 

  13. [13]

    R. Goddard, S. M. Goodwin, On commuting varieties of parabolic subalgebras, J. Pure Appl. Algebra 222 (2018), no. 3, 481–507.

    MathSciNet  Article  Google 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.

    MathSciNet  Article  Google 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.

    MathSciNet  Article  Google 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.

    MathSciNet  Article  Google Scholar 

  17. [17]

    M. E. C. Jordan, Traité des Substitutions et des Équations Algébriques, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1989.

  18. [18]

    A. G. Keeton, Commuting Varieties Associated with Symmetric Pairs, PhD thesis, University of California, San Diego, 1996.

  19. [19]

    S. H. Murray, Conjugacy classes in maximal parabolic subgroups of general linear groups, J. Algebra 233 (2000), 135–155.

    MathSciNet  Article  Google Scholar 

  20. [20]

    H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, University Lecture Series, Vol. 18, American Mathematical Society, 1999.

  21. [21]

    A. Premet, Nilpotent commuting varieties of reductive lie algebras, Invent. Math. 154 (2003), no. 3, 653–683.

    MathSciNet  Article  Google Scholar 

  22. [22]

    R. W. Richardson, Commuting varieties of semisimple lie algebras and algebraic groups, Compositio Math. 38 (1979), no. 3, 311–327.

    MathSciNet  MATH  Google 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.

    MathSciNet  Article  Google Scholar 

  24. [24]

    G. Röhrle, On the modality of parabolic subgroups of linear algebraic groups, Manuscripta Math. 98 (1999), no. 1, 9–20.

    MathSciNet  Article  Google Scholar 

  25. [25]

    J-P. Serre, Espaces fibrés algébriques, Séminaire Claude Chevalley 3 (1958), 1–37.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to M. BULOIS.

Additional information

Supported by Ruhr-University Bochum; benefited from a one month “professeur invité” position at University of Saint-Étienne.

Supported by Université Jean Monnet, Labex MILYON/ANR-10-LABX-0070 and ANR Grant GeoLie/ANR-15-CE40-0012.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

BOOS, M., BULOIS, M. PARABOLIC CONJUGATION AND COMMUTING VARIETIES. Transformation Groups 24, 951–986 (2019). https://doi.org/10.1007/s00031-018-9507-4

Download citation