The (cyclic) enhanced nilpotent cone via quiver representations

  • Gwyn Bellamy
  • Magdalena BoosEmail author
Open Access


The \({{\,\mathrm{GL}\,}}(V)\)-orbits in the enhanced nilpotent cone \(V\times \mathcal {N}\) are (essentially) in bijection with the orbits of a certain parabolic \(P\subseteq {{\,\mathrm{GL}\,}}(V)\) (the mirabolic subgroup) in the nilpotent cone \(\mathcal {N}\). We give a new parameterization of the orbits in the enhanced nilpotent cone, in terms of representations of the underlying quiver. This parameterization generalizes naturally to the enhanced cyclic nilpotent cone. Our parameterizations are different from the previous ones that have appeared in the literature. Explicit translations between the different parametrizations are given.

Mathematics Subject Classification

Primary 16G20 Secondary 16G60 17B08 



The authors would like to thank C. Johnson for his very precise and valuable ideas regarding the translation between our parametrization and his original parametrization of orbits. We also thank K. Bongartz and M. Reineke for helpful remarks on the subject. The first author was partially supported by EPSRC Grant EP/N005058/1.


  1. 1.
    Achar, P.N., Henderson, A.: Orbit closures in the enhanced nilpotent cone. Adv. Math. 219(1), 27–62 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras, vol. 1, Volume 65 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (2006). (Techniques of representation theory)Google Scholar
  3. 3.
    Bellamy, G., Boos, M.: Semi-simplicity of the category of admissible D-modules. Preprint, arXiv:1709.08986 (2017)
  4. 4.
    Bernstein, J.N.: \(P\)-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-Archimedean case). In: Lie Group Representations, II (College Park, Md., 1982/1983), Volume 1041 of Lecture Notes in Mathematics, pp. 50–102. Springer, Berlin (1984)Google Scholar
  5. 5.
    Bongartz, K.: Algebras and quadratic forms. J. Lond. Math. Soc. (2) 28(3), 461–469 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bongartz, K.: Minimal singularities for representations of Dynkin quivers. Comment. Math. Helv. 69(4), 575–611 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bongartz, K.: On degenerations and extensions of finite-dimensional modules. Adv. Math. 121(2), 245–287 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Boos, M.: Finite parabolic conjugation on varieties of nilpotent matrices. Algebr. Represent. Theory 17(6), 1657–1682 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brüstle, T., de la Peña, J.A., Skowroński, A.: Tame algebras and Tits quadratic forms. Adv. Math. 226(1), 887–951 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    de la Peña, J.A., Skowroński, A.: The Tits forms of tame algebras and their roots. In: Representations of Algebras and Related Topics, EMS Series of Congress Reports, pp. 445–499. European Mathematical Society, Zürich (2011)Google Scholar
  11. 11.
    Drozd, Y.A.: Tame and wild matrix problems. In Representation Theory, II (Proceedings of the Second International Conference, Carleton University, Ottawa, Ontario, 1979), Volume 832 of Lecture Notes in Mathematics, pp. 242–258. Springer, Berlin (1980)Google Scholar
  12. 12.
    Ginzburg, V., Dobrovolska, G., Travkin, R.: Moduli spaces, indecomposable objects and potentials over a finite field. Preprint, arXiv:1612.01733v1 (2016)
  13. 13.
    Gabriel, P.: The universal cover of a representation-finite algebra. In: Representations of Algebras (Puebla, 1980), Volume 903 of Notes in Mathematics, pp. 68–105. Springer, Berlin (1981)Google Scholar
  14. 14.
    Johnson, C.P.: Enhanced Nilpotent Representations of a Cyclic Quiver. ProQuest LLC, Ann Arbor (2010). Thesis (Ph.D.)–The University of UtahGoogle Scholar
  15. 15.
    Kempken, G.: Darstellung, Eine, des Köchers \({\tilde{\text{A}}}_{k}\). Bonner Mathematische Schriften [Bonn Mathematical Publications], 137. Universität Bonn, Mathematisches Institut, Bonn, 1982. Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn (1982)Google Scholar
  16. 16.
    Mautner, C.: Affine pavings and the enhanced nilpotent cone. Proc. Am. Math. Soc. 145(4), 1393–1398 (2017)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Nörenberg, R., Skowroński, A.: Tame minimal non-polynomial growth simply connected algebras. Colloquium Math. 73(2), 301–330 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Serre, J.-P.: Espaces fibrés algébriques (d’après André Weil). In: Séminaire Bourbaki, Vol. 2, pp. Exp. No. 82, 305–311. Société Mathématique de France, Paris (1995)Google Scholar
  19. 19.
    Skowroński, A.: Simply connected algebras and Hochschild cohomologies [ MR1206961 (94e:16016)]. In: Representations of Algebras (Ottawa, ON, 1992), Volume 14 of CMS Conference on Proceedings, pp. 431–447. American Mathematical Society, Providence, RI (1993)Google Scholar
  20. 20.
    Skowroński, A.: Tame algebras with strongly simply connected galois coverings. Colloquium Math. 72(2), 335–351 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sun, M.: Point stabilisers for the enhanced and exotic nilpotent cones. J. Group Theory 14(6), 825–839 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Travkin, R.: Mirabolic Robinson–Schensted–Knuth correspondence. Selecta Math. (N.S.) 14(3–4), 727–758 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Unger, L.: The concealed algebras of the minimal wild, hereditary algebras. Bayreuth. Math. Schr. 31, 145–154 (1990)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Vinberg, È.B.: The Weyl group of a graded Lie algebra. Izv. Akad. Nauk SSSR Ser. Mat. 40(3), 488–526 (1976)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Zwara, G.: Degenerations for modules over representation-finite algebras. Proc. Am. Math. Soc. 127(5), 1313–1322 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zwara, G.: Degenerations of finite-dimensional modules are given by extensions. Compos. Math. 121(2), 205–218 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018
corrected publication 2019

OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK
  2. 2.Fakultät für MathematikRuhr-Universität Bochum, Universitätsstraße 150BochumGermany

Personalised recommendations