Mathematische Annalen

, Volume 327, Issue 4, pp 671–721

Reflection functors for quiver varieties and Weyl group actions



We introduce reflectionfunctors on quiver varieties. They are hyper-Kähler isometries between quiver varieties with different parameters, related by elements in the Weyl group. The definition is motivated by the origial reflection functor given by Bernstein-Gelfand-Ponomarev [1], but they behave much nicely. They are isomorphisms and satisfy the Weyl group relations. As an application, we define Weyl group representations of homology groups of quiver varieties. They are analogues of Slodowy’s construction of Springer representations of the Weyl group.


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  1. 1.
    Bernstein, I.N., Gelfand, I.M., Ponomarev, V.A.: Coxeter functors and Gabriel’s theorem. Uspekhi Math. Nauk 28, 19–33Google Scholar
  2. 2.
    Crawley-Boevey, W., Holland, M.P.: Noncommutative deformations of Kleinian singularities. Duke Math. 92, 605–635 (1998)MathSciNetMATHGoogle Scholar
  3. 3.
    Gocho, T., Nakajima, H.: Einstein-Hermitian connections on hyper-Kähler quotients. J. Math. Soc. Japan 44, 43–51 (1992)MathSciNetMATHGoogle Scholar
  4. 4.
    Hitchin, N.J., Karlhede, A., Lindström, U., Roček, M.: Hyperkähler metrics and supersymmetry. Comm. Math. Phys 108, 535–589 (1987)MathSciNetMATHGoogle Scholar
  5. 5.
    Hotta, R.: On Springer’s representations. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28, 863–876 (1981)MathSciNetMATHGoogle Scholar
  6. 6.
    King, A.: Moduli of representations of finite dimensional algebras. Quarterly J. Math. 45, 515–530 (1994)MathSciNetMATHGoogle Scholar
  7. 7.
    Kronheimer, P.B.: The construction of ALE spaces as a hyper-Kähler quotients. J. Diff. Geom. 29, 665–683 (1989)MathSciNetMATHGoogle Scholar
  8. 8.
    Kronheimer, P.B., Nakajima, H.: Yang-Mills instantons on ALE gravitational instantons. Math. Ann. 288, 263–307 (1990)MathSciNetMATHGoogle Scholar
  9. 9.
    Lusztig, G.: Green polynomials and singularities of unipotent classes. Adv. Math. 42, 169–178 (1981)MathSciNetMATHGoogle Scholar
  10. 10.
    Lusztig, G.: On quiver varieties. Adv. Math 136, 141–182 (1998)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Lusztig, G.: Quiver varieties and Weyl group actions. Ann. Inst. Fourier (Grenoble) 50, 461–489 (2000)MathSciNetMATHGoogle Scholar
  12. 12.
    Lusztig, G.: Remarks on quiver varieties. Duke Math. J 105, 239–265 (2000)MathSciNetMATHGoogle Scholar
  13. 13.
    Maffei, A.: A remark on quiver varieties and Weyl groups. Preprint math.AG/0003159Google Scholar
  14. 14.
    Nakajima, H.: Moduli spaces of anti-self-dual connections on ALE gravitational instantons. Invent. Math. 102, 267–303 (1990)MathSciNetMATHGoogle Scholar
  15. 15.
    Nakajima, H.: Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. 76, 365–416 (1994)MathSciNetMATHGoogle Scholar
  16. 16.
    Nakajima, H.: Instanton on ALE spaces and canonical bases. In: ‘‘Proceeding of Symposium on Representation Theory, Yamagata, 1992’’ (in Japanese)Google Scholar
  17. 17.
    Nakajima, H.: Quiver varieties and Kac-Moody algebras. Duke Math. 91, 515–560 (1998)MathSciNetMATHGoogle Scholar
  18. 18.
    Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. Univ. Lect. Ser. 18, AMS, (1999)Google Scholar
  19. 19.
    Slodowy, P.: Four lectures on simple groups and singularities. Comm. Math. Inst. Rijksuniv. Utrecht 11, (1980)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsKyoto UniversityKyotoJapan

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