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Reflection functors for quiver varieties and Weyl group actions

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Abstract

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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References

  1. Bernstein, I.N., Gelfand, I.M., Ponomarev, V.A.: Coxeter functors and Gabriel’s theorem. Uspekhi Math. Nauk 28, 19–33

    Google Scholar 

  2. Crawley-Boevey, W., Holland, M.P.: Noncommutative deformations of Kleinian singularities. Duke Math. 92, 605–635 (1998)

    MathSciNet  MATH  Google Scholar 

  3. Gocho, T., Nakajima, H.: Einstein-Hermitian connections on hyper-Kähler quotients. J. Math. Soc. Japan 44, 43–51 (1992)

    MathSciNet  MATH  Google Scholar 

  4. Hitchin, N.J., Karlhede, A., Lindström, U., Roček, M.: Hyperkähler metrics and supersymmetry. Comm. Math. Phys 108, 535–589 (1987)

    MathSciNet  MATH  Google Scholar 

  5. Hotta, R.: On Springer’s representations. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28, 863–876 (1981)

    MathSciNet  MATH  Google Scholar 

  6. King, A.: Moduli of representations of finite dimensional algebras. Quarterly J. Math. 45, 515–530 (1994)

    MathSciNet  MATH  Google Scholar 

  7. Kronheimer, P.B.: The construction of ALE spaces as a hyper-Kähler quotients. J. Diff. Geom. 29, 665–683 (1989)

    MathSciNet  MATH  Google Scholar 

  8. Kronheimer, P.B., Nakajima, H.: Yang-Mills instantons on ALE gravitational instantons. Math. Ann. 288, 263–307 (1990)

    MathSciNet  MATH  Google Scholar 

  9. Lusztig, G.: Green polynomials and singularities of unipotent classes. Adv. Math. 42, 169–178 (1981)

    MathSciNet  MATH  Google Scholar 

  10. Lusztig, G.: On quiver varieties. Adv. Math 136, 141–182 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lusztig, G.: Quiver varieties and Weyl group actions. Ann. Inst. Fourier (Grenoble) 50, 461–489 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Lusztig, G.: Remarks on quiver varieties. Duke Math. J 105, 239–265 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Maffei, A.: A remark on quiver varieties and Weyl groups. Preprint math.AG/0003159

  14. Nakajima, H.: Moduli spaces of anti-self-dual connections on ALE gravitational instantons. Invent. Math. 102, 267–303 (1990)

    MathSciNet  MATH  Google Scholar 

  15. Nakajima, H.: Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. 76, 365–416 (1994)

    MathSciNet  MATH  Google Scholar 

  16. Nakajima, H.: Instanton on ALE spaces and canonical bases. In: ‘‘Proceeding of Symposium on Representation Theory, Yamagata, 1992’’ (in Japanese)

  17. Nakajima, H.: Quiver varieties and Kac-Moody algebras. Duke Math. 91, 515–560 (1998)

    MathSciNet  MATH  Google Scholar 

  18. Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. Univ. Lect. Ser. 18, AMS, (1999)

  19. Slodowy, P.: Four lectures on simple groups and singularities. Comm. Math. Inst. Rijksuniv. Utrecht 11, (1980)

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Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Hiraku Nakajima

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  1. Hiraku Nakajima
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Correspondence to Hiraku Nakajima.

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Mathematics Subject Classification (2000): Primary 53C26; Secondary 14D21, 16G20, 20F55, 33D80

Supported by the Grant-in-aid for Scientific Research (No.11740011), the Ministry of Education, Japan.

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Nakajima, H. Reflection functors for quiver varieties and Weyl group actions. Math. Ann. 327, 671–721 (2003). https://doi.org/10.1007/s00208-003-0467-0

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