Advertisement

Mathematische Annalen

, Volume 335, Issue 1, pp 31–46 | Cite as

Quiver varieties and Demazure modules

  • Alistair Savage
Article

Abstract

Using subvarieties, which we call Demazure quiver varieties, of the quiver varieties of Nakajima, we give a geometric realization of Demazure modules of Kac-Moody algebras with symmetric Cartan data. We give a natural geometric characterization of the extremal weights of a representation and show that Lusztig's semicanonical basis is compatible with the filtration of a representation by Demazure modules. For the case of Open image in new window , we give a characterization of the Demazure quiver variety in terms of a nilpotency condition on quiver representations and an explicit combinatorial description of the Demazure crystal in terms of Young pyramids.

Mathematics Subject Classification (2000)

Primary 16G20 17B37 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Foda, O., Misra, K.C., Okado, M.: Demazure modules and vertex models: the Open image in new window(2) case. J. Math. Phys., 39(3), 1601–1622 (1998)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Frenkel, I., Khovanov, M., Schiffmann, O.: Homological realization of Nakajima varieties and Weyl group actions. arXiv:math.QA/0311485Google Scholar
  3. 3.
    Geiss, C., Leclerc, B., Schröer, J.: Semicanonical bases and preprojective algebras. arXiv:math.RT/0402448Google Scholar
  4. 4.
    Grojnowski, I., Lusztig, G.: A comparison of bases of quantized enveloping algebras. In Linear algebraic groups and their representations (Los Angeles, CA, 1992), vol. 153 of Contemp. Math. Amer. Math. Soc. Providence, RI, pp 11–19, 1993Google Scholar
  5. 5.
    Kang, S.-J.: Crystal bases for quantum affine algebras and combinatorics of Young walls. Proc. London Math. Soc. (3) 86(1), 29–69 (2003)Google Scholar
  6. 6.
    Kang, S.-J., Lee, H.: Higher level affine crystals and young walls. arXiv:math.QA/0310430Google Scholar
  7. 7.
    Kashiwara, M.: The crystal base and Littelmann's refined Demazure character formula. Duke Math. J. 71(3), 839–858 (1993)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Kashiwara, M., Saito, Y.: Geometric construction of crystal bases. Duke Math. J. 89(1), 9–36 1997CrossRefMathSciNetGoogle Scholar
  9. 9.
    Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math. Soc. 4(2), 365–421 (1991)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Lusztig, G.: Affine quivers and canonical bases. Inst. Hautes Études Sci. Publ. Math. (76), 111–163 (1992)Google Scholar
  11. 11.
    Lusztig, G.: Semicanonical bases arising from enveloping algebras. Adv. Math. 151(2), 129–139 (2000)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Nakajima, H.: Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76(2), 365–416 (1994)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Nakajima, H.: Quiver varieties and Kac-Moody algebras. Duke Math. J. 91(3), 515–560 (1998)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Saito, Y.: Crystal bases and quiver varieties. Math. Ann. 324(4), 675–688 (2002)CrossRefGoogle Scholar
  15. 15.
    Sanderson, Y.B.: Dimensions of Demazure modules for rank two affine Lie algebras. Compositio Math. 101(2), 115–131 (1996)MathSciNetGoogle Scholar
  16. 16.
    Sanderson, Y.B.: Real characters for Demazure modules of rank two affine Lie algebras. J. Algebra 184(3), 985–1000 (1996)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Savage, A.: Geometric and combinatorial realizations of crystal graphs. arXiv:math.RT/ 0310314, to appear in Algebr. Represent. Theory.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.The Fields Institute for Research in Mathematical Sciences and The University of TorontoTorontoCanada

Personalised recommendations