Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 003, 37 pages      arXiv:0809.2605      https://doi.org/10.3842/SIGMA.2009.003
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Quiver Varieties and Branching

Hiraku Nakajima a, b
a) Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
b) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Received September 15, 2008, in final form January 05, 2009; Published online January 11, 2009

Abstract
Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group Gaff [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of Gcpt-instantons on R4/Zr correspond to weight spaces of representations of the Langlands dual group GaffÚ at level r. When G = SL(l), the Uhlenbeck compactification is the quiver variety of type sl(r)aff, and their conjecture follows from the author's earlier result and I. Frenkel's level-rank duality. They further introduce a convolution diagram which conjecturally gives the tensor product multiplicity [Braverman A., Finkelberg M., Private communication, 2008]. In this paper, we develop the theory for the branching in quiver varieties and check this conjecture for G = SL(l).

Key words: quiver variety; geometric Satake correspondence; affine Lie algebra; intersection cohomology.

pdf (563 kb)   ps (343 kb)   tex (53 kb)

References

  1. Bando S., Einstein-Hermitian metrics on noncompact Kähler manifolds, in Einstein Metrics and Yang-Mills connections (Sanda, 1990), Lecture Notes in Pure and Appl. Math., Vol. 145, Dekker, New York, 1993, 27-33.
  2. Baranovsky V., Moduli of sheaves on surfaces and action of the oscillator algebra, J. Differential Geom. 55 (2000), 193-227, math.AG/9811092.
  3. Borho W., MacPherson R., Partial resolutions of nilpotent varieties, in Analysis and Topology on Singular Spaces, II, III (Luminy, 1981), Astérisque, Vol. 101, Soc. Math. France, Paris, 1983, 23-74.
  4. Braverman A., Finkelberg M., Pursuing the double affine Grassmannian I: transversal slices via instantons on Ak-singularities, arXiv:0711.2083.
  5. Braverman A., Finkelberg M., Private communication, 2008.
  6. Braverman A., Kazhdan D., The spherical Hecke algebra for affine Kac-Moody groups I, arXiv:0809.1461.
  7. Chriss N., Ginzburg V., Representation theory and complex geometry, Birkhäuser Boston Inc., Boston, MA, 1997.
  8. Crawley-Boevey W., Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), 257-293.
  9. Frenkel I.B., Representations of affine Lie algebras, Hecke modular forms and Korteweg-de Vries type equations, in Lie Algebras and Related Topics (New Brunswick, N.J., 1981), Lecture Notes in Math., Vol. 933, Springer, Berlin - New York, 1982, 71-110.
  10. Grojnowski I., Instantons and affine algebras. I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), 275-291, alg-geom/9506020.
  11. Hasegawa K., Spin module versions of Weyl's reciprocity theorem for classical Kac-Moody Lie algebras - an application to branching rule duality, Publ. Res. Inst. Math. Sci. 25 (1989), 741-828.
  12. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
  13. Kashiwara M., Saito Y., Geometric construction of crystal bases, Duke Math. J. 89 (1997), 9-36, q-alg/9606009.
  14. King A.D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 515-530.
  15. Kronheimer P.B., The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), 665-683.
  16. Kronheimer P.B., Nakajima H., Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), 263-307.
  17. Lusztig G., Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), 169-178.
  18. Lusztig G., Canonical bases arising from quantized enveloping algebras. II, in Common Trends in Mathematics and Quantum Field Theories (Kyoto, 1990), Progr. Theoret. Phys. Suppl. (1990), no. 102, 175-201.
  19. Malkin A., Tensor product varieties and crystals: the ADE case, Duke Math. J. 116 (2003), 477-524, math.AG/0103025.
  20. Mirkovic I., Vilonen K., Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett. 7 (2000), 13-24, math.AG/9911050.
  21. Nagao K., Quiver varieties and Frenkel-Kac construction, math.RT/0703107.
  22. Nakajima H., Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), 365-416.
  23. Nakajima H., Varieties associated with quivers, in Representation Theory of Algebras and Related Topics (Mexico City, 1994), CMS Conf. Proc., Vol. 19, Amer. Math. Soc., Providence, RI, 1996, 139-157.
  24. Nakajima H., Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), 515-560.
  25. Nakajima H., Lectures on Hilbert schemes of points on surfaces, University Lecture Series, Vol. 18, American Mathematical Society, Providence, RI, 1999.
  26. Nakajima H., Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), 145-238, math.QA/9912158.
  27. Nakajima H., Quiver varieties and tensor products, Invent. Math. 146 (2001), 399-449, math.QA/0103008.
  28. Nakajima H., Geometric construction of representations of affine algebras, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 423-438, math.QA/0212401.
  29. Nakajima H., Reflection functors for quiver varieties and Weyl group actions, Math. Ann. 327 (2003), 671-721.
  30. Nakajima H., Sheaves on ALE spaces and quiver varieties, Mosc. Math. J. 7 (2007), 699-722.
  31. Nakanishi T., Tsuchiya A., Level-rank duality of WZW models in conformal field theory, Comm. Math. Phys. 144 (1992), 351-372.
  32. Rudakov A., Stability for an abelian category, J. Algebra 197 (1997), 231-245.
  33. Saito Y., Crystal bases and quiver varieties, Math. Ann. 324 (2002), 675-688, math.QA/0111232.


Previous article   Next article   Contents of Volume 5 (2009)