Transformation Groups

, Volume 22, Issue 3, pp 793–844 | Cite as


  • TOSHIAKI SHOJIEmail author


Let V be an n-dimensional vector space over an algebraic closure of a finite field F q , and G = GL(V). A variety Open image in new window is called an enhanced variety of level r. Let Open image in new window be the unipotent variety of Open image in new window . We have a partition Open image in new window indexed by r-partitions λ of n In the case where r = 1 or 2, X λ is a single G-orbit, but if r ≥ 3, X λ is, in general, a union of infinitely many G-orbits. In this paper, we prove certain orthogonality relations for the characteristic functions (over F q ) of the intersection cohomology \( \mathrm{I}\mathrm{C}\left({\overline{X}}_{\boldsymbol{\uplambda}},{\overline{\mathbf{Q}}}_l\right) \), and show some results, which suggest a close relationship between those characteristic functions and Kostka functions associated to the complex reflection group Open image in new window .


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AH]
    P. N. Achar, A. Henderson, Orbit closures in the enhanced nilpotent cone, Adv. In Math. 219 (2008), no. 1, 27–62, Corrigendum, ibid. 228 (2011), 2984–2988.Google Scholar
  2. [K1]
    S. Kato, An exotic Deligne-Langlands correspondence for symplectic groups, Duke Math. J. 148 (2009) 306–371.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [K2]
    S. Kato, An algebraic study of extension algebras, preprint, arXiv:1207.4640 (2013).Google Scholar
  4. [L1]
    G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), 169–178.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [L2]
    G. Lusztig, Character sheaves II, Adv. in Math. 57 (1985), 226–265.Google Scholar
  6. [L3]
    G. Lusztig, Character sheaves V, Adv. in Math. 61 (1986), 103–155.Google Scholar
  7. [L4]
    G. Lusztig, Character sheaves on disconnected groups III, Represent. Theory 8 (2004), 125–144.Google Scholar
  8. [M]
    I. G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1995.zbMATHGoogle Scholar
  9. [MS]
    J. G. Mars, T. A. Springer, Hecke algebra representations related to spherical varieties, Represent. theory 2 (1989), 33–69.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [S1]
    T. Shoji, Green functions associated to complex reflection groups, J. Algebra 245 (2001), 650–694.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [S2]
    T. Shoji, Green functions attached to limit symbols, in: Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., Vol. 40, Math. Soc. Japan, Tokyo, 2004, pp. 443–467.Google Scholar
  12. [S3]
    T. Shoji, Exotic symmetric spaces of higher level - Springer correspondence for complex reflection groups, Transform. Groups 21 (2016), 197–264.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [SS1]
    T. Shoji, K. Sorlin, Exotic symmetric space over a finite field, I, Transform. Groups 18 (2013), 877–929.Google Scholar
  14. [SS2]
    T. Shoji, K. Sorlin, Exotic symmetric space over a finite field, II, Transform. Groups 19 (2014), 887–926.Google Scholar
  15. [T]
    R. Travkin; Mirabolic Robinson–Schensted–Knuth correspondence, Selecta Math. 14 (2009), 727–758.Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsTongji UniversityShanghaiP.R. China

Personalised recommendations