Transformation Groups

, Volume 6, Issue 4, pp 353–370 | Cite as

Fourier transform, parabolic induction, and nilpotent orbits

  • Anthony Henderson


We prove that in the symmetric space setting the functors of Fourier transform (in the sense of Deligne) and parabolic induction (in the sense of Lusztig) commute. We derive two consequences: the first is a new proof of Lusztig's description of the intersection cohomology of nilpotent orbit closures for GL n , and the second is an analogous description for GL2n/Sp2n.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    E. Bannai, N. Kawanaka, S.-Y. Song,The character table of the Hecke algebra \(\mathcal{H}(GL_{2n} (\mathbb{F}q),Sp_{2n} (\mathbb{F}q))\), J. Algebra129 (1990), 320–366.Google Scholar
  2. [2]
    A. A. Beilinson, J. Bernstein, P. Deligne,Faisceaux pervers, Astérisque100 (1982).Google Scholar
  3. [3]
    J. Bernstein, P. Lunts,Equivariant Sheaves and Functors, Lecture Notes in Math., Vol. 1578, Springer-Verlag, 1994.Google Scholar
  4. [4]
    J.-L. Brylinski,Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque140–141 (1986), 3–134.Google Scholar
  5. [5]
    I. Grojnowski,Character Sheaves on Symmetric Spaces, PhD thesis, MIT, 1992.Google Scholar
  6. [6]
    N. Katz, G. Laumon,Transformation de Fourier et majoration de sommes exponentielles, Publ. Math. IHES62 (1985), 361–418.Google Scholar
  7. [7]
    G. I. Lehrer,The space of invariant functions on a finite Lie algebra, Trans. Amer. Math. Soc.348 (1996), 31–50.Google Scholar
  8. [8]
    G. Lusztig,Green polynomials and singularities of unipotent classes, Adv. Math.42 (1981), 169–178.Google Scholar
  9. [9]
    G. Lusztig,Character sheaves, I, Adv. Math.56 (1985), 193–237; II, Adv. Math.57 (1985), 226–265; III, Adv. Math.57 (1985), 266–315; IV, Adv. Math.59 (1986), 1–63; V, Adv. Math.61 (1986), 103–155.Google Scholar
  10. [10]
    G. Lusztig, Fourier transforms on a semisimple Lie algebra over\(\mathbb{F}q\), in:Algebraic Groups Utrecht 1986, Lecture Notes in Math., Vol. 1271, 1987, 177–188.Google Scholar
  11. [11]
    G. Lusztig,Quivers, perverse sheaves and enveloping algebras, J. Amer. Math. Soc.4 (1991), 365–421.Google Scholar
  12. [12]
    G. Lusztig,Study of perverse sheaves arising from graded Lie algebras, Adv. in Math.112 (1995), 147–217.Google Scholar
  13. [13]
    I. G. Macdonald,Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Univ. Press, New York, 1995.Google Scholar
  14. [14]
    T. Shoji,Geometry of orbits and Springer correspondence, Astérisque168 (1988), 61–140.Google Scholar
  15. [15]
    M. Vergne,Instantons et correspondance de Kostant-Sekiguchi, C. R. Acad. Sci. Paris, Sér. I Math.320 (1995), 901–906.Google Scholar

Copyright information

© Birkhäuser Boston 2001

Authors and Affiliations

  • Anthony Henderson
    • 1
  1. 1.School of Mathematics and StatisticsUniversity of SydneyAustralia

Personalised recommendations