Dixmier Algebras for Classical Complex Nilpotent Orbits via Kraft-Procesi Models I

  • Ranee Brylinski
Part of the Progress in Mathematics book series (PM, volume 213)


We attach a Dixmier algebra \(\mathcal{B}\) to the closure of \(\bar{\mathcal{O}}\) any nilpotent orbit of G where G is GL(nℂ), O(nℂ) or Sp(2n ℂ). This algebra \(\mathcal{B}\) is a noncommutative analog of the coordinate ring \(\mathcal{R} of \bar{\mathcal{O}}\) in the sense that \(\mathcal{B}\) has a G-invariant algebra filtration and \(gr\mathcal{B} = \mathcal{R}\).

We obtain \(\mathcal{B}\) by making a noncommutative analog of the Kraft-Procesi construction which modeled \(\bar{\mathcal{O}}\) as the algebraic symplectic reduction of a finite-dimensional symplectic vector space L. Indeed \(\mathcal{B}\) is a subquotient of the Weyl algebra for L.

\(\mathcal{B}\) identifies with the quotient of \(\mathcal{U}{\text{(}}\mathfrak{g})\) by a two-sided ideal J, where \(\mathfrak{g} = Lie (G)\). Then gr J is the ideal \(\Im (\bar{\mathcal{O}})\) in \(S(\mathfrak{g})\) of functions vanishing on \(\bar{\mathcal{O}}\). In every case where \(\mathcal{O}\) is connected, J is a completely prime primitive ideal.


Filtration Manifold Tral 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Adams and D. Barbasch, Reductive dual pair correspondence for complex groups J. Funct. Anal.132 (1995), 1–42.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    A. Astashkevich and R. Brylinski, Non-Local Equivariant Star Product on the Minimal Nilpotent OrbitAdv. Math 171(2002), 86–102.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    D. Barbasch, The unitary dual for complex classical Lie groupsInv. Math. 96(1989), 103–176.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    D. Barbasch, Orbital Integrals of Nilpotent Orbits, inThe Mathematical Legacy of Harish-ChandraR. S. Doran and V. S. Varadarajan, eds., Proc. Symp. Pure Math., Vol. 68, AMS, 2000, 97–110.CrossRefGoogle Scholar
  5. [5]
    D. Barbasch and D.A. Vogan, Primitive ideals and orbital integrals in complex classical groupsMath. Ann. 259(1982), 152–199.MathSciNetCrossRefGoogle Scholar
  6. [6]
    D. Barbasch and D.A. Vogan, Unipotent representations of complex semisimple groupsAnn. of Math. 121(1985), 41–110.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    R. Brylinski, Equivariant Deformation Quantization for the Cotangent Bundle of a Flag ManifoldAnn. Inst. Fourier52:3 (2002), 881–897.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    R. Brylinski, Dixmier Algebras for Classical Complex Nilpotent Orbits via Kraft-Procesi Models II, in preparation.Google Scholar
  9. [9]
    R. BrylinskiQuantization of Classical Complex Nilpotent Orbitsin preparation.Google Scholar
  10. [10]
    M. Cahen and S. Gutt, An algebraic construction of * product on the regular orbits of semi simple Lie groups, inGravitation and GeometryW. Rindler and A. Trautman eds., 73–82, Bibliopolis, 1987.Google Scholar
  11. [11]
    D.H. Collingwood and W.M. McGovernNilpotent Orbits in Semisimple Lie AlgebrasVan Nostrand Reinhold, 1993.MATHGoogle Scholar
  12. [12]
    J. DixmierEnveloping AlgebrasNorth-Holland, 1977.Google Scholar
  13. [13]
    M. Duflo, Représentations unitaires irréductibles des groupes simples complexes de rang deuxBull. Soc. Math. France 107(1979), 55–96.MathSciNetMATHGoogle Scholar
  14. [14]
    C. Duval, P. Lecomte and V. Ovsienko, Conformally equivariant quantization: existence and uniquenessAnn. Inst. Fourier 49:6(1999), 1999–2029.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    R. Godement, Théorie des Faisceaux, Hermann, Paris, Third edition, 1973.MATHGoogle Scholar
  16. [16]
    K.I. Gross, The dual of a parabolic subgroup and a degenerate principal series ofSp(n C) Amer. J. Math 93(1971), 399–428.CrossRefGoogle Scholar
  17. [17]
    R. Hartshorne, Algebraic Geometry, Graduate Texts in Math 52, Springer-Verlag New York, 1977.MATHCrossRefGoogle Scholar
  18. [18]
    R. Howe, Transcending classical invariant theoryJ. Amer. Math. Soc. 2(1989), 535–552.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    A. Joseph, The minimal orbit in a simple Lie algebra and associated maximal idealAnn. Scient. Ec. Norm. Sup. 9(1976), 1–30.MATHGoogle Scholar
  20. [20]
    A. Knapp and D.A. VoganCohomological Induction and Unitary RepresentationsPrinceton University Press, 1995.Google Scholar
  21. [21]
    P.Z. Kobak and A. Swann, Classical nilpotent orbits as hyperkähler quotientsInt. J. Math. 7(1996), 193–210.MathSciNetCrossRefGoogle Scholar
  22. [22]
    H. Kraft and C. Procesi, Closures of conjugacy classes of matrices are normalInv. Math. 53(1979), 227–247.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    H. Kraft and C. Procesi, On the geometry of conjugacy classes in clasical groupsComment. Math. Helv. 57(1982), 539–602.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    T. Levasseur and J.T. StaffordRings of differential operators on classical rings of invariantsMem. Amer. Math. Soc., no. 412, 1989.Google Scholar
  25. [25]
    J-S. Li, Singular unitary representations of classical groupsInv. Math. 97(1989), 237–255.MATHCrossRefGoogle Scholar
  26. [26]
    W. McGovernQuantization of nilpotent orbits and their covers in complex classical groups, preprintYale University (1989).Google Scholar
  27. [27]
    W. McGovernCompletely prime maximal ideals and quantization, Mem. Amer. Math. Soc., no. 519, 1994.Google Scholar
  28. [28]
    C. Moeglin, Idéaux complètement premiers de l’algèbre enveloppante degl n (C) J. Alg. 87(1987), 287–366.MathSciNetCrossRefGoogle Scholar
  29. [29]
    C. Moeglin, Correspondance de Howe pour les paires réductives duales, quelques calculs dans le cas ArchimedienJ. Funct. Anal. 85(1989), 1–85.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    M. Van den Bergh, Differential operators on semi-invariants for tori and weighted projective spacesTopics in Invariant Theory(Paris 1989/90), 255–272, Lecture Notes in Math. 1478, Springer, Berlin 1991.Google Scholar
  31. [31]
    D.A. Vogan, Noncommutative algebras and unitary representationsThe Mathematical Heritage of Hermann Weyl35–60, Proc. Symp. Pure Math., vol. 48, Amer. Math. Soc., Providence, 1988.MathSciNetCrossRefGoogle Scholar
  32. [32]
    D.A. Vogan, Dixmier algebras, sheets and representation theory, Operator Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory, 333–396, Progress in Math, vol. 92, Birkhäuser, 1990.Google Scholar
  33. [33]
    D.A. Vogan, Associated varieties and unipotent representations, in Harmonic Analysis on Reductive Lie Groups, 315–388, Progress in Math, vol. 101, Birkhäuser, 1991.Google Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Ranee Brylinski
    • 1
  1. 1.Department of MathematicsPenn State UniversityUniversity ParkUSA

Personalised recommendations