Advertisement

Inventiones mathematicae

, Volume 69, Issue 3, pp 437–476 | Cite as

Differential operators on homogeneous spaces. I

Irreducibility of the associated variety for annihilators of induced modules
  • Walter Borho
  • Jean-Luc Brylinski
Article

Summary

In this paper, we extend recent work of one of us [Br] to investigate an old problem of the other one [B2]. Given a connected semisimple complex Lie-groupG with Lie-algebrag, we study the representation\(\psi _X :U(\mathfrak{g}) \to D(X)\) of the enveloping algebra of\(\mathfrak{g}\) by global differential operators on a complete homogeneous spaceX=G/P. It turns out that the kernelI x of ψ X is the annihilator of a generalizedVerma-module. On the other hand, we study the associated graded ideal grI x , and relate it to the geometry of a generalizedSpringer-resolution, that is a map\(\pi _X :T^* (X) \to \mathfrak{g}\) of the cotangent-bundle ofX onto a nilpotent variety in\(\mathfrak{g}\), as studied e.g. in [BM1]. We prove, for instance, that grI x is prime if and only if π X is birational with normal image. In general, we show that\(\sqrt {grI_X }\) is prime. Equivalently, the associated variety ofI x in\(\mathfrak{g}\) is irreducible: In fact, it is the closure of theRichardson-orbit determined byP. For the homogeneous spaceY=G/(P, P), we prove that the analogous idealI y has for associated variety the closure of theDixmier-sheet determined byP. From this main result, we derive as a corollary, that for any module induced from a finitedimensional LieP-module the associated variety of the annihilator is irreducible, proving an old conjecture [B2], 2.5. Finally, we give some applications to the study of associated varieties of primitive ideals.

Keywords

Recent Work Differential Operator Homogeneous Space Normal Image Primitive Ideal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AMa] Abraham, R., Marsden, J.E.: Foundations of mechanics, 2. edition. Benjamin/Cummings Publ. Co., Reading, Ma. 1978Google Scholar
  2. [Al] Alekse'evski, A.V.: Component groups of centralizer for unipotent elements in semisimple algebraic groups (in Russian). Trudy Tbilisskogo Matematicheskogo Instituta62, 5–28 (1979)Google Scholar
  3. [Av] Alvis, D.: Induce/restrict matrices for exceptional Weyl groups. Manuscript, M.I.T., Cambridge, MA, 1981Google Scholar
  4. [AvL] Alvis, D., Lusztig, G.: On Springer's correspondence for simple groups. Preprint, M.I.T., Cambridge, MA, 1981Google Scholar
  5. [BaV1] Barbasch, D., Vogan, D.: Primitive ideals and orbital integrals in complex classical groups. Math. Ann.259, 153–199 (1982)Google Scholar
  6. [BaV2] Barbasch, D., Vogan, D.: Primitive ideals and orbital integrals in complex exceptional groups. Preprint, M.I.T. 1981Google Scholar
  7. [BaV3] Barbasch, D., Vogan, D.: The local structure of characters. J. of Functional Analysis37, 27–55 (1980)Google Scholar
  8. [BeBe] Beilinson, A., Bernstein, J.: Localisation de g-modules. C. R. Acad. Sc. Paris292, 15–18 (1981)Google Scholar
  9. [Be] Bernstein, I.N.: Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients. Funct. Anal. Apl.5, 98–101 (1971)Google Scholar
  10. [Bj] Björk, J.-E.: Rings of differential operators. North-Holland Pub. Co., Amsterdam-New York-Oxford 1979Google Scholar
  11. [B1] Borho, W.: Berechnung der Gelfand-Kirillov-Dimension bei induzierten Darstellungen. Math. Ann.225, 177–194 (1977)Google Scholar
  12. [B2] Borho, W.: Recent advances in enveloping algebras of semi-simple Lie-algebras. Sémin. Bourbaki, Exposé 489 (1976); Lecture Notes in Math., vol.677. Berlin-Heidelberg-New York. Springer 1978Google Scholar
  13. [B3] Borho, W.: Definition einer Dixmier-Abbildung für\(\mathfrak{s}\mathfrak{l}(n,\mathbb{C})\). Invent. Math.40, 143–169 (1977)Google Scholar
  14. [B4] Borho, W.: Über Schichten halbeinfacher Lie-Algebren; Invent. Math.65, 283–317 (1981)Google Scholar
  15. [BJ] Borho, W., Jantzen, J.-C.: Über primitive Ideale in der Einhüllenden einer halbeinfachen Lie-Algebra. Invent. Math.39, 1–53 (1977)Google Scholar
  16. [BK1] Borho, W., Kraft, H.: Über die Gelfand-Kirillov-Dimension. Math. Ann.220, 1–24 (1976)Google Scholar
  17. [BK2] Borho, W., Kraft, H.: Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen. Comment. Math. Helvet.54, 61–104 (1979)Google Scholar
  18. [BM1] Borho, W., MacPherson, R.: Représentations des groupes de Weyl et homologie d'intersection pour les variétés nilpotentes. Note C.R.A.S. Paris t.29 (27 april 1981). pp. 707–710Google Scholar
  19. [BM2] Borho, W., MacPherson, R.: Partial resolutions of nilpotent varieties; Proceedings of the conference on “Analyse et Topologie sur les variétés singulières”, organized by Teissier and Verdier at Marseille-Luminy 1981 (to appear in Astérisque)Google Scholar
  20. [Br] Brylinski, J.-L.: Differential Operators on the Flag Varieties; Proceedings of the Conference on “Young tableaux and Schur functors in Algebra and Geometry” at Torun (Poland) 1980 (to appear)Google Scholar
  21. [BrKa] Brylinski, J.-L., Kashiwara, M.: Kazhdan-Lusztig conjecture and holonomic systems. Invent. math.64, 387–410 (1981)Google Scholar
  22. [CD] Conze-Berline, N., Duflo, M.: Sur les reprśentations induites des groupes semi-simples complexes. Compositio math.34, 307–336 (1977)Google Scholar
  23. [Di] Dixmier, J.: Algèbres enveloppantes. Paris: Gauthier-Villars 1974Google Scholar
  24. [D] Duflo, M.: Sur la classification des idéaux primitifs dans l'algèbre enveloppante d'une algèbre de Lie semisimple. Annals of Math.105, 107–120 (1977)Google Scholar
  25. [EGA] Dieudonné, J., Grothendieck, A.: Eléments des Géométrie Algébrique III (Étude cohomologique des faisceaux cohérentes): Publ. Math. IHES n0 11 (1961) and n0 17 (1963)Google Scholar
  26. [EP] Elashvili, A.G., Panov, A.N.: Polarizations in semisimple Lie-algebras (in Russian). Bull. Acad. Sci. Georgian SSR87, 25–28 (1977)Google Scholar
  27. [E1] Elkik, R.: Désingularisation des adhérences d'orbites polarisables et des nappes dans les algèbres de Lie réductives. Preprint, Paris 1978Google Scholar
  28. [GKi] Gelfand, I.M., Kirillov, A.A.: The structure of the Lie field connected with a split semisimple Lie algebra. Funct. Anal. Applic.3, 6–21 (1969)Google Scholar
  29. [Gi] Ginsburg, V.: Symplectic geometry and representations. Preprint, Moscow State University, 1981Google Scholar
  30. [Ha] Hartshorne, R.: Residues and duality. Lecture Notes in Math., Vol. 20. Berlin-Heidelberg-New York: Springer 1966Google Scholar
  31. [He] Hesselink, W.H.: Polarization in the classical groups. Math. Z.160, 217–234 (1978)Google Scholar
  32. [HS] Hotta, R., Springer, T.A.: A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups. Invent. math.41, 113–127 (1977)Google Scholar
  33. [H] Hotta, R.: The Weyl groups as monodromies and nilpotent orbits (after M. Kashiwara); preprint, Tohoku University, 1982Google Scholar
  34. [J] Jantzen, J.-C.: Moduln mit einem höchsten Gewicht; Lecture Notes in Math., vol. 750. Berlin-Heidelberg-New York: Springer 1979Google Scholar
  35. [Ja] Jategaonkar, A.V.: Solvable Lie algebras, polycyclic-by-finite groups, and bimodule Krull dimension. Communications in Algebra10, 19–69 (1982)Google Scholar
  36. [Jo1] Joseph, A.: Towards the Jantzen conjecture I, II. Compositio math.40, 35–67, 69–78 (1980)Google Scholar
  37. [Jo2] Joseph, A.: Towards the Jantzen conjecture III. Compositio math.41, 23–30 (1981)Google Scholar
  38. [Jo3] Joseph, A.: Kostant's problem, Goldie-rank, and the Gelfand-Kirillov conjecture. Invent. math.56, 191–213 (1980)Google Scholar
  39. [Jo4] Joseph, A.: Goldie rank in the enveloping algebra of a semisimple Lie algebra I, II. J. of Algebra65, 269–306 (1980)Google Scholar
  40. [Jo5] Joseph, A.:W-module structure in the primitive spectrum of the enveloping algebra of a semisimple Lie algebra. In: Noncommutative Harmonic Analysis, Lecture Notes in Mathematics, vol. 728, pp. 116–135. Berlin-Heidelberg-New York: Springer 1978Google Scholar
  41. [KKS] Kazhdan, D., Kostant, B., Sternberg, S.: Hamitonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math.31, 481–507 (1978)Google Scholar
  42. [Ke1] Kempf, G.: The Grothendieck Cousin complex of an induced representation. Advances in Math.29, 310–396 (1978)Google Scholar
  43. [Ke2] Kempf, G.: The geometry of homogeneous spaces versus induced representations. In: Supplement. American J. Math.: Algebraic Geometry, J.-I. Igusa ed., The Johns Hopkins centen. Lect., Symp. Baltimore/Maryland 1976, 1–5 (1977)Google Scholar
  44. [Kk] Kempken, G.: Induced conjugacy classes in classical Lie-algebras. Abh. Math. Sem. Univ. Hamburg (to appear)Google Scholar
  45. [Ko1] Kostant, B.: Quantization and unitary representation. In: Lectures in Modern Analysis and Applications III. Lecture Notes in Mathematics, vol. 170, pp. 87–208, Berlin-Heidelberg-New York: Springer 1976Google Scholar
  46. [Ko2] Kostant, B.: Quantization and representation theory; In: Representation Theory of Lie Groups. Proceedings, London Math. Soc. Lecture Notes Series34, 287–316 (1979)Google Scholar
  47. [KP1] Kraft, H., Procesi, C.: Closures of Conjugacy Classes of Matrices are Normal. Invent. math.53, 227–247 (1979)Google Scholar
  48. [KP2] Kraft, H., Procesi, C.: On the geometry of conjugacy classes in classical groups. Comment. Math. Helvet. (to appear)Google Scholar
  49. [L] Lusztig, G.: A class of irreducible representations of a Weyl group. Proc. Kon. Nederl. Akad. Wetensch.82, 323–335 (1979)Google Scholar
  50. [Md] Macdonald, I.G.: Some irreducible representations of Weyl groups. Bull. London Math. Soc.4, 148–150 (1972)Google Scholar
  51. [Mi] Mizuno, K.: The conjugate classes of unipotent elements of the Chevalley groupsE 7 andE 8. Tokyo J. Math.3, 391–459 (1980)Google Scholar
  52. [Mü] Müller, B.J.: Localizations in non-commutative noetherian rings. Can. J. Math.28, 600–610 (1976)Google Scholar
  53. [R] Richardson, R.-W.: Conjugacy classes in parabolic subgroups of semi-simple algebraic groups. Bull. London Math. Soc.6, 21–24 (1974)Google Scholar
  54. [Ŝ] Ŝapovalov, N.N.: On a conjecture of Gel'fand-Kirillov. Funct. Anal. Applic.7, 165–166 (1973)Google Scholar
  55. [Sh] Shoji, T.: The conjgacy classes of Chevalley groups of type (F 4) over finite fields of characteristicp≠2. J. Fac. Sci. Univ. Tokyo21, 1–17 (1974)Google Scholar
  56. [So] Souriau, J.M.: Structure des systèmes dynamiques. Paris: Dunod 1970Google Scholar
  57. [Sp] Spaltenstein, N.: The fixed point set of a unipotent transformation on the flag manifold. Proc. Kon. Nederl. Akad. Wetensch.79, 452–456 (1976)Google Scholar
  58. [Sp2] Spaltenstein, N.: A property of special representions of Weyl groups; preprint, Warwick 1982Google Scholar
  59. [S1] Springer, T.A.: The unipotent variety of a semisimple group. Proc. Colloqu. Alg. Geom. Tata Inst., Bombay 1968, 373–391 (1969)Google Scholar
  60. [S2] Springer, T.A.: Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. math.36, 173–207 (1976)Google Scholar
  61. [S3] Springer, T.A.: A construction of representations of Weyl groups. Invent. math.44, 279–293 (1978)Google Scholar
  62. [SXt] Steinberg, R.: Conjugacy classes in algebraic groups. Lecture Notes in Mathematics, vol. 366. Berlin-Heidelberg-New York: Springer 1974Google Scholar
  63. [W] Wallach, N.: On the Enright-Varadarajan modules. A construction of the discrete series. Ann. Sci. Ec. Norm. Sup. 4e, sér.9, 81–102 (1976)Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Walter Borho
    • 1
    • 2
  • Jean-Luc Brylinski
    • 3
    • 4
  1. 1.FB 7 MathematikUniversität GH WuppertalWuppertal 1
  2. 2.Max-Planck-Institut für MathematikBonn 3Federal Republic of Germany
  3. 3.Centre de MathématiqueEcole PolytechniquePalaiseau - CedexFrance
  4. 4.Department of MathematicsBrown UniversityProvidenceUSA

Personalised recommendations

Cite article

Buy options