Inventiones mathematicae

, Volume 69, Issue 3, pp 437–476

Differential operators on homogeneous spaces. I

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

DOI: 10.1007/BF01389364

Cite this article as:
Borho, W. & Brylinski, JL. Invent Math (1982) 69: 437. doi:10.1007/BF01389364


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 kernelIx of ψX is the annihilator of a generalizedVerma-module. On the other hand, we study the associated graded ideal grIx, 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 grIx is prime if and only if πX is birational with normal image. In general, we show that\(\sqrt {grI_X }\) is prime. Equivalently, the associated variety ofIx 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 idealIy 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.

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
  5. 5.MP1 für MathematikBonn 3

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