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Differential operators on homogeneous spaces. I

Identifieur interne : 000747 ( France/Analysis ); précédent : 000746; suivant : 000748

Differential operators on homogeneous spaces. I

Auteurs : Walter Borho [Allemagne] ; Jean-Luc Brylinski [France, États-Unis]

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RBID : ISTEX:929DAB0A1AE5702924465AFDC1963B63A1DA14B4

English descriptors

Abstract

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.

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DOI: 10.1007/BF01389364


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ISTEX:929DAB0A1AE5702924465AFDC1963B63A1DA14B4

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<term>Joseph representation</term>
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<term>London math</term>
<term>Main result</term>
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<term>Subset</term>
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<term>Verma</term>
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<front>
<div type="abstract" xml:lang="en">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.</div>
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