Differential operators on homogeneous spaces. I
Identifieur interne : 002972 ( Main/Merge ); précédent : 002971; suivant : 002973Differential operators on homogeneous spaces. I
Auteurs : Walter Borho [Allemagne] ; Jean-Luc Brylinski [France, États-Unis]Source :
- Inventiones mathematicae [ 0020-9910 ] ; 1982-10-01.
English descriptors
- KwdEn :
- Affine, Affine space, Algebra, Algebraic, Algebraic groups, Annihilator, Barbasch, Base point, Bav2, Birational, Borel subgroup, Borho, Brylinski, Canonical, Clan, Classical groups, Closure, Compositio math, Cone representation, Conjecture, Conjugacy classes, Cotangent, Cotangent bundle, Dense orbit, Differential operators, Exceptional groups, Fibre, Filtration, Finite type, Finitely, First author, Flag varieties, Flag variety, Generalized verma module, Generic point, Good filtration, Highest weight, Homogeneous, Homogeneous polynomials, Homogeneous space, Homogeneous spaces, Inclusion, Inclusion method, Irreducible, Irreducible components, Irreducible representations, Joseph representation, Kempken, Lecture notes, Linear form, London math, Main result, Math, Module, More detail, Natural filtration, Nilpotent elements, Nilpotent orbit, Nilpotent orbits, Normal image, Notation, Orbit, Other hand, Other words, Parabolic, Parabolic subgroup, Parabolic subgroups, Positive roots, Preprint, Present paper, Primitive ideals, Rational functions, Representation theory, Resp, Richardson orbit, Richardson orbits, Semisimple, Sheaf, Simultaneous resolution, Spaltenstein, Special case, Special orbit, Special orbits, Special representations, Springer, Subalgebra, Subgroup, Subset, Various points, Vector fields, Verma, Vogan, Wave front sets, Weyl, Weyl group, Weyl group representations, Weyl groups.
- Teeft :
- Affine, Affine space, Algebra, Algebraic, Algebraic groups, Annihilator, Barbasch, Base point, Bav2, Birational, Borel subgroup, Borho, Brylinski, Canonical, Clan, Classical groups, Closure, Compositio math, Cone representation, Conjecture, Conjugacy classes, Cotangent, Cotangent bundle, Dense orbit, Differential operators, Exceptional groups, Fibre, Filtration, Finite type, Finitely, First author, Flag varieties, Flag variety, Generalized verma module, Generic point, Good filtration, Highest weight, Homogeneous, Homogeneous polynomials, Homogeneous space, Homogeneous spaces, Inclusion, Inclusion method, Irreducible, Irreducible components, Irreducible representations, Joseph representation, Kempken, Lecture notes, Linear form, London math, Main result, Math, Module, More detail, Natural filtration, Nilpotent elements, Nilpotent orbit, Nilpotent orbits, Normal image, Notation, Orbit, Other hand, Other words, Parabolic, Parabolic subgroup, Parabolic subgroups, Positive roots, Preprint, Present paper, Primitive ideals, Rational functions, Representation theory, Resp, Richardson orbit, Richardson orbits, Semisimple, Sheaf, Simultaneous resolution, Spaltenstein, Special case, Special orbit, Special orbits, Special representations, Springer, Subalgebra, Subgroup, Subset, Various points, Vector fields, Verma, Vogan, Wave front sets, Weyl, Weyl group, Weyl group representations, Weyl groups.
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.
Url:
DOI: 10.1007/BF01389364
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ISTEX:929DAB0A1AE5702924465AFDC1963B63A1DA14B4Le document en format XML
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<term>Annihilator</term>
<term>Barbasch</term>
<term>Base point</term>
<term>Bav2</term>
<term>Birational</term>
<term>Borel subgroup</term>
<term>Borho</term>
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<term>Clan</term>
<term>Classical groups</term>
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<term>Compositio math</term>
<term>Cone representation</term>
<term>Conjecture</term>
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<term>Cotangent bundle</term>
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<term>Differential operators</term>
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<term>Homogeneous spaces</term>
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<term>London math</term>
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<term>Notation</term>
<term>Orbit</term>
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<term>Other words</term>
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<term>Parabolic subgroup</term>
<term>Parabolic subgroups</term>
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<term>Preprint</term>
<term>Present paper</term>
<term>Primitive ideals</term>
<term>Rational functions</term>
<term>Representation theory</term>
<term>Resp</term>
<term>Richardson orbit</term>
<term>Richardson orbits</term>
<term>Semisimple</term>
<term>Sheaf</term>
<term>Simultaneous resolution</term>
<term>Spaltenstein</term>
<term>Special case</term>
<term>Special orbit</term>
<term>Special orbits</term>
<term>Special representations</term>
<term>Springer</term>
<term>Subalgebra</term>
<term>Subgroup</term>
<term>Subset</term>
<term>Various points</term>
<term>Vector fields</term>
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<term>Vogan</term>
<term>Wave front sets</term>
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<term>Algebraic groups</term>
<term>Annihilator</term>
<term>Barbasch</term>
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<term>Bav2</term>
<term>Birational</term>
<term>Borel subgroup</term>
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<term>Brylinski</term>
<term>Canonical</term>
<term>Clan</term>
<term>Classical groups</term>
<term>Closure</term>
<term>Compositio math</term>
<term>Cone representation</term>
<term>Conjecture</term>
<term>Conjugacy classes</term>
<term>Cotangent</term>
<term>Cotangent bundle</term>
<term>Dense orbit</term>
<term>Differential operators</term>
<term>Exceptional groups</term>
<term>Fibre</term>
<term>Filtration</term>
<term>Finite type</term>
<term>Finitely</term>
<term>First author</term>
<term>Flag varieties</term>
<term>Flag variety</term>
<term>Generalized verma module</term>
<term>Generic point</term>
<term>Good filtration</term>
<term>Highest weight</term>
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<term>Homogeneous polynomials</term>
<term>Homogeneous space</term>
<term>Homogeneous spaces</term>
<term>Inclusion</term>
<term>Inclusion method</term>
<term>Irreducible</term>
<term>Irreducible components</term>
<term>Irreducible representations</term>
<term>Joseph representation</term>
<term>Kempken</term>
<term>Lecture notes</term>
<term>Linear form</term>
<term>London math</term>
<term>Main result</term>
<term>Math</term>
<term>Module</term>
<term>More detail</term>
<term>Natural filtration</term>
<term>Nilpotent elements</term>
<term>Nilpotent orbit</term>
<term>Nilpotent orbits</term>
<term>Normal image</term>
<term>Notation</term>
<term>Orbit</term>
<term>Other hand</term>
<term>Other words</term>
<term>Parabolic</term>
<term>Parabolic subgroup</term>
<term>Parabolic subgroups</term>
<term>Positive roots</term>
<term>Preprint</term>
<term>Present paper</term>
<term>Primitive ideals</term>
<term>Rational functions</term>
<term>Representation theory</term>
<term>Resp</term>
<term>Richardson orbit</term>
<term>Richardson orbits</term>
<term>Semisimple</term>
<term>Sheaf</term>
<term>Simultaneous resolution</term>
<term>Spaltenstein</term>
<term>Special case</term>
<term>Special orbit</term>
<term>Special orbits</term>
<term>Special representations</term>
<term>Springer</term>
<term>Subalgebra</term>
<term>Subgroup</term>
<term>Subset</term>
<term>Various points</term>
<term>Vector fields</term>
<term>Verma</term>
<term>Vogan</term>
<term>Wave front sets</term>
<term>Weyl</term>
<term>Weyl group</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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