Nilpotent orbits, primitive ideals, and characteristic classes: a geometric perspective in ring theory
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
1989
|
| Series: | Progress in mathematics
78 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 131 S. |
| ISBN: | 0817634738 3764334738 |
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MARC
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| 245 | 1 | 0 | |a Nilpotent orbits, primitive ideals, and characteristic classes |b a geometric perspective in ring theory |c W. Borho; J.-L. Brylinski; R. MacPherson |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 1989 | |
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| 650 | 4 | |a Orbite nilpotente | |
| 650 | 4 | |a Théorème Joseph | |
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Record in the Search Index
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| adam_text | CONTENTS
GENERAL INTRODUCTION 1
Si. A DESCRIPTION OF SPRINGER S WEYL GROUP
REPRESENTATIONS IN TERMS OF
CHARACTERISTIC CLASSES OF CONE BUNDLES 10
1.1 Segre classes of cone bundles 10
1.2 Characteristic class of a subvariety of a vector bundle 11
1.3 Characteristic class determined by a sheaf on a bundle 12
1.4 Comparison of the two definitions for Q 13
1.5 Homology of the flag variety 15
1.6 Cohomology of the flag variety 17
1.7 Orbital cone bundles on the flag variety 18
1.8 Realization of Springer s Weyl group representation 19
1.9 Reformulation in terms of intersection homology 21
1.10 The Weyl group action 22
1.11 Reduction to a crucial lemma 23
1.12 Completion of the proof of theorem 1.8 25
1.13 Comparison with Springer s original construction 26
1.14 Theorem: The maps in the diagram are W equivariant 28
1.15 Hotta s transformation formulas 29
§ 2. GENERALITIES ON EQUIVARIANT K THEORY 31
2.1 Algebraic notion of fibre bundle 31
2.2 Equivariant vector bundles and definition of Kp(X) 32
2.3 Equivariant homogeneous vector bundles 33
2.4 Functoriality in the group G 34
2.5 Functoriality in the space X 34
2.6 The sheaf theoretical point of view 35
2.7 Existence of equivariant locally free resolutions 36
2.8 Remarks on Gysin homomorphisms in terms of coherent sheaves 38
2.9 Equivariant K—theory on a vector bundle: Basic restriction
techniques 39
2.10 Filtrations on I G(X) 41
2.11 Representation rings for example 42
2.12 Application of equivariant K—theory to ^ modules 42
§ 3. EQUIVARIANT K THEORY OF TORUS ACTIONS AND FORMAL
CHARACTERS 45
3.1 The completed representation ring of a torus 45
3.2 Formal characters of T—modules 46
3.3 Example 47
3.4 T equivariant modules with highest weight 48
3.5 Projective and free cyclic highest weight modules 49
3.6 Formal characters of equivariant coherent sheaves 51
3.7 Restriction to the zero point 52
3.8 Computation of 7 degree 54
3.9 Character polynomials 54
3.10 Degree of character polynomial equals codimension of support 56
3.11 Positivity property of character polynomials 56
3.12 Division by a nonzero divisor 57
3.13 Proof of theorem 3.10 and 3.11 57
3.14 Determination of character polynomials by supports 59
3.15 The theory of Hilbert—Samuel polynomials as a special case 60
3.16 Restriction to one parameter subgroups 62
3.17 A lemma on the growth of coefficients of a power series 66
3.18 An alternative proof of theorem 3.10 67
§ 4. EQUIVARIANT CHARACTERISTIC CLASSES OF ORBITAL CONE
BUNDLES 68
4.1 Borel pictures of the cohomology of a flag variety 68
4.2 Description in terms of harmonic polynomials on a Cartan subalgebra 69
4.3 Equivariant K theory on T X 70
4.4 Restriction to a fibre of T X 72
4.5 Definition of equivariant characteristic classes 72
4.6 Comparison to the characteristic classes defined in §1 73
4.7 Equivariant characteristic classes of orbital cone bundles 77
4.8 Comparison with Joseph s notion of characteristic polynomials 78
4.9 Generalization to the case of sheaves 81
4.10 Equivariance under a Levi subgroup 83
4.11 Multiple cross section of a unipotent action 85
4.12 For example SL« equivariance 89
4.13 Completing the proof of theorem 4.7.2 89
4.14 Reproving Hotta s transformation formula 91
4.15 On explicit computations of our characteristic classes 92
4.16 Example 93
4.17 Remark 94
§ 5. PRIMITIVE IDEALS AND CHARACTERISTIC CLASSES 95
5.1 Characteristic class attached to a g module 95
5.2 Translation invariance 96
5.3 Characteristic variety of a Harish—Chandra bimodule 97
5.4 Homogeneous Harish Chandra bimodules 99
5.5 Characteristic cycle and class of a Harish Chandra bimodule 101
5.6 Identification with a character polynomial 102
5.7 Harmonicity of character polynomial 103
5.8 Equivariant characteristic class for a Harish—Chandra bimodule 105
5.9 Alternative proof of identification with character polynomials 106
5.10 Some non commutative algebra 108
5.11 Definition of the polynomials Pw 110
5.12 Relation to primitive ideals Ill
5.13 Irreducibility of Joseph s Weyl group representation 114
5.14 Irreducibility of associated varieties of primitive ideals 116
5.15 Evaluation of character polynomials 117
5.16 Computation of Goldie ranks 119
5.17 Joseph King factorization of polynomials pw 120
5.18 Goldie ranks of primitive ideals 122
BIBLIOGRAPHY 124
|
| any_adam_object | 1 |
| author | Borho, Walter 1945- Brylinski, Jean-Luc MacPherson, Robert |
| author_GND | (DE-588)1013288300 |
| author_facet | Borho, Walter 1945- Brylinski, Jean-Luc MacPherson, Robert |
| author_role | aut aut aut |
| author_sort | Borho, Walter 1945- |
| author_variant | w b wb j l b jlb r m rm |
| building | Verbundindex |
| bvnumber | BV002523884 |
| callnumber-first | Q - Science |
| callnumber-label | QA247 |
| callnumber-raw | QA247 |
| callnumber-search | QA247 |
| callnumber-sort | QA 3247 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 300 SK 340 SK 230 SK 240 |
| classification_tum | MAT 170f |
| ctrlnum | (OCoLC)20355039 (DE-599)BVBBV002523884 |
| dewey-full | 512/.4 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.4 |
| dewey-search | 512/.4 |
| dewey-sort | 3512 14 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV002523884 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:46:16Z |
| institution | BVB |
| isbn | 0817634738 3764334738 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001623430 |
| oclc_num | 20355039 |
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| physical | 131 S. |
| publishDate | 1989 |
| publishDateSearch | 1989 |
| publishDateSort | 1989 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spelling | Borho, Walter 1945- Verfasser (DE-588)1013288300 aut Nilpotent orbits, primitive ideals, and characteristic classes a geometric perspective in ring theory W. Borho; J.-L. Brylinski; R. MacPherson Boston [u.a.] Birkhäuser 1989 131 S. txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 78 Anneau Anneaux (Algèbre) ram Classe caractéristique Drapeau Groupe Weyl Idéal primitif K-théorie Orbite nilpotente Théorème Joseph Rings (Algebra) Halbeinfache Lie-Gruppe (DE-588)4122188-6 gnd rswk-swf Charakteristische Klasse (DE-588)4194231-0 gnd rswk-swf Irreduzible Darstellung (DE-588)4162430-0 gnd rswk-swf Äquivariante K-Theorie (DE-588)4207962-7 gnd rswk-swf Weyl-Gruppe (DE-588)4065886-7 gnd rswk-swf Nilpotenter Orbit (DE-588)4304229-6 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Primitives Ideal (DE-588)4261523-9 gnd rswk-swf Halbeinfache Lie-Gruppe (DE-588)4122188-6 s Geometrie (DE-588)4020236-7 s DE-604 Äquivariante K-Theorie (DE-588)4207962-7 s Weyl-Gruppe (DE-588)4065886-7 s Irreduzible Darstellung (DE-588)4162430-0 s Nilpotenter Orbit (DE-588)4304229-6 s Primitives Ideal (DE-588)4261523-9 s Charakteristische Klasse (DE-588)4194231-0 s Brylinski, Jean-Luc Verfasser aut MacPherson, Robert Verfasser aut Progress in mathematics 78 (DE-604)BV000004120 78 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001623430&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Borho, Walter 1945- Brylinski, Jean-Luc MacPherson, Robert Nilpotent orbits, primitive ideals, and characteristic classes a geometric perspective in ring theory Progress in mathematics Anneau Anneaux (Algèbre) ram Classe caractéristique Drapeau Groupe Weyl Idéal primitif K-théorie Orbite nilpotente Théorème Joseph Rings (Algebra) Halbeinfache Lie-Gruppe (DE-588)4122188-6 gnd Charakteristische Klasse (DE-588)4194231-0 gnd Irreduzible Darstellung (DE-588)4162430-0 gnd Äquivariante K-Theorie (DE-588)4207962-7 gnd Weyl-Gruppe (DE-588)4065886-7 gnd Nilpotenter Orbit (DE-588)4304229-6 gnd Geometrie (DE-588)4020236-7 gnd Primitives Ideal (DE-588)4261523-9 gnd |
| subject_GND | (DE-588)4122188-6 (DE-588)4194231-0 (DE-588)4162430-0 (DE-588)4207962-7 (DE-588)4065886-7 (DE-588)4304229-6 (DE-588)4020236-7 (DE-588)4261523-9 |
| title | Nilpotent orbits, primitive ideals, and characteristic classes a geometric perspective in ring theory |
| title_auth | Nilpotent orbits, primitive ideals, and characteristic classes a geometric perspective in ring theory |
| title_exact_search | Nilpotent orbits, primitive ideals, and characteristic classes a geometric perspective in ring theory |
| title_full | Nilpotent orbits, primitive ideals, and characteristic classes a geometric perspective in ring theory W. Borho; J.-L. Brylinski; R. MacPherson |
| title_fullStr | Nilpotent orbits, primitive ideals, and characteristic classes a geometric perspective in ring theory W. Borho; J.-L. Brylinski; R. MacPherson |
| title_full_unstemmed | Nilpotent orbits, primitive ideals, and characteristic classes a geometric perspective in ring theory W. Borho; J.-L. Brylinski; R. MacPherson |
| title_short | Nilpotent orbits, primitive ideals, and characteristic classes |
| title_sort | nilpotent orbits primitive ideals and characteristic classes a geometric perspective in ring theory |
| title_sub | a geometric perspective in ring theory |
| topic | Anneau Anneaux (Algèbre) ram Classe caractéristique Drapeau Groupe Weyl Idéal primitif K-théorie Orbite nilpotente Théorème Joseph Rings (Algebra) Halbeinfache Lie-Gruppe (DE-588)4122188-6 gnd Charakteristische Klasse (DE-588)4194231-0 gnd Irreduzible Darstellung (DE-588)4162430-0 gnd Äquivariante K-Theorie (DE-588)4207962-7 gnd Weyl-Gruppe (DE-588)4065886-7 gnd Nilpotenter Orbit (DE-588)4304229-6 gnd Geometrie (DE-588)4020236-7 gnd Primitives Ideal (DE-588)4261523-9 gnd |
| topic_facet | Anneau Anneaux (Algèbre) Classe caractéristique Drapeau Groupe Weyl Idéal primitif K-théorie Orbite nilpotente Théorème Joseph Rings (Algebra) Halbeinfache Lie-Gruppe Charakteristische Klasse Irreduzible Darstellung Äquivariante K-Theorie Weyl-Gruppe Nilpotenter Orbit Geometrie Primitives Ideal |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001623430&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000004120 |
| work_keys_str_mv | AT borhowalter nilpotentorbitsprimitiveidealsandcharacteristicclassesageometricperspectiveinringtheory AT brylinskijeanluc nilpotentorbitsprimitiveidealsandcharacteristicclassesageometricperspectiveinringtheory AT macphersonrobert nilpotentorbitsprimitiveidealsandcharacteristicclassesageometricperspectiveinringtheory |