Author | Borho, W. author |
---|---|
Title | Nilpotent Orbits, Primitive Ideals, and Characteristic Classes [electronic resource] : A Geometric Perspective in Ring Theory / by W. Borho, J-L. Brylinski, R. MacPherson |
Imprint | Boston, MA : Birkhรคuser Boston, 1989 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4558-2 |
Descript | VIII, 134 p. online resource |
General Introduction -- ยง 1. A Description of Springerโs Weyl Group Representations in Terms of Characteristic Classes of Cone Bundles -- 1.1 Segre classes of cone bundles -- 1.2 Characteristic class of a subvariety of a vector bundle -- 1.3 Characteristic class determined by a sheaf on a bundle -- 1.4 Comparison of the two definitions for Q -- 1.5 Homology of the flag variety -- 1.6 Cohomology of the flag variety -- 1.7 Orbital cone bundles on the flag variety -- 1.8 Realization of Springerโs Weyl group representation -- 1.9 Reformulation in terms of intersection homology -- 1.10 The Weyl group action -- 1.11 Reduction to a crucial lemma -- 1.12 Completion of the proof of theorem 1.8 -- 1.13 Comparison with Springerโs original construction -- 1.14 Theorem: The maps in the diagram are W equivariant -- 1.15 Hottaโs transformation formulas -- ยง 2. Generalities on Equivariant K-Theory -- 2.1 Algebraic notion of fibre bundle -- 2.2 Equivariant vector bundles and definition of KG(X) -- 2.3 Equivariant homogeneous vector bundles -- 2.4 Functoriality in the group G -- 2.5 Functoriality in the space X -- 2.6 The sheaf theoretical point of view -- 2.7 Existence of equivariant locally free resolutions -- 2.8 Remarks on Gysin homomorphisms in terms of coherent sheaves -- 2.9 Equivariant K-theory on a vector bundle: Basic restriction techniques -- 2.10 Filtrations on KG(X) -- 2.11 Representation rings for example -- 2.12 Application of equivariant K-theory to D-modules -- ยง 3. Equivariant K-Theory of Torus Actions and Formal Characters -- 3.1 The completed representation ring of a torus -- 3.2 Formal characters of T-modules -- 3.3 Example -- 3.4 T-equivariant modules with highest weight -- 3.5 Projective and free cyclic highest weight modules -- 3.6 Formal characters of equivariant coherent sheaves -- 3.7 Restriction to the zero point -- 3.8 Computation of ? degree -- 3.9 Character polynomials -- 3.10 Degree of character polynomial equals codimension of support -- 3.11 Positivity property of character polynomials -- 3.12 Division by a nonzero divisor -- 3.13 Proof of theorem 3.10 and 3.11 -- 3.14 Determination of character polynomials by supports -- 3.15 The theory of HilbertโSamuel polynomials as a special case -- 3.16 Restriction to one parameter subgroups -- 3.17 A lemma on the growth of coefficients of a power series -- 3.18 An alternative proof of theorem 3.10 -- ยง 4. Equivariant Characteristic Classes of Orbital Cone Bundles -- 4.1 Borel pictures of the cohomology of a flag variety -- 4.2 Description in terms of harmonic polynomials on a Cartan subalgebra -- 4.3 Equivariant Kโtheory on T*X -- 4.4 Restriction to a fibre of T*X -- 4.5 Definition of equivariant characteristic classes -- 4.6 Comparison to the characteristic classes defined in ยง1 -- 4.7 Equivariant characteristic classes of orbital cone bundles -- 4.8 Comparison with Josephโs notion of โcharacteristic polynomialsโ -- 4.9 Generalization to the case of sheaves -- 4.10 Equivariance under a Levi subgroup -- 4.11 Multiple cross section of a unipotent action -- 4.12 For example SL2 equivariance -- 4.13 Completing the proof of theorem 4.7.2 -- 4.14 Reproving Hottaโs transformation formula -- 4.15 On explicit computations of our characteristic classes -- 4.16 Example -- 4.17 Remark -- ยง 5. Primitive Ideals and Characteristic Classes -- 5.1 Characteristic class attached to a g module -- 5.2 Translation invariance -- 5.3 Characteristic variety of a HarishโChandra bimodule -- 5.4 Homogeneous Harish-Chandra bimodules -- 5.5 Characteristic cycle and class of a HarishโChandra bimodule -- 5.6 Identification with a character polynomial -- 5.7 Harmonicity of character polynomial -- 5.8 Equivariant characteristic class for a HarishโChandra bimodule -- 5.9 Alternative proof of identification with character polynomials -- 5.10 Some nonโcommutative algebra -- 5.11 Definition of the polynomials PW -- 5.12 Relation to primitive ideals -- 5.13 Irreducibility of Josephโs Weyl group representation -- 5.14 Irreducibility of associated varieties of primitive ideals -- 5.15 Evaluation of character polynomials -- 5.16 Computation of Goldie ranks -- 5.17 JosephโKing factorization of polynomials PW -- 5.18 Goldie ranks of primitive ideals