Nilpotent Orbits, Primitive Ideals, and Characteristic Classes: A Geometric Perspective in Ring Theory [Hardcover]

1. The Subject Matter. Consider a complex semisimple Lie group G with Lie algebra g and Weyl group W. In this book, we present a geometric perspective on the following circle of ideas: polynomials The vertices of this graph are some of the most important objects in representation theory. Each has a theory in its own right, and each has had its own independent historical development. - A nilpotent orbit is an orbit of the adjoint action of G on g which contains the zero element of g in its closure. (For the special linear group 2 G = SL(n,C), whose Lie algebra 9 is all n x n matrices with trace zero, an adjoint orbit consists of all matrices with a given Jordan canonical form; such an orbit is nilpotent if the Jordan form has only zeros on the diagonal. In this case, the nilpotent orbits are classified by partitions of n, given by the sizes of the Jordan blocks.) The closures of the nilpotent orbits are singular in general, and understanding their singularities is an important problem. - The classification of irreducible Weyl group representations is quite old.General Introduction.- ? 1. A Description of Springers 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 Springers 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 Springers original construction.- 1.14 Theorem: The maps in the diagram are W equivariant.- 1.15 Hottas transformation formulas.- ? 2. Generalities on Equivariant K-Theory.- 2.1 Algebraic notion of fibre bundle.- 2.2l£,