Nilpotent orbits, primitive ideals, and characteristic classes. A geometric perspective in ring theory
type: Item
instance of: scholarly article
title: Nilpotent orbits, primitive ideals, and characteristic classes. A geometric perspective in ring theory
zbMATH Open document ID: 0697.17006
author: Walter Borho
author: Jean-Luc Brylinski
author: Robert MacPherson
published in: Progress in Mathematics
publication date: 1992-09-17T00:00:00Z
review text: Let G be a semisimple complex Lie group with Lie algebra \({\mathfrak g}\) and Weyl group W. In this book, geometric methods (e.g. equivariant K-theory, \({\mathcal D}\)-modules, intersection homology) are used to study and explain the connections between the following 3 important objects in representation theory: 1. Nilpotent orbits 2. Primitive ideals in U(\({\mathfrak g})\) (the enveloping algebra of \({\mathfrak g})\) 3. Irreducible representations of W. Let B denote a Borel subgroup of G and consider the moment map \(\pi\) from the cotangent bundle on G/B into \({\mathfrak g}\). By an orbital cone bundle the authors understand an irreducible component of \(\pi^{-1}({\mathcal O})\), \({\mathcal O}^ a \)nilpotent orbit in \({\mathfrak g}\). For such a bundle \({\mathcal K}\) they study the associated characteristic class in the cohomology ring H(G/B). This class can be computed via the Segre class of \({\mathcal K}\), and through the usual identification of H(G/B) with the space of W-invariant polynomials on the Cartan subalgebra \({\mathfrak t}\) of \({\mathfrak g}\) the classes of orbital cone bundles allows a direct construction of the Springer representation associated to \({\mathcal O}.\) Similarly one obtains characteristic classes in the Grothendieck K- subgroups K(G/B) as well as in equivariant K-theory. Actually, classes may be associated to any \({\mathcal D}\)-module on G/B. The class of a \({\mathcal D}\)-module is a polynomial on \({\mathfrak t}\) which may be thought of as its formal character. In fact, in the case of a localization of U(\({\mathfrak g})/I\), I a primitive ideal, it corresponds to the character of the corresponding highest weight module. This leads to a geometric formulation of the classification of primitive ideals, with integral central character. The above three sets of objects and their interrelations have of course been studied for a long time by a number of people (Barbasch, Borel, Duflo, Jantzen, Joseph, Vogan, to mention jus…
reviewed by: Henning Haahr Andersen
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17-02
Mathematics Subject Classification ID: 22-02
Mathematics Subject Classification ID: 22E47
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 44229
zbMATH Keywords: semisimple complex Lie group
zbMATH Keywords: Weyl group
zbMATH Keywords: equivariant K-theory
zbMATH Keywords: representation theory
zbMATH Keywords: Nilpotent orbits
zbMATH Keywords: Primitive ideals
zbMATH Keywords: enveloping algebra
zbMATH Keywords: Irreducible representations
zbMATH Keywords: characteristic classes
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 44229