Nilpotent orbits, primitive ideals, and characteristic classes. A geometric perspective in ring theory
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 just a few). Most of the results in this book are not new. Instead the aim has been to use geometric (in place of the more traditional algebraic) methods in the constructions and proofs. This sheds new lights on the close connection between the three topics. The book is not self-contained. It relies heavily on previous work by the authors as well as on many basic facts both from algebraic groups, topology and representation theory. However, the authors have taken great care to make the book readable to people without complete background in these theories.
scientific article; zbMATH DE number 44229