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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Introduction

Abstract
Consider a complex semisimple Lie group G with Lie algebra g and Weyl group W.
W. Borho, J-L. Brylinski, R. MacPherson

### §1. A Description of Springer’s Weyl Group Representations in Terms of Characteristic Classes of Cone Bundles

Abstract
Our “algebraic varieties” are reduced, and defined over an algebraically closed base field k of characteristic 0. We restrict attention to the case k = whenever we find it convenient for topological interpretations. If not otherwise stated, we consider de Rham (co)—homology etc. with coefficients in k.
W. Borho, J-L. Brylinski, R. MacPherson

### §2. Generalities on equivariant K—theory

Abstract
For the convenience of those readers not familiar with equivariant K—theory, we have collected here in some detail the general facts needed from this theory as prerequisites for subsequent chapters. In the present chaper, G may be an arbitrary linear algebraic group over k.
W. Borho, J-L. Brylinski, R. MacPherson

### §3. Equivariant K—theory of torus actions and formal characters

Abstract
In this paragraph, we consider a torus T, that is a commutative connected reductive group over k, and a linear action of T on a vector space E of finite dimension r over k. We shall assume that all weights of T in E are positive with respect to some linear partial ordering ≥. We assume that the semi—group of positive (integral) weights in finitely generated. For example, T might be the group of homotheties of E. In the applications in subsequent chapters, T will be the maximal torus in a semisimple group, E will be the nilradical of a Borel subalgebra.
W. Borho, J-L. Brylinski, R. MacPherson

### §4. Equivariant characteristic classes of orbital cone bundles

Abstract
In this chapter, G is a connected semisimple algebraic group over k, and T a maximal torus in G. We use the notations introduced in 1.5. So, in particular, U is a maximal unipotent subgroup normalized by T, and B = TU = UT is a fixed Borel subgroup; g,b,t,u are the Lie algebras of G,B,T,U, etc.
W. Borho, J-L. Brylinski, R. MacPherson

### §5. Characteristic Classes and Primitive Ideals

Abstract
In this chapter, G is again a semisimple group with Lie algebra g, and we use the notations introduced in 1.5. As in chapter 3 and 4, we denote by Λ the lattice in t* of integral weights (3.1) of our maximal torus T ⊂ G, by Ω ⊂ Λ the “dominant integral weights” (3.16) with respect to the ordering fixed by our choice of a Borel subgroup B ⊃ T, and by p ∈ Ω half the sum of weights in b. We furthermore denote by U(g) the enveloping algebra of g, that is the ring of differential operators on G invariant under right translations. Our purpose is to study g—modules, that is to say U(g)—modules. In particular, we are interested in the annihilators in U(g) of simple g—modules, called primitive ideals. We denote by L(λ) the simple g-module of highest weight λ (which is defined as the unique simple quotient of the universal (or Verma-) module M(λ) = U(g) ⨂ U (b)kλ, where kλ is a one—dimensional b—module of weight λ). Then the center of U(g) is a polynomial ring in dim (T) variables (Harish—Chandra, Chevalley); this center acts by a character on L(λ) which is denoted xλ; we note that by Harish—Chandra’s theorem, x λ =x µ if and only if µ = w.λ, for some Weyl group element w ∈ W, where the “shifted Weyl group action” w.λ:= w(λ+ρ)-ρ is used. Finally, it will be convenient to identify g* with g, and t* with t by means of the Killing form. We apply analogous notations to the group G×G, so for instance (λ,/µ) ∈ t* × t* defines a central character x(λ,µ) of U(g × g) etc..
W. Borho, J-L. Brylinski, R. MacPherson

### Backmatter

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