Electronic ISBN:  9781470403218 
Product Code:  MEMO/153/728.E 
List Price:  $54.00 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 153; 2001; 90 ppMSC: Primary 22; 18;
The problem of producing geometric constructions of the linear representations of a real connected semisimple Lie group with finite center, \(G_0\), has been of great interest to representation theorists for many years now. A classical construction of this type is the BorelWeil theorem, which exhibits each finite dimensional irreducible representation of \(G_0\) as the space of global sections of a certain line bundle on the flag variety \(X\) of the complexified Lie algebra \(\mathfrak g\) of \(G_0\).
In 1990, Henryk Hecht and Joseph Taylor introduced a technique called analytic localization which vastly generalized the BorelWeil theorem. Their method is similar in spirit to Beilinson and Bernstein's algebraic localization method, but it applies to \(G_0\) representations themselves, instead of to their underlying HarishChandra modules. For technical reasons, the equivalence of categories implied by the analytic localization method is not as strong as it could be.
In this paper, a refinement of the HechtTaylor method, called equivariant analytic localization, is developed. The technical advantages that equivariant analytic localization has over (nonequivariant) analytic localization are discussed and applications are indicated.ReadershipGraduate students and research mathematicians interested in topological groups, Lie groups, category theory, and homological algebra.

Table of Contents

Chapters

Introduction

1. Preliminaries

2. The category $\mathcal {T}$

3. Two equivalences of categories

4. The category $D^b_{G_0}(\mathcal {D}_X)$

5. Descended structures

6. The category $D^b_{G_0}(\mathcal {U}_0(\mathfrak {g}))$

7. Localization

8. Our main equivalence of categories

9. Equivalence for any regular weight $\lambda $


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The problem of producing geometric constructions of the linear representations of a real connected semisimple Lie group with finite center, \(G_0\), has been of great interest to representation theorists for many years now. A classical construction of this type is the BorelWeil theorem, which exhibits each finite dimensional irreducible representation of \(G_0\) as the space of global sections of a certain line bundle on the flag variety \(X\) of the complexified Lie algebra \(\mathfrak g\) of \(G_0\).
In 1990, Henryk Hecht and Joseph Taylor introduced a technique called analytic localization which vastly generalized the BorelWeil theorem. Their method is similar in spirit to Beilinson and Bernstein's algebraic localization method, but it applies to \(G_0\) representations themselves, instead of to their underlying HarishChandra modules. For technical reasons, the equivalence of categories implied by the analytic localization method is not as strong as it could be.
In this paper, a refinement of the HechtTaylor method, called equivariant analytic localization, is developed. The technical advantages that equivariant analytic localization has over (nonequivariant) analytic localization are discussed and applications are indicated.
Graduate students and research mathematicians interested in topological groups, Lie groups, category theory, and homological algebra.

Chapters

Introduction

1. Preliminaries

2. The category $\mathcal {T}$

3. Two equivalences of categories

4. The category $D^b_{G_0}(\mathcal {D}_X)$

5. Descended structures

6. The category $D^b_{G_0}(\mathcal {U}_0(\mathfrak {g}))$

7. Localization

8. Our main equivalence of categories

9. Equivalence for any regular weight $\lambda $