**Memoirs of the American Mathematical Society**

2001;
90 pp;
Softcover

MSC: Primary 22; 18;

Print ISBN: 978-0-8218-2725-3

Product Code: MEMO/153/728

List Price: $54.00

Individual Member Price: $32.40

**Electronic ISBN: 978-1-4704-0321-8
Product Code: MEMO/153/728.E**

List Price: $54.00

Individual Member Price: $32.40

# Equivariant Analytic Localization of Group Representations

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*Laura Smithies*

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 Borel-Weil
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 Borel-Weil 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 Harish-Chandra 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 Hecht-Taylor method, called
equivariant analytic localization, is developed. The technical advantages that
equivariant analytic localization has over (non-equivariant) analytic
localization are discussed and applications are indicated.

#### Readership

Graduate students and research mathematicians interested in topological groups, Lie groups, category theory, and homological algebra.

#### Table of Contents

# Table of Contents

## Equivariant Analytic Localization of Group Representations

- Contents ix10 free
- Introduction 112 free
- Chapter 1. Preliminaries 920 free
- Chapter 2. The Category T 1930
- Chapter 3. Two Equivalences of Categories 3142
- Chapter 4. The Category D[sup(b)][sub(Go)](D[sub(x)]) 4152
- Chapter 5. Descended Structures 4960
- Chapter 6. The Category D[sup(b)][sub(Go)](u[sub(0)](g)) 6172
- Chapter 7. Localization 6778
- Chapter 8. Our Main Equivalence of Categories 7384
- Chapter 9. Equivalence for Any Regular Weight λ 8192
- Bibliography 89100