# Equivariant Analytic Localization of Group Representations (Memoirs of the American Mathematical Society)

By: Laura Smithies (author)Paperback

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### Description

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.

### Contents

Introduction Preliminaries The category ${\mathcal T}$ Two equivalences of categories The category $D^b {G 0}({\mathcal D} X)$ Descended structures The category $D^b {G 0}({\mathcal U} 0(\mathfrak g))$ Localization Our main equivalence of categories Equivalence for any ergular weight $\lambda$ Bibliography.

### Product Details

• ISBN13: 9780821827253
• Format: Paperback
• Number Of Pages: 90
• ID: 9780821827253
• ISBN10: 0821827251

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