Equivariant analytic localization of group representations
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Bibliographic Information
Equivariant analytic localization of group representations
（Memoirs of the American Mathematical Society, no. 728）
American Mathematical Society, 2001
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"September 2001, volume 153, number 728 (fourth of 5 numbers)"
Includes bibliographical references (p. 89-90)
Description and Table of Contents
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.
Table of 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.
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