Equivariant analytic localization of group representations
Author(s)
Bibliographic Information
Equivariant analytic localization of group representations
（Memoirs of the American Mathematical Society, no. 728）
American Mathematical Society, 2001
Available at 16 libraries
Note
"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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