| eBook ISBN: | 978-1-4704-0321-8 |
| Product Code: | MEMO/153/728.E |
| List Price: | $54.00 |
| MAA Member Price: | $48.60 |
| AMS Member Price: | $32.40 |
| eBook ISBN: | 978-1-4704-0321-8 |
| Product Code: | MEMO/153/728.E |
| List Price: | $54.00 |
| MAA Member Price: | $48.60 |
| AMS Member Price: | $32.40 |
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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 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.
ReadershipGraduate students and research mathematicians interested in topological groups, Lie groups, category theory, and homological algebra.
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Table of Contents
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Chapters
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Introduction
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1. Preliminaries
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2. The category $\mathcal {T}$
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3. Two equivalences of categories
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4. The category $D^b_{G_0}(\mathcal {D}_X)$
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5. Descended structures
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6. The category $D^b_{G_0}(\mathcal {U}_0(\mathfrak {g}))$
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7. Localization
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8. Our main equivalence of categories
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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 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.
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 $
