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Equivariant Analytic Localization of Group Representations

Laura Smithies Kent State University, Kent, OH
Available Formats:
Electronic 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 Click above image for expanded view Equivariant Analytic Localization of Group Representations Laura Smithies Kent State University, Kent, OH Available Formats:  Electronic 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
• Book Details

Memoirs of the American Mathematical Society
Volume: 1532001; 90 pp
MSC: 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.

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 $\lambda$
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1532001; 90 pp
MSC: 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.

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 $\lambda$
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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