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Equivariant Analytic Localization of Group Representations
Laura Smithies, Kent State University, OH
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Memoirs of the American Mathematical Society
2001; 90 pp; softcover
Volume: 153
ISBN-10: 0-8218-2725-1
ISBN-13: 978-0-8218-2725-3
List Price: US$51 Individual Members: US$30.60
Institutional Members: US\$40.80
Order Code: MEMO/153/728

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.

• The category $${\mathcal T}$$
• The category $$D^b_{G_0}({\mathcal D}_X)$$
• The category $$D^b_{G_0}({\mathcal U}_0(\mathfrak g))$$
• Equivalence for any regular weight $$\lambda$$