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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
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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.

Readership

Graduate students and research mathematicians interested in topological groups, Lie groups, category theory, and homological algebra.

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 regular weight \(\lambda\)
  • Bibliography
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