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