Equivariant analytic localization of group representations

The problem of producing geometric constructions of the linear representations of a real connected semisimple Lie group with finite center, Go, 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 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 Go 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.