Localization and standard modules for real semisimple Lie groups I: The duality theorem

In this paper we relate two constructions of representations of semisimple Lie groups constructions that appear quite different at first glance. Homogeneous vector bundles are one source of representations: if a real semisimple Lie group Go acts on a vector bundle E --* M over a quotient space M=Go/Ho, then Go acts also on the space of sections C~(M, E), and on any subspace VcC~(M, E) defined by a Go-invariant system of differential equations. Ordinary induction, so-called cohomological induction and the construction of representations by "quantizat ion" all fit into the framework of homogeneous vector bundles. For any complex semisimple Lie algebra g, there is an equivalence of categories, due to Beilinson-Bernstein [1], between ~-modules on the one hand, and sheaves of ~-modules over the flag variety X of g on the other. In the context of real semisimple Lie groups this equivalence of categories associates infinitesimal representations to orbits in the flag variety of the complexified Lie algebra orbits not of the group Go itself, but of the complexification of maximal compact subgroup Ko c Go. We shall show that these Beilinson-Bernstein modules are naturally dual to modules attached to certain homogeneous vector bundles. In the special case of a compact group, both the Beilinson-Bernstein construction and the construction via homogeneous vector bundles reduce to the Borel-Weil-Bott theorem; our duality theorem is then a particular instance of Serre duality.