Cohomology of Standard Modules on Partial Flag Varieties

Abstract

Cohomological induction gives an algebraic method for constructing representations of a real reductive Lie group $G$ from irreducible representations of reductive subgroups. Beilinson-Bernstein localization alternatively gives a geometric method for constructing Harish-Chandra modules for $G$ from certain representations of a Cartan subgroup. The duality theorem of Hecht, Mili\vci\'c, Schmid and Wolf establishes a relationship between modules cohomologically induced from minimal parabolics and the cohomology of the $\ms{D}$-modules on the complex flag variety for $G$ determined by the Beilinson-Bernstein construction. The main results of this paper give a generalization of the duality theorem to partial flag varieties, which recovers cohomologically induced modules arising from nonminimal parabolics.